People who offer "proofs" otherwise don't understand math.

That's a harsh comment, but I think it's fair. Consider this common "proof" the two equal one another.

.999=x

10x=9.999

10x - x = 9x

9x=9

1x=1.

.999 = 1.

The hand-waving is obvious. How does one multiply an infinite series of 9s by 10? What happens to the zero you'd get when multiplying by ten? Are we to believe it just disappears?

Of course not. The proof is invalid. It's just an optical illusion relying on tricking the reader by hoping they don't notice the hand-waving. The reality is no "proof" can address the issue better than simply looking at the two values. If the two are equal to one another, subtracting one from the other must give an answer of 0.

When we do that, we see there is a difference - 0.000...1. It's infinitely small, but it is real. Therefore, the two numbers aren't equal. Why then do so many people believe they are? Because they say so.

Literally. They say so, so it's true. That's it. You see, there is a thing in math called an axiom. It's a statement **assumed** to be true without proof. One axiom underlying the real number system basically says:

Non-zero infinitesimals do not exist.

Which means 0.000...1 does not exist. Why? Because we say so.

Only **we** don't say so. A person who thinks 0.999... doesn't equal 1 obviously believes infinitesimals exist. They don't accept that axiom. They'd use a different one, like many mathematicians who work with infinitesimals on a regular basis.

That's right. There's an entire field of math which uses infinitesimals. It's just as valid as the real number system. Which one you use is merely a matter of preference. Whether 0.999... and 1 are equal is based on the completely arbitrary choice of whether one uses the real number system or a different one.

All the "proofs" the two are equal are meaningless. Implicit in all of them is the statement, "Using the real number system." That's begging the question. It's tricking people by assuming there could be no difference between the two numbers then concluding there is no difference between the two numbers.

Anyone who understands how math works should know 0.999... equals 1 only if you choose for it to. "Proofs" the two are equal tell us nothing about the subject but everything about the speaker. Namely, they don't know what they're talking about.

If you feel 0.999... does not equal 1, you're right. If you feel it does equal one, you're right too. Which answer is "right" just depends on which type of math you feel most comfortable with. It's purely a matter of personal choice.

I wrote this post because of a (in-person) discussion which stemmed from an offhand comment I made about an offhand remark in a comment by William Connolley on this blog post. As such, I felt it was appropriate to share a link to this post over there. Credit where credit is due, and whatnot.

Connolley takes issue with this post. I'm not sure he'll want to comment here, so I'm copying what he said so people can see his disagreement:

I have no idea what he thinks is wrong with this post. I've suggested he could comment here to explain, but in the meantime, I feel I should point out the link he presented largely agrees with me. It openly states there are algebraic frameworks in which 0.999 does not equal one. The only real area of disagreement is it points out there are frameworks which allow for infinitesimals where the two numbers are still equal.

That's true. One can design axioms which allow for infinitesimals to exist yet have those two numbers be equal. It primarily happens when one allows certain types of infinitesimals to exist but not others. It doesn't change anything about the point of this post, but it is does make some of the statements I made technically untrue. That said, the link has a discussion clearly in favor of the point I'm making:

I'm not sure how William Connolley thinks a link which has respected mathematicians basically making the same point I'm making "might help" me realize I'm wrong. Maybe he'll come over and explain.

William Connolley is amusing. He just argued against the existence of (non-zero) infinitesimals, saying:

Notice he says "on the non-negative reals." That's right. He explicitly limits the discussion to the real number system. As I pointed out in this post:

You have to admire his chutzpah. How many people do you think could respond to a criticism of an argument by repeating the argument and claiming that proves the criticism wrong?

I was annoyed by William Connolley's latest response to me. I don't think I'll be pursuing a conversation with him any further as he's shown himself to be a horrible person when it comes to having discussions. I challenge anyone to read the exchange and believe Connolley has done anything to discredit this post. In the meantime, I'll copy my response to him:

Wow! You managed to make William Connelley look smart and knowledgeable!

You don't even understand the concept of an infinite (as opposed to finite) series. Any high-school level textbook on the subject will tell you that

a + a*r + a*r^2 + a*r^3 + ... (to inifinity) = a / (1 - r)

(for magnitudes of r less than 1)

In this case, a = 0.9 and r = 0.1, so we have

0.9 + 0. 09 + 0.009 + 0.0009 + (to infinity) = 0.9 / (1 - 0.1) = 1.0

That's not approximately equals; it's exactly equals. This isn't even controversial. It's taught to high school kids the world over.

It's remarkable how often people introduce themselves into discussions with insults of the form, "You're an idiot." I had considered a moderation rule whereby such remarks are forbidden, but I decided they impinge upon the speaker far more than the target. If people want to paint themselves as unreasonable from the get go, why save them the trouble?

With an unnecessary and insulting introduction to match yours out of the way, we can consider what you actually said. You claim I don't understand the concept of an infinite series, and you base this claim on the notion any "high-school level textbook" would teach me I'm wrong. However, the entire argument you advance is one based upon the idea of treating .9 repeating as a convergent series. That is tied to the concept of limits which is in turn tied to the axiom I discuss in this blog post - that non-zero infinitesimals do not exist.

In other words, your claim I don't know what I'm talking about rests entirely upon assuming an axiom is true despite this post clearly showing there is no inherent reason to assume that axiom is true. Your argument is, quite literally, the exact argument I rebutted in this post. That makes your argument nothing but an argument by assertion. Ironically, you seem completely unaware of that, thus making you guilty of the very offense you accuse me of.

You'd have been better off replacing your comment with:

That would have at least made it less obvious you didn't even try to understand what was said in this post.

Hmm... You can dish it out, but you cannot take it. You called the standard exposition used in textbooks around the world completely wrong -- not just saying that there could be another way of looking at it that would yield different results. You said yourself that it was harsh.

You used an example of a finite series to argue about an infinite series and I called you on it.

And no, the argument of equality does not rest on Archimedes concept of no non-zero infinitesimals.

There's an incredibly simple, yet completely valid, proof of the equality that most high school students understand immediately:

1/3 = 0.333...

(1/3) * 3 = 1

0.333... = 0.999...

Ergo:

0.999... = 1

The fundamental concept is that an infinite number of infinitesimals can yield a finite value.

That's:

0.333... * 3 = 0.999...

Ed Bo, just about everything you said in your response to me is wrong:

You claim I "cannot take it," yet I did nothing which indicated I was bothered by your insult. Pointing out you look like imbecile for saying something doesn't indicate I'm bothered by you saying it.

This is stupid. If two contradictory answers to a question exist, it is wrong to say the answer to the question is only one of them. It's no better than telling people the square root of four is -2. The fact an answer can be true does not mean it can be given to the exclusion of all others.

Moreover, a person who doubts the two values are equal is clearly questioning the premise behind saying they are. That premise, which is nothing but an arbitrary assumption one need not make, is what should be discussed. Throwing "proofs" around that do nothing but assume the answer doesn't address their concerns. It does nothing to address the premise behind the question. All it does is mislead them.

I did nothing of the sort. You're just making this up.

This is pure hand-waving. It's a bold claim you offer no support for. Instead, you offer yet another proof which assumes there are no non-zero infinitesimals, suggesting you have no idea what you just said isn't true.

This is a proof which depends upon assuming non-zero infinitesimals do not exist. Under a different system, 0.333… * 3 could equal 1, not 0.999….

In other words, your argument is still assuming something is true then "proving" it built upon that assumption - the exact point this post is about.

This is entirely nonsensical for your argument, yet you claim it is "[t]he fundamental concept." That makes it clear you have no idea what you're talking about.

If 0.99999... is not equal to 1, then that means there is a real number between them.

But if there are an infinite number of 9's, then the only way a number could be between 0.999999... and 1 is if it were INFINITELY SMALL.

If a number is infinitely small, then there cannot be a smaller number than it.

HOWEVER, it is a well known fact of the real numbers that THERE IS NO SMALLEST POSITIVE NUMBER.

Hence, BY CONTRADICTION, 0.999999.... = 1.

You're Welcome, Brandon.

Lord Byron, you just used the phrases "real number" and "real numbers" in response claiming to prove this post wrong. This post explains the real number system is not the only system. That means you not only have you failed to prove this post wrong, you've failed to even address what the post says.

Non-standard calculus does not provide a number system. The adjacent alternative systems to the real number system are the the rational numbers and the complex numbers. In both of these numbers systems 0.999... = 1. Other number systems include the integers (under which 0.999... does not exist and hence nothing can be said about it), the natural numbers (ditto), quaternions (under which 0.999... = 1 also), etc.

There are NO number systems where both (1) 0.999.. exists AND (2) it does not equal 1. [Under the felicitous assumption that we're using decimal systems, obviously.] So basically, if 0.999... exists in a [decimal] number system, then it will = 1. Always. If it doesn't, then you're not dealing with an actual number system.

I agree that William Connolley is a useless idiot and I laud your efforts to expose climate extremism, but this post in particular is beyond moronic and merely provides fodder to discredit your entire blog.

Ha! Maybe I should have read the full wikipedia article. I graciously withdraw my previous comment. Infinitesimals can be included in number systems. A warning to the commenters everywhere: do your homework before posting.

SebZear, I have no idea what makes you believe what you just wrote. Not only is it easy to find number systems given by non-standard analysis (three of the most obvious being hyperreal, superreal and surreal), non-standard analysis must create new number systems as a matter of definition.

All you've done is hand-wave away a multitude of mathematical fields and numerical systems anyone could find in a matter of seconds with Google. Doing that in order to claim "this post in particular is beyond moronic" is, well, beyond moronic.

To be clear, standard calculus does not accept the existence of (non-zero) infinitesimals. Non-standard calculus does accept the existence of (non-zero) infinitesimals. The two cannot use the same number system.

I didn't see your follow-up comment before submitting mine. Sorry about that. It's good you caught your own mistake. You don't need me piling on.

There are two main issues I have with this post. Because of its prominence and usefulness in the real world, assuming that we are using the real number system is reasonable. When I say 1+1=2, I am using the real number system. I could also be using many other number systems and it's still true, but unless specified we use the real number system. I could say that 1+1>2, and then just way I'm using a number system where that's true. While I'm technically right, it's a useless statement. I do agree with your point that it technically begs the question, but there are plenty of people who don't believe it even when you specify that you are only using the real number system - and they still claim 'it just feels wrong' or the other similar arguments. This goes completely against your statement regarding that people are right whichever way they feel.

Secondly, you state that the choice of number system is a matter of personal preference. This is to an extent true in this discussion, but not true in practical fields such as engineering or computer science. And plenty of people in these fields don't believe this concept within they're numbering system, which by necessity (for consistency with other practitioners to allow their systems to work) must be the real number system.

Richard, I'm afraid I can't agree with you when you say:

In the real world, most people don't use the real number system. Most people don't use formalized math. The difference between standard and non-standard analysis is completely irrelevant to what they do. Assuming people use a particular number system even when that number system contributes nothing to what they do is patently begging the question.

When you say "1+1=2," there's no reason to assume you're using a real number system. All that does is add a series of unnecessary complications to the situation. Why say you're using the real number system instead of the rational number system? Or integer system? Or whole number system? Or natural number system? These systems are all nested within one another, and the normal way of approaching problems is to pick whichever system is simplest. If you know all the numbers in a problem are going to be integers, there's simply no need to think about the rational or real number system.

I'm pretty sure if that's true, it's only because the explanation they were given was poor. For instance, telling people "you are only using the real number system" likely won't change how they feel if they don't understand what that statement actually means. That would explain the reaction you describe. It wouldn't make sense, however, for a person to react that way if they actually understood the explanation like I've laid out in this post. Once a person understands the decision to allow or not allow non-zero infinitesimals to exist is arbitrary, it would be nonsensical for them to say, "It feels wrong that if you assume they don't exist, the two numbers are equal." What would make sense is for them to say, "It feels like the two numbers shouldn't be equal, so I won't make the assumption which makes them equal."

The real number system doesn't allow us to do math we couldn't do under a different system. More interestingly, infinitesimals have been used when studying problems even within the real number system. They were seen as a temporary solution to be used until a more robust one could be found. I doubt there is any practical necessity to avoid using them given they have been used, successfully.

But again, this goes back to how number systems are nested. If you really wanted to, you could create a superset which encompassed real numbers and allowed for non-zero infinitesimals to exist while maintaining all the calculations we can currently do in the real number system. It would be similar to how real numbers encompass rational and irrational numbers. Just like how the real number system can still allow you to do all the calculationg you might have done in the rational number system, this new number system could allow us to do all the normal calculations we might do within the real number plus more.

Of course, the new number system wouldn't serve much purpose for most problems so it would likely not be used that much. But in the same way you can say you're using the real number system when you say "1+1=2," you could say you're using this new number system even if it isn't necessary for the problem you're solving.

Hey, I don't have anything to add, since your explanation already does a perfect job. I just want to compliment you on how you handled yourself, in literally every situation. You handled bad posts politely and reservedly, even bringing in Connelly's remarks from on a different site. You didn't get quick-tempered with people who seem clearly wrong, and ignoring everything you'd previously said. This page should be the role model of the internet. Good on you, my man.

Thanks Owen McGee!

I am only a sophomore in high school, and even though I am a member of my school math team, I am not familiar with all of the terminology used in some of your responses and the post itself. I must say I do disagree with you on this topic. The simplest proof of 1/3=0.333... and 3*1/3=1 and 0.333...=0.999... therefore means that 0.999...=1 should be more than suffice to show that these two values are equal. This can also be proved using the infinite sum formula and algebraicly by setting 0.999..=x, multiplying by ten and subtracting the two equations giving us 9x=9 and x=1. Based on all of these proofs I am unsure of how you could see that these are not the same value.

Dennis Whitener, I appreciate you taking the time to comment here, but I am afraid you have missed the point of this post. To demonstrate, ask yourself this. Why does .999... * 10 = 9.999...? Why does it not equal 9.999...0? The answer in "standard" mathematics is there cannot be anything after the ellipsis because the ellipsis represents an infinite string. Coming after an infinite string of decimal points would make a number infinitely small. Infinitely small numbers are called "infinitesimals." In "standard" analysis, infinitesimals (aside from 0) are declared not to exist by fiat. The non-existence of such infinitesimals is an axiom or postulate, something assumed to be true without any proof in order to further an analysis.

But what if we did not make that assumption? What if we allowed infinitesimals to exist? In that case, there could be a genuine difference between numbers like .999...0 and .999.5. There are fields of mathematics where that is true. People do analyses on a daily basis using non-zero infinitesimals. There is no inherent reason they should not. The proscription against non-zero infinitesimals is one of tradition rather than logic. Early mathematicians made the rule which said non-zero infinitesimals do not exist (even as some of them used such infinitesimals in proofs as kludges). Since then, most people have just gone with it. After the issue was re-examined in the last couple centuries, and especially over the last few decades, the mathematical community has come to accept the use of non-zero infinitesimals is perfectly valid in fields of math which rely on different assumptions than "standard" analysis. Some mathematicians switch between "standard" and "non-standard" analysis depending on the type of problem they're working on. It's more a matter of preference/ease of use than anything else.

People who intuitively question the idea that .999... equals 1 simply because .999... is "infinitely close" to 1 are not making any sort of mistake. They are simply not thinking of the "real" number system. The "real" number system is just one of many numerical systems though, and there is nothing inherently wrong with using a different one. The only thing which is wrong is using proofs, especially arithmetic ones, which only work in the "real" number system in response to people who intuitively question the "real" number system

There is no question .999... equals 1 within the "real" number system. The question is, why must people use the "real" number system? The answer is, they shouldn't have to. There's no mathematical basis for forcing them to. It's just tradition.

Ok thank you for clarifying. This makes much more sense. As I stated I am only in 10th grade so I’ve only ever needed real and imaginary numbers in my lifetime. I appreciate you explaining more thouroghly the concept of infinitesimals and how they differ from the number systems I am accustomed to using. I can now say that we agree since I misinterpreted your point to be that 0.999... is never equal to 1. This is my fault for skimming the article. I’m sure I will be able to understand this concept even further when I take pre calculus next school year. Have a great day.

Oh and in response to your question about why 0.999...=9.999... is what I understand to be because when multiplying by ten you move the decimal point to the right one time, not just add a zero to the end to put it simply. This is why 0.1*10 is 1.0 not 0.10.

Dennis Whitener, I'm glad to hear it. If you'd like a better writeup than mine, I think this person did a better job than I did.

But how do you move the decimal point to the right in an infinite series? To start, there are an infinite number of 9s after the decimal place. You then take one of those 9s away and put it before the decimal point. You then have an infinite number of 9s after the decimal point, less one. How do you say those are the same?

The answer in "standard" analysis is to say there is no difference between "infinity" and "infinity minus one." That's fine. However, in "non-standard" analysis, there can be a difference between the two. That's also fine. Mathematics is not a single, absolute science taking from physical properties. It's just a form of logic that allows people to create different number systems using different axioms.

Unfortunately, most people never realize that about math because they aren't exposed to the formal approach in which the number systems they use are created. Most people who go through high school might hear about "natural" and "real" number systems, but they will also be taught one builds upon the other. The idea you can build different mathematical frameworks by making different choices is something most people will never even consider.

1/9 = 0.1111111

2/9 = 0.2222222

(If we carry on the pattern...)

8/9 = 0.8888888

9/9 = 0.9999999 = 1

If 0.99999 is not equal to 1, then why does this pattern occur. I am only 14, so please explain.

Just to clarify,

9/9 =1, but according to the pattern it is also equal to 0.99999.

Vikash Hamilton, I can't tell you why a pattern occurs yet might not continue on indefinitely. What I can tell you is nine entries is a pattern means little in math. In various aspects of math, you can find the same series of nine numbers show up in many different patterns. Without more data, you can't tell which pattern is actually present. Similarly, you can find many instances where there seems to be a pattern but the pattern doesn't hold.

In this case, part of what you're seeing is an optical illusion created by the number system we use. As you know, we use the numbers 0-9 in our number system. As you probably know, at the lowest level computers (more or less) only use the numbers 0-1. That is called binary, or base 2. Another common number system goes from 0-F, with A, B, C, D, E, F following the number 9. That is called hexadecimal, or base 16. In each of these number systems (and the infinite other systems we could choose to work with), the pattern you see will be different. In base 2, the "pattern" will be a single item prior to the "proof": .111.... In hexadecimal, it would consist of 15 (.AAA, .BBB, etc.).

How many times do you think a pattern has to repeat before you accept it as "proof"? Whatever the answer, you can change your numerical system to fit it. You could work in base 100 and have it repeat 99 times. But at the same time, you could pick a more exotic number system and have the pattern not show up at all (sort of).

This is why inductive reasoning like looking for patterns and assuming they'll continue indefinitely is risky. If you don't understand why those patterns happen, you can easily draw incorrect conclusions. That is why in math deductive reasoning reigns supreme It is only by using rigorous, deductive analysis we can tell what does and does not hold true for our numerical systems. Inductive reasoning is limited to being a useful tool for exploring idea, not proving them,

Yet the pattern carries on infinitely

10/9 = 1.11111

11/9 = 1.22222

12/9 = 1.33333

...

17/9 = 1.88888

18/9 = 1.99999 = 2

...

89/9 = 9.88888

90/9 = 9.99999 = 10

91/9 = 10.11111

So if this pattern carries on without a finite ending, why mustn't 1.99999 = 2 and 9.99999 = 10 (etc)

Vikash Hamilton, you can only claim that pattern goes on infinitely if you assume what you are trying to prove. If the question is whether or not .9 repeating equals 1, you can't include .999... = 1 in your pattern as proof of anything. If one believed the two numbers did not equal one another, then the pattern you describe would hold for nine items, break for one, then repeat for nine more.

While in base 16 (hexadecimal) it'd repeat for 15 before breaking. And in base two (binary) it'd never repeat at all. That what you see changes based solely upon the choice of base you use shows why inductive reasoning like this is risky. If you want to pursue this line of thought, what I'd recommend doing is examining why this pattern happens at all. Once you understand why the pattern occurs, you can understand why the pattern would or would not continue in the manner you suggest.

(I lack the mathematical chops to be comfortable explaining it myself, but if you pursue the question, what you'll ultimately find is the same thing I point out in this post. Namely, whether or not the pattern holds as you suggest will ultimately depend upon whether or not your mathematical framework accepts the existence of non-zero infinitesimals.)

Simple reason why 1/3 is NOT equal to .333... and 1/9 is NOT equal to .111... This is because when you do long division with 1/3 you get .333... WITH A REMAINDER OF 1 which everyone forgets about. This remainder will also be equal be equal to 1/(3∞). This remainder is also the difference between .999... and 1 divided by 3, which makes sense as .333... is one third of .999...

The implication by others that nothing is smaller than a infinitesimal is not correct, like saying that nothing is bigger than infinity. Take 2∞=∞. If nothing is bigger than infinity than these 2 numbers should be equal to each other. You can test this by either subtracting or dividing infinity from each side to get either ∞=0 or 2=1 which are both nonsense. The 2=1 proof is even using the real number system so their should be no argument from anyone else. This leads to the fact that one (1/∞) or one infinitesimal is the difference between .999... and 1 while (1/3)-.333...=1/(3∞).

A simple formula using the help of long division can be (Quotient)=(Partial Quotient)+(Remainder)/((Divisor)*10^(number of steps done)) An example of this could be 5/8=.6+2/(8*10^1)=.625 or maybe 1/3=.333...+1/(3*10^∞). It might help if you do long division on paper then you might understand better as this is can be very confusing to understand without a picture (writing from school distributed computer and do not know how to upload picture). Note: I am also only in 7th grade

A question for anyone out there thinking that .9 repeating equals 1 or not: is .999...*∞=...999 or ...999.999...? I am with the ...999 just to say

Your Logic is so off

First you CAN multiply a infinite # of 9s by 10

Just move the place value down 1,(how to multiply by ten)

So the first example is right

Second there is a infinite # of 9s so when you multiply by 10 the “0” is never even there, because of the fact that it’s INFINITE, meaning that it has no other #, the 0 is replace by a nine.

What do you mean “because we say so”

you can’t add something to infinity. So you can’t have .000…1 If you think otherwise your not so good at math

Plus there are a lot of different proofs of this in lot of different types of math.

http://www.purplemath.com/modules/howcan1.htm

I’m only doing this because I want you to see that you can be wrong

Also Luke Kennedy

“This is because when you do long division with 1/3 you get .333... WITH A REMANDER OF ONE”

That is COMPLETELY INCORRECT

When doing long division You can’t have an infinite # with a remainder as an answer because the remainder is there to make it so that 1st graders don’t have to find the fraction of decimal.

One last word

The proof that has worked the best in my case is that

1-.9=.1

1-.99=.01

1-.999=.001

So for every 9 you subtract 1 and that is how many 0s you will have plus a one

An equation for it is

N-1=Z

With N being the amount of nines

And Z being the amount of 0s in the answer

If N= Infinity ,what does Z equal

Hint the answer isn’t (infinity-1)

Please don’t take something I say in this post and butcher it, if you need me to clear something up let me know.

I have convinced everyone I’ve shown my evidence to and I’m not stopping with you

Ps. Please don’t mention anything about nonzero infinitesimals cause I’m not saying anything about that.

Dayne Meyes:

Well, if you refuse to say anything about the central issue of the topic being discussed, I'm not sure what there is to discuss. The entire crux of this disagreement, one you've done nothing to address, is how people should handle non-zero infinitesimals. To refuse to even look at the issue makes your contribution to the discussion... not very useful.

The truth is your comments are clearly based upon the premise non-zero infinitesimals do not exist. If you refuse to discuss the idea they could exist, then all you're doing is demanding people start with your assumptions so they can reach your conclusions.

I’m not saying nonzero infintesimals aren’t real

I’m saying there is a lot of evidence pointing towards the 2 being equal and you are changeant the topic and not adressing what I really think that this post is about

That .999... doesn’t equal 1

I’m stating my case on how why I think that statement is false. I’m very sorry about the earlier posts if they came out a little rude

When you respond to this post I want to hear why you think that my math is wrong not that something I say is wrong.

That seems familiar

“Please don’t take something I say in this post and butcher it”

Thank you for taking this debate up with me

Now I will address everything from the newer version of the post

"How does one multiply an infinite series of 9s by 10?"

Easy. You apply the linearity property of multiplication, and then you

prove it for the infinite case by using the principle of mathematical

induction.

"What happens to the zero you'd get when multiplying by ten?"

There is no zero. Multiplying by ten only yields an extra 0 digit when

the number being multipled is expressed as an integer within base-10

notation.

"Of course not. The proof is invalid. It's just an optical illusion

relying on tricking the reader by hoping they don't notice the

hand-waving."

There is no hand-waving involved. You just happen to not understand how

arithmetic multiplication works. This is what this comes down to.

"When we do that, we see there is a difference - 0.000...1. It's

infinitely small, but it is real."

No, it is not real, and it is not infinitesimally small, because it is

not a number. By definition, if there is a series of infinite digits,

then this series is unending, hence there cannot be a different digit

after.

"Why then do so many people believe they are? Because they say so."

No, we KNOW they are equal because we can PROVE they are equal. The

Wikipedia article on the subject provides many proofs, all of which rely

on very different ideas. To reject the proofs is unreasonable.

"Only we don't say so. A person who thinks 0.999... doesn't equal 1

obviously believes infinitesimals exist. They don't accept that axiom.

They'd use a different one, like many mathematicians who work with

infinitesimals on a regular basis."

Again, I'm uncertain of how many times I should repeat myself. Any

system which uses infinitesimal numbers also has the equality 0.999... =

1 anyway, and infinitesimals cannot be represented by decimal expansion,

so the 0.000...1 you showed me earlier is still nonsense. So, no,

believing in infinitesimals does not justify the denial of the equality.

All it demonstrates is a lack of understanding of what infinitesimals

are and of what a decimal expansion is.

"All the "proofs" the two are equal are meaningless. Implicit in all of

them is the statement, "Using the real number system." That's begging

the question. It's tricking people by assuming there could be no

difference between the two numbers then concluding there is no

difference between the two numbers."

No, this is not tricking anyone. You said it yourself, the equality

implies the field of real numbers. This is called a convention. Claiming

this is misleading shows you don;t understand how conventions work. You

know, you made this remarkable statement at the beginning of the article

saying WE do not understand math, but clearly, what the actual sentences

show is that you're the one who does not understand it.

"Anyone who understands how math works should know 0.999... equals 1

only if you choose for it to. "Proofs" the two are equal tell us nothing

about the subject but everything about the speaker. Namely, they don't

know what they're talking about."

Another ad hominem.

"It's remarkable how often people introduce themselves into discussions

with insults of the form, "You're an idiot.""

Which is exactly what you did, you moron. I have no reason to treat you

civilly when you're the first one, literally the first one to insult.

LITERALLY the first one. Disgusting piece of hypocrite that you are.

"It's no better than telling people the square root of four is -2."

You clearly don't understand how conventions work. This is the second

claim you make that suggests this to me. I was giving you the benefit of

the doubt, but right now I cannot anymore.

"Richard, I'm afraid I can't agree with you when you say:

Because of its prominence and usefulness in the real world, assuming

that we are using the real number system is reasonable. When I say

1+1=2, I am using the real number system. I could also be using many

other number systems and it’s still true, but unless specified we use

the real number system.

In the real world, most people don't use the real number system."

No, most people do use the real number system in most applications.

"Most people don't use formalized math."

Whether the math is formalized or not has literally nothing to do with

whether they use the system or not. Non-sequitur. Jesu Christ. All the

fallacies I've had to address today.

"When you say "1+1=2," there's no reason to assume you're using a real

number system. All that does is add a series of unnecessary

complications to the situation. Why say you're using the real number

system instead of the rational number system? Or integer system? Or

whole number system? Or natural number system?"

Because literally in almost every application, the real numbers

specifically are used. Not the rationals. Not the integers. The f***ing

real numbers, Brandon. Is this really so difficult a concept to

understand? In the real world, you will practically never encounter the

equation 1 + 1 = 2 in isolation, so the fact that this equation can

exist in multiple common systems is not an argument.

"I'm pretty sure if that's true, it's only because the explanation they

were given was poor. For instance, telling people "you are only using

the real number system" likely won't change how they feel if they don't

understand what that statement actually means."

It doesn't matter. Mathematics is not about feelings, and it is not

about intutition. I could not care les about your feelings concerning

the number 1. If something is true, then it is true. Period.

"The real number system doesn't allow us to do math we couldn't do under

a different system. More interestingly, infinitesimals have been used

when studying problems even within the real number system. They were

seen as a temporary solution to be used until a more robust one could be

found. I doubt there is any practical necessity to avoid using them

given they have been used, successfully."

The hyperreal numbers have solved some problems, but relatively minor

ones, and they stil by no means encompass all applications. Plus, the

hyperreal number system is significantly harder to understand, and even

more difficult at that to use, and such difficulties and confusions are

not worth the effort of solving problems in the domain of real number

problems when they can be solved by just the real numbers with relative

ease. Besides, the fact is that the real numbers are still the most

applicable system by far.

"But again, this goes back to how number systems are nested. If you

really wanted to, you could create a superset which encompassed real

numbers and allowed for non-zero infinitesimals to exist while

maintaining all the calculations we can currently do in the real number

system. It would be similar to how real numbers encompass rational and

irrational numbers. Just like how the real number system can still allow

you to do all the calculationg you might have done in the rational

number system, this new number system could allow us to do all the

normal calculations we might do within the real number plus more."

Almost every non-Archimedean number system is like this. The only

practical ones are definitely like this. So that goes to undermine the

very main point of the post, because even with these systems, 0.99... =

1. If you truly want something different, then use the p-adic numbers,

but those are significantly more impractical.

"Dennis Whitener, I appreciate you taking the time to comment here, but

I am afraid you have missed the point of this post. To demonstrate, ask

yourself this. Why does .999... * 10 = 9.999...? Why does it not equal

9.999...0?"

I've already answered this question. It comes down to fractions, it does

not have anything to do with infinitesimals. Surely, 0.999...0*10 =

9.999...0, but 0.999...*10 = 9.999... even if infiitesimals exist. There

is no digit at the end of the strings, not even in systems where

infitesimals are allowed, because that is nonsense and defies the very

definition of what infinity is.

"Coming after an infinite string of decimal points would make a number

infinitely small."

No, it doesn't make it infinitely small. You should study what an

infinitesimal is and how it actually is represented before spouting

nonsense you literally know nothing about.

"Early mathematicians made the rule which said non-zero infinitesimals

do not exist (even as some of them used such infinitesimals in proofs as

kludges)"

They had plenty of good reason for it.

These are the comments I'm addressing from your post in the old

WordPress.

"It just doesn’t seem to make sense that 0.anything could be equal to

1.0."

0.999... equals 1, not 0.888... or any other decimal.

"It’s only by making unintuitive (to many people) assumptions, like

“non-zero infinitesimals don’t exist,” that one can declare the

two numbers equal."

1. Intuition is wrong more often than not, so unintuitive assumptions

are not a problem. The vast majority of science and mathematics are

unintuitive. The truth doesn't care about your naive intuition.

2. Your claim is false. In the surreal numbers, where non-zero

infinitesimal numbers exist, 0.999... = 1 is also true anyway, and it

can be prove very easily.

"SebZear, I have no idea what makes you believe what you just wrote. Not

only is it easy to find number systems given by non-standard analysis

(three of the most obvious being hyperreal, superreal and surreal),

non-standard analysis must create new number systems as a matter of

definition. All you’ve done is hand-wave away a multitude of

mathematical fields and numerical systems anyone could find in a matter

of seconds with Google. Doing that in order to claim “this post in

particular is beyond moronic” is, well, beyond moronic. To be clear,

standard calculus does not accept the existence of (non-zero)

infinitesimals. Non-standard calculus does accept the existence of

(non-zero) infinitesimals. The two cannot use the same number system."

Not relevant, since I already stated that in the surreal number system,

and in fact, in any number system with infinitesimals, 0.999... = 1 is

still true, and this can be proven in complete generality.

"This is a proof which depends upon assuming non-zero infinitesimals do

not exist. Under a different system, 0.333… * 3 could equal 1, not

0.999…."

No, it doesn't require this at all. Also, under any number system in

which arithmetic multiplication exists AND in which 0.333... exists,

0.333... * 3 will always imply 0.999... = 1, and this can be proven by

applying the linearity property of multiplication.

"That premise, which is nothing but an arbitrary assumption one need not

make, is what should be discussed"

No. The premise is not based on an assumption. The premise can be

proven.

"Throwing “proofs” around that do nothing but assume the answer

doesn’t address their concerns. It does nothing to address the premise

behind the question. All it does is mislead them."

No one is using these proofs.

"However, the entire argument you advance is one based upon the idea of

treating .9 repeating as a convergent series. That is tied to the

concept of limits which is in turn tied to the axiom I discuss in this

blog post – that non-zero infinitesimals do not exist."

Actually, it is not tied to this idea, at all. Limits are well defined

even in the surreal number axioms, and in the surreal numbers, you can

use either the limits or the actual infinite quantities proposed by the

axioms. Either way, you will still always conclude that 0.999... = 1.

"In other words, your claim I don’t know what I’m talking about

rests entirely upon assuming an axiom is true despite this post clearly

showing there is no inherent reason to assume that axiom is true."

Your post didn't show there is no inherent reason to assume the axiom is

true. In fact, there is an inherent reason to assume the axiom is true,

and it underlies the burden of proof. Since there is no proof that these

numbers exist, I have no reason to assume they exist, so the axiom is

the only reasonable conclusion. Now, I can assert their existence as an

axiom, but this changes the rules and creates a new form of mathematic

which may not be compatible, and which may or may not be useful. Hence

why the burden of proof is necessary to begin with.

"0.999… does not equal 1. People who offer “proofs” otherwise

don’t understand math.“

People who reject proofs arbitrarily without providing a reason have no

understanding of mathe either, and this includes you. Also, what an ad

hominem. Clearly unable to prove your stance, so you decide to call us

ignorant, even though some of us literally have degrees in math. The

irony.

"It openly states there are algebraic frameworks in which 0.999 does not

equal one. The only real area of disagreement is it points out there are

frameworks which allow for infinitesimals where the two numbers are

still equal."

Yes, all of which change the very definition of summation itself, in

which case it may not even be true to say that 1 + 1 = 2. To say that

0.999... does not = 1 based on a fringe number system which has little

to no applications from the real world and which differs from any normal

notion of what numbers are and addition is, it is completely dishonest,

and such willing mislead and dishonesty deserves the hate this post has

gotten. Period.

Now I will address everything else in this page.

"How does one multiply an infinite series of 9s by 10?"

Easy. You apply the linearity property of multiplication, and then you

prove it for the infinite case by using the principle of mathematical

induction.

"What happens to the zero you'd get when multiplying by ten?"

There is no zero. Multiplying by ten only yields an extra 0 digit when

the number being multipled is expressed as an integer within base-10

notation.

"Of course not. The proof is invalid. It's just an optical illusion

relying on tricking the reader by hoping they don't notice the

hand-waving."

There is no hand-waving involved. You just happen to not understand how

arithmetic multiplication works. This is what this comes down to.

"When we do that, we see there is a difference - 0.000...1. It's

infinitely small, but it is real."

No, it is not real, and it is not infinitesimally small, because it is

not a number. By definition, if there is a series of infinite digits,

then this series is unending, hence there cannot be a different digit

after.

"Why then do so many people believe they are? Because they say so."

No, we KNOW they are equal because we can PROVE they are equal. The

Wikipedia article on the subject provides many proofs, all of which rely

on very different ideas. To reject the proofs is unreasonable.

"Only we don't say so. A person who thinks 0.999... doesn't equal 1

obviously believes infinitesimals exist. They don't accept that axiom.

They'd use a different one, like many mathematicians who work with

infinitesimals on a regular basis."

Again, I'm uncertain of how many times I should repeat myself. Any

system which uses infinitesimal numbers also has the equality 0.999... =

1 anyway, and infinitesimals cannot be represented by decimal expansion,

so the 0.000...1 you showed me earlier is still nonsense. So, no,

believing in infinitesimals does not justify the denial of the equality.

All it demonstrates is a lack of understanding of what infinitesimals

are and of what a decimal expansion is.

"All the "proofs" the two are equal are meaningless. Implicit in all of

them is the statement, "Using the real number system." That's begging

the question. It's tricking people by assuming there could be no

difference between the two numbers then concluding there is no

difference between the two numbers."

No, this is not tricking anyone. You said it yourself, the equality

implies the field of real numbers. This is called a convention. Claiming

this is misleading shows you don;t understand how conventions work. You

know, you made this remarkable statement at the beginning of the article

saying WE do not understand math, but clearly, what the actual sentences

show is that you're the one who does not understand it.

"Anyone who understands how math works should know 0.999... equals 1

only if you choose for it to. "Proofs" the two are equal tell us nothing

about the subject but everything about the speaker. Namely, they don't

know what they're talking about."

Another ad hominem.

"It's remarkable how often people introduce themselves into discussions

with insults of the form, "You're an idiot.""

Which is exactly what you did, you moron. I have no reason to treat you

civilly when you're the first one, literally the first one to insult.

LITERALLY the first one. Disgusting piece of hypocrite that you are.

"It's no better than telling people the square root of four is -2."

You clearly don't understand how conventions work. This is the second

claim you make that suggests this to me. I was giving you the benefit of

the doubt, but right now I cannot anymore.

"Richard, I'm afraid I can't agree with you when you say:

Because of its prominence and usefulness in the real world, assuming

that we are using the real number system is reasonable. When I say

1+1=2, I am using the real number system. I could also be using many

other number systems and it’s still true, but unless specified we use

the real number system.

In the real world, most people don't use the real number system."

No, most people do use the real number system in most applications.

"Most people don't use formalized math."

Whether the math is formalized or not has literally nothing to do with

whether they use the system or not. Non-sequitur. Jesu Christ. All the

fallacies I've had to address today.

"When you say "1+1=2," there's no reason to assume you're using a real

number system. All that does is add a series of unnecessary

complications to the situation. Why say you're using the real number

system instead of the rational number system? Or integer system? Or

whole number system? Or natural number system?"

Because literally in almost every application, the real numbers

specifically are used. Not the rationals. Not the integers. The f***ing

real numbers, Brandon. Is this really so difficult a concept to

understand? In the real world, you will practically never encounter the

equation 1 + 1 = 2 in isolation, so the fact that this equation can

exist in multiple common systems is not an argument.

"I'm pretty sure if that's true, it's only because the explanation they

were given was poor. For instance, telling people "you are only using

the real number system" likely won't change how they feel if they don't

understand what that statement actually means."

It doesn't matter. Mathematics is not about feelings, and it is not

about intutition. I could not care les about your feelings concerning

the number 1. If something is true, then it is true. Period.

"The real number system doesn't allow us to do math we couldn't do under

a different system. More interestingly, infinitesimals have been used

when studying problems even within the real number system. They were

seen as a temporary solution to be used until a more robust one could be

found. I doubt there is any practical necessity to avoid using them

given they have been used, successfully."

The hyperreal numbers have solved some problems, but relatively minor

ones, and they stil by no means encompass all applications. Plus, the

hyperreal number system is significantly harder to understand, and even

more difficult at that to use, and such difficulties and confusions are

not worth the effort of solving problems in the domain of real number

problems when they can be solved by just the real numbers with relative

ease. Besides, the fact is that the real numbers are still the most

applicable system by far.

"But again, this goes back to how number systems are nested. If you

really wanted to, you could create a superset which encompassed real

numbers and allowed for non-zero infinitesimals to exist while

maintaining all the calculations we can currently do in the real number

system. It would be similar to how real numbers encompass rational and

irrational numbers. Just like how the real number system can still allow

you to do all the calculationg you might have done in the rational

number system, this new number system could allow us to do all the

normal calculations we might do within the real number plus more."

Almost every non-Archimedean number system is like this. The only

practical ones are definitely like this. So that goes to undermine the

very main point of the post, because even with these systems, 0.99... =

1. If you truly want something different, then use the p-adic numbers,

but those are significantly more impractical.

"Dennis Whitener, I appreciate you taking the time to comment here, but

I am afraid you have missed the point of this post. To demonstrate, ask

yourself this. Why does .999... * 10 = 9.999...? Why does it not equal

9.999...0?"

I've already answered this question. It comes down to fractions, it does

not have anything to do with infinitesimals. Surely, 0.999...0*10 =

9.999...0, but 0.999...*10 = 9.999... even if infiitesimals exist. There

is no digit at the end of the strings, not even in systems where

infitesimals are allowed, because that is nonsense and defies the very

definition of what infinity is.

"Coming after an infinite string of decimal points would make a number

infinitely small."

No, it doesn't make it infinitely small. You should study what an

infinitesimal is and how it actually is represented before spouting

nonsense you literally know nothing about.

"Early mathematicians made the rule which said non-zero infinitesimals

do not exist (even as some of them used such infinitesimals in proofs as

kludges)"

They had plenty of good reason for it.

"People who intuitively question the idea that .999... equals 1 simply because .999... is "infinitely close" to 1 are not making any sort of mistake. They are simply not thinking of the "real" number system. The "real" number system is just one of many numerical systems though, and there is nothing inherently wrong with using a different one. The only thing which is wrong is using proofs, especially arithmetic ones, which only work in the "real" number system in response to people who intuitively question the "real" number system"

Except THEY are mistaken, because even in the number systems where infinitesimals do exist, the equality is true. There are number systems in which the equality is not true, but they do not use infinitesimals. and actually they change the very definition of addition and multiplication. So this is incommensurate, because in such systems, decimal representation does not even work the same anyway. Also, most people are not questioning the real number system, as Richard clarified. And they're not being taught poorly either. Some people are being taught poorly, but most people questioning this are actually very knowledgeable. I can tell you this as someone who has a degree. They just fail to understand the very idea of what infinity means, and the way decimals work.

"The question is, why must people use the "real" number system? The answer is, they shouldn't have to. There's no mathematical basis for forcing them to."

There is plenty of logical and practical basis for forcing it. You know, you can go ahead and ask every mathematician you see, but also every scientist, every doctor, every economist, every demographist, and even foundation philosophers. Most of them will agree with me. There is a reason the real number system is the convention.

"But how do you move the decimal point to the right in an infinite series? To start, there are an infinite number of 9s after the decimal place. You then take one of those 9s away and put it before the decimal point. You then have an infinite number of 9s after the decimal point, less one. How do you say those are the same?"

Because infinite numbers remain unchanged when subtracting a finite number or adding a finite number to them. In fact, this is the defining property of infinite numbers. If they do not satisfy this property, then you can prove that they are not infinite.

"The answer in "standard" analysis is to say there is no difference between "infinity" and "infinity minus one." That's fine. However, in "non-standard" analysis, there can be a difference between the two."

No, there is not. Look, it is completely acceptable to propose alternative axioms, but you literally don't even understand the very axioms you're proposing. Have you actually studied non-standard analysis? i don't think so.

"Vikash Hamilton, I can't tell you why a pattern occurs yet might not continue on indefinitely. What I can tell you is nine entries is a pattern means little in math."

The pattern continues necessarily because of the principle of mathematical induction.

"In various aspects of math, you can find the same series of nine numbers show up in many different patterns. Without more data, you can't tell which pattern is actually present. Similarly, you can find many instances where there seems to be a pattern but the pattern doesn't hold."

You don't need more data if you have parametrized the series by a map with a variable input, which is actually what one does in the principle of mathematical induction.

"In this case, part of what you're seeing is an optical illusion created by the number system we use. As you know, we use the numbers 0-9 in our number system. As you probably know, at the lowest level computers (more or less) only use the numbers 0-1. That is called binary, or base 2. Another common number system goes from 0-F, with A, B, C, D, E, F following the number 9. That is called hexadecimal, or base 16. In each of these number systems (and the infinite other systems we could choose to work with), the pattern you see will be different. In base 2, the "pattern" will be a single item prior to the "proof": .111.... In hexadecimal, it would consist of 15 (.AAA, .BBB, etc.)."

Yes, and whenever these number systems are being used, the base is specified as a subscript. So unless there is a subscript, all of this is irrelevant, good sir.

"How many times do you think a pattern has to repeat before you accept it as "proof"? Whatever the answer, you can change your numerical system to fit it. You could work in base 100 and have it repeat 99 times. But at the same time, you could pick a more exotic number system and have the pattern not show up at all (sort of)."

That isn't really how patterns work, but in any case, this has nothing to do with number systems anyway.

"This is why inductive reasoning like looking for patterns and assuming they'll continue indefinitely is risky. If you don't understand why those patterns happen, you can easily draw incorrect conclusions."

I'm afraid you don't understand the principle of mathematical induction, then. Also, all of our real world knowledge is based almost exclusively inductive reasoning. Inductive reasoning works.

"That is why in math deductive reasoning reigns supreme It is only by using rigorous, deductive analysis we can tell what does and does not hold true for our numerical systems. Inductive reasoning is limited to being a useful tool for exploring idea, not proving them,..."

The principle of mathematical induction is actually a form of deductive reasoning, ironically. So, your argument doesn't hold.

"If one believed the two numbers did not equal one another, then the pattern you describe would hold for nine items, break for one, then repeat for nine more."

No, it wouldn't actually. Instead, the pattern would hold for exactly 0 items. You can prove that if it does not hold for 1 item, then it cannot hold for the previous items either. I'm not doing it, though, because it isn't an important point, and given how dogmatic you are, there is not worth in proving it.

"That what you see changes based solely upon the choice of base you use shows why inductive reasoning like this is risky. If you want to pursue this line of thought, what I'd recommend doing is examining why this pattern happens at all. Once you understand why the pattern occurs, you can understand why the pattern would or would not continue in the manner you suggest."

Let me repeat myself: the principle of mathematical induction. Enough said.

"Namely, whether or not the pattern holds as you suggest will ultimately depend upon whether or not your mathematical framework accepts the existence of non-zero infinitesimals."

No, it doesn't. You keep spreading this false information without even understanding the axioms to build these infinitesimals. You literally pretended that these infinitesimals can be expressed as decimals. Again, if you actually read on these things instead of being a willful liar, you will realize that in the hyperreal numbers, 0.9... = 1 is still true, as is the case with the surreal number too, which are a generalization of the hyperreal numbers. I know how to use the hyperreal numbers at the very least. I could bother to explaining it to you if you were respectful, but your response to me in the old blog post shows that you're not even willing to admit you're wrong, even though any mathematician will tell you you are wrong. The dogma is too real.

"Well, if you refuse to say anything about the central issue of the topic being discussed, I'm not sure what there is to discuss. The entire crux of this disagreement, one you've done nothing to address, is how people should handle non-zero infinitesimals."

No, it isn't, actually. I've explained about a hundred times already. The crux of the issue is the definition of addition and how to define it whenever the space of inputs is infinite-dimensional. You're free to change the way addition works completely, but there is no reason to do so, and it is wrong to do so unless you specify that you're deviating from the convention. There is a good reason conventions exist.

I could have been more polite presenting all this information, but not only did you insult mathematicians and people who are educated, but you are willfully spreading misinformation about non standard analysis and about how axioms work, and are also completely rejecting conventions without presenting a valid reason. So I'm just direct.

I've addressed every point. if you're annoyed by me, then perhaps you should stop spreading misinformation.

Wow. What kind of person feels the need to post five comments in a row, repeating the same ideas over and over just so they can make sure they respond to every last single word? I mean, if the first time you talk to someone you find yourself saying:

You've probably taken the wrong approach. Repeating yourself a hundred times before getting a response is a reason people should be exasperated with you, not a reason you should be exasperated with them.

Honestly, I can't imagine why anyone would bpther reading all that. Heck, as a moderation policy, it's spam. There's no reason to post 5000 words so you can repeat yourself a hundred times without anyone else speaking.

DO NOT look ahead in this text as it will bias you while you are reading through. Think about each sentence before you move along to the next. And please just ignore any errors that i may have made in syntax or any possible writing errors.

There is a certain matter on the way you perceive the term ".9 repeating" in this case. There are two ways to think about the term. Think about having any number, most preferably use 1. Divide that number by 2. Divide it again. Keep doing this until you get an infinitesimally small number. Is this number a real number? You cannot get to 0 by dividing, and hopefully you already knew this. Now take the number 1 again. Instead of dividing it by 2, this time divide it by 10. Divide it again. Keep doing so until you have reached an infinitesimally that only contains a single digit, the digit 1. What number do you think this is? Do you believe that it is zero? Do you believe that it is real?

Really think about this seemingly simple statement, and in all honesty, it is very simple and easy to understand for even the most inexperienced of minds that have came to this page to seek in what they hope is the end to their disputed mind. So do you believe that a number such as this, 1/10^∞, can even exist. Some of you may have heard the term asymptote. An asymptote of a curve is a line such that the distance between the line and the curve tends or approaches to zero without ever touching. This is the same situation with .9 repeating and 1. The curve will be the sigma equation for .9 repeating. The extended version will look like this: 9/10^1+9/10^2+9/10^3...+9/10^n. Now imagine the asymptote being equal to 1. The curve gets closer to the asymptote by .9. Then the asymptote gets closer by .09. Then the line gets closer again by .009, then .0009, then .00009, and so on. The main point of the idea is, the 2 functions will NEVER touch.

Now here is the other way to think of .9 repeating, just another way to write 1. Most mathematicians will say that the two numbers will converge, like in the extended sigma notation that I presented earlier, or any other sigma notation that converges. There is just no such thing as .9 repeating. It is an arbitrary number that just doesn't exist. An infinite sum is not finding the sum of the numbers, but having a difference that you shrink down and keep shrinking until the equation is equal to the actual value. Take .9+.09+.009+... for example. Using this equation, the difference between .9 and 1 is .1. Then the difference keep getting smaller and smaller until the number reaches its real value. There is no such thing as the number .9 repeating and numbers that do not have an actual value like .83 or 9.5, even i is more real then .9 repeating.

Through this perspectivial analysis of .9 repeating, it is determined that is is either differing by the infinitesimally small amount that some say cannot exist, or .9 repeating was something created that has somehow survived the trails of speculation by the public who have not even considered that it may just not even be real in the theoretical world. But for the people saying that adding 9's over and over again is equal to 1, you have not yet explored the different perspectives and ways of thinking for such a unique but simple number which troubled so many to an extent in which I have spent half an hour to write this, or all you others who many have even spent HOURS to argue your point. But my idea is to never waste and be as effective and efficient as I can, which is why I have given you these different ways to think of the problem, "is .999... repeating equal to 1?"

"the zero you'd get when multiplying by ten"? A zero at the end of a number, after a decimal separator?

0.4 * 10 = 4

No zero needed. One can add it (make it 4.0) but that doesn't chance the number. Your zero argument is false.

Hein, I'm afraid you did absolutely nothing to address what I said. Selectively quoting a person so you can pretend to respond to something they said while in reality you don't respond to what they said may make you feel warm and fuzzy inside, but it won't do anything to help discussions progress.

If you don't mind talking to yourself, feel free to stick with your current approach. If you want to talk to others though, I recommend you try reading what they say. If nothing else, the fact you had to cut off a portion of the sentence you're responding to ought to be enough to tell you what you were doing was wrong.

I sympathize with your complaint about the claims that .999... = 1. Most of the replies provided are not adequate, or are rather nasty in tone. I also laud your demand for clarity on this issue. That said, I am perplexed by your complaint about 10x=9.999... That somehow this is an invalid operation. Do you also not accept Root(2)*Root(3)=Root(6)? If you think finding a place for the 0 in 10*.999... is a problem how do you begin to multiply numbers where neither have an end? I would think the better argument against this so-called proof is the line 9x = 9. 9*.999... = 8.999... to say that it equals 9, requires one to accept that 8.999... = 9 but subtracting 8 from both sides leaves .999... = 1, which is what was supposed to be proved.

As for 10x = 9.999... The reason we can multiply like this is that mathematicians worked very hard to show that these numbers can be represented as infinite series, and that the operation of multiplication works in infinite series. This justifies the use of algebraic formulas on symbols representing these numbers. So just as the algebra for Root(2)*Root(3)=Root(6), so too does the algebra of 10x=9.999... (where x=.999...). .999... is simply the geometric series:

.999...=9(1/10)^1 + 9(1/10)^2 + 9(1/10)^3 + ...

(I really wish there was a way to write this in mathematical type--unfortunately this blog doesn't support MathML or MathJax).

The same operations that were shown to work on infinite series like Root(2) (called Cauchey Series) also work on geometric series.

There is another proof that shows that a geometric series can be formulated as (a/(1-r)). This turns out to be another way to see why .999... = 1. See https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-series-optional/v/deriving-geometric-series-sum-formula

and

https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-series-optional/v/infinite-geometric-series

The formal proof however uses limits, which from comments on here, some feel are suspect. There is a way to derive the formula using Taylor series, and then proving convergence, but that is too complicated to do here, and I couldn't find a link for it anywhere on the web.

That said, I don't think this is the best way to understand why .999...= 1, and why we want that equality in the real number system.

The hand-waving you complain about can only be gotten rid of by rigorous work in infinite series. It is there that you will find why the algebra works.

Ironically, your statement about the difference between .999... and 1 is "real" is technically false. What you should have said is that the difference is Hyper-real.

The reason why mathematicians, and anyone SHOULD, accept that .999... = 1, isn't based on some arbitrary assertion (i.e. they say so). It is based on the results of a complete system that allows for precise and non-contradictory measurements. Most people cannot use the Hyper-real system, and therefore use the "Real Number system." In order to do that the system uses numbers defined in a consistent way. So, for those numbers to give non-contradictory results over the operations of arithmetic certain rules must be followed. Those are the axioms. They weren't discovered and placed into a complete system arbitrarily. It took centuries of development.

You seem to think that a person can simply change axiomatic systems at will. Robinson's Hyper-real system has to accept indeterminacy. And now there isn't just finite and infinite. There's this new category called infinitesimal that cannot interact with the other two. In other words, there are new rules that create other confusing statements--until you understand why and how they work. You've isolated on ONE part of non-standard analysis and applied it to the real numbers.

Have you thought about what could be added to .999... in order to get it equal to 1 in the Hyper-real system? Nothing! So, not only does .999... not equal 1, you can't add anything to it to make it equal to 1. An uncountable infinite number of Hyper-reals can be added to .999..., and it would not equal 1. So guess what that means. 1 - .999... is undetermined! Is that more intuitive than the real number system proving that .999... = 1? (I prove it below).

Mathematics is such a valuable area of study because it is so useful. Denying some aspect of it prevents you from using the results. We, of course, want to make sure that the equivalences are in fact equivalent; that's what the proofs are for. If we're simply faking them, or asserting them, then math, or parts of it, begin to be unusable.

Even though an axiom cannot be 'proven' (which literally means showing that it is the result of an axiom or several axioms), it can be given validity by demonstrating that it's denial, would require it's use, or some corollary's use. To deny the Least Upper Bound Axiom, you need to use its corollary the Archimedian property, and if you want to deny the Axiom of Archimedes, you would have to its corollary, the L.U.B. theorem to deny it. These two properties are deeply intertwined, and adopting one, allows proof of the other. Denying both requires the adoption of one of them in order to attempt to invalidate their use. This is why adopting one is acceptable--its denial would require a form of itself in order to deny it--like cutting off a branch of a tree that you are sitting on. Note: there is a difference between dropping an axiom, and saying it is invalid to use it.

The axioms that complete the real number system are essential for the use of the arithmetic operations we need in order to make measurements. Without those axioms problems arise--as you are aware of from the history of mathematics (the Pythagoreans discovering Root(2) incommensurability).

And it is in the axioms, and their corollaries, that we see why .999... must equal 1.

From the standard real number axioms two key theorems emerge. One is the Archimedian property, and from the Archimedian property, the Density of the Rational numbers.

The Archimedian property says that if a and b are positive real numbers, then there exists some positive integer n such that n*a > b. I'll explain the value of this after the proof:

PROOF: Suppose that n*a > b did NOT hold for any integer n, thus for all integers n we instead have n*a <= b. Then let the set A = {n*a|n is an element of N} (where N is the natural numbers). This set A is bounded above by b. By the Least Upper Bound axiom there exists a least upper bound for A. Call this l.

Since a is positive, (l - a) < l; and thus (l - a) is not an upper bound for the set A. So, for some integer m it must be the case that (l - a) < m*a, or equivalently, l < (m + 1)*a. But this contradicts the choice of l as an upper bound for A.

q.e.d.

So what does this give us? It gives us the fabulous result that every magnitude of the same type can be compared! No two lengths are incommensruable anymore. No more Root(2) problems.

Using the Archimedian result we can prove the density of the Rational numbers, which states: If a and b are real numbers with a < b, then there is a Rational number r such that a < r < b.

PROOF: Both 1 and (b - a) are positive, so the Archimedian property is available for use. For some integer m we have 1 < m*(b - a). This can be rewritten as

m*a + 1 < m*b

Let n be the largest integer such that n <= m*a. Add 1 to both sides to give

n + 1 <= m*a + 1 < m*b.

But since n is the largest integer less than or equal to m*a, then m*a < n + 1 which means

m*a < n + 1 < m*b

rewritten as

a < (n+1)/m < b,

we create a Rational number between a and b.

So if .999... < 1, then there MUST be a rational number that will fit between .999... and 1.

Intuitively I think you might be able to see now what this result leads to. A rational number is some finite value created by two integers, say u and v in the form of u/v. This number is necessarily finite. But the number .999... is created by a geometric series which is inifinite in its construction. No finite value will fit between .999... and 1, yet the Density of the Rational numbers theorem says there HAS to be a rational number between any two distinct real numbers. Thus .999... can't be less than 1. I will prove for you that not a real OR a rational number fits between .999... and 1.

PROOF: Let 0.999... + e represent a number between 0.999... and 1. so 0.999... + e 0.

Let f(k) = 0.999...9 (where the last 9 is the kth 9)

for all k in N (the Natural Numbers), f(k) 10^-k, then 0.999... + e > f(k) + e > 1. That can't happen so for all k in N, e 0 though, so if -k = log(e) (all use of logs here is in base 10), which will eventually happen, then

10^log(e) < e, but 10^log(e) = e. so we have a contradiction.

q.e.d.

This isn't arbitrary. To deny this result comes at the pain of giving up a lot of the results worked for in the real number system. The beauty of this result is that we never have to worry about something popping up between .999... and 1. No mathematical operation will EVER produce a value between them.

The interest in non-standard analysis for most people is that some results are easier to prove than in standard analysis. It was thought that using the hyper-reals to teach Calculus would be easier for students. However, if you look at any "infinitesimal" calculus textbook you will see either a huge gloss (i.e. hand-waving) over the development of ultrafilters and translation, or an explanation that is far too complicated for calculus students to understand.

I am not saying there isn't any value to the study of non-standard analysis, but it's value is really tagged to the development of standard analysis, and one cannot talk of hyper-reals while in the real number system. It's like mixing cricket rules with baseball rules. Pitch in baseball is not Pitch in Cricket, and when dealing with .999... and 1 and saying they aren't equal because there are hyper-reals between them, you can't use real number proofs from that system to prove or disprove anything; hyper-reals don't exist in the real number system, and non-standard analysis isn't a "better" system to analyze the statement .999.. = 1, than standard analysis. Pointing to a Hyper-real is like saying "I see a ghost," and when someone asks "where?" you point to a video game. Ok, they exist in the video game, but they don't exist out here with the finite and infinite numbers. They are simply different systems.

Note: There isn't an axiom in the real number system stating that hyper-reals don't exist. However, it is a consequence of those axioms that they don't exist in that system (otherwise there would be contradictions).

For a long time I too was not satisfied with the statement that .999... = 1 in the real number system. It was only after taking real analysis, and working it out down to excruciating detail, that I understood why .999... does in fact equal 1. If it helps (it did for me), you can start by understanding why no contradiction arises, or will ever arise, by treating these number representations as if they were equal in the real number system. If you don't see that, then you need to ask questions until you understand why. It has been laid bare here.

Trevor, thanks for the response. You say:

The issue is when you shift the decimal point to the right with that operation, the result is there is an infinite number of 9s after the decimal point in one number while there are an infinite number less one on the other side. Within the real number system, that distinction is irrelevant because of the axioms of the number system. In other words, it's no different than offering a proof which says, "Non-zero infinitesimals do not exist therefore .9 repeating equals 1."

That's why I labeled proofs like that optical illusions. They don't prove the issue in question. The issue in question when a person asks why the two numbers are equal is, "Why should we accept the axiom that non-zero infinitesimals do not exist?" Any answer which presupposes we accept that axiom is always going to be begging the question. That's why the first part of your comment misses the mark, because the issue in this post is not that

limitsare being questioned, but that the real number system itself is.This simply isn't true. Before I get into that though, I should stress the hyperreal number system is not the only "alternate" number system one could come up with for the real number system (e.g. the surreal number system is an extension of the hyperreal). This is not a binary choice. That said, people can easily use the hyperreal number system. Kids are routinely taught the natural number system before they're taught integers, both of which are taught before the rational number system, all three of which are taught before the real number system. Each system is taught to people built upon the previous. There is no reason the hyperreal number system could not be treated the same.

When a young child asks, "Why can't we have numbers smaller than 0," you don't say, "Because people can't use whole numbers." When a child asks, "Why can't we have numbers smaller than 1," you don't say, "Because people can't use real numbers." In the same way, when a person asks, "Why can't there be infinitely small numbers," you shouldn't say, "Because people can't use the hyperreal numbers."

No, I haven't. I haven't applied anything to the real number system. I've said when a person asks a question directly challenging an axiom of the real number system, it is highly inappropriate to offer proofs which ultimately say nothing more than, "Because that's how the real number system works." Even if you think people in general can't use hyperreal numbers for some reason, when a person directly asks about the possibility of non-zero infinitesimals, they clearly have some grasp of the idea of numbers beyond teh real number system.

This section confuses me. 1 - .9 repeating is not undetermined any more than saying 0 < x < 1 leaves the value of x undetermined. Yes, there is an infinite range of values x could be, but why should that be confusing? And if it is not confusing, why should one range containing an infinite number of values be more confusing than another? It's quite simple really. There can be an infinite number of infinitely small numbers between real numbers. When one doesn't need to deal with infinitesimals, they take the standard part of the hyperreals and their values simplify to the reals. Which is the same as taking a limit. It's also basically the same as rounding a real number into a whole number.

I can't begin to understand your reasoning here. I mean that literally. I can't see the slightest explanation justifying your claim rejecting the Archimedean property would require using the least-upper bound property. This claim, as well as its inverse, is the entire basis for everything you say here, but you do nothing to justify it.

And it doesn't make any sense. It is false by basic logic. You cannot assert rejecting one axiom automatically accept a different axiom. That's just not how logic works. I can only assume you meant something other than what you wrote as this paragraph is completely incoherent to me. I have to stop my response here because I literally cannot come up with words to adequately explain how wrong this paragraph is.

Rejecting one assumption does not require a person automatically accept a similar and highly related assumption. I can't understand why you would say it does.

Hi Brandon,

Let me make some clarifications. The multiplication operation you describe is an algorithm designed for integers. There are many algorithms that can be used to achieve the same answer correctly. So just because you can't find a place for a zero doesn't mean the multiplication can't be done, it just means that algorithm doesn't work in this case. The work in infinite series shows that the multiplication can be done, just not in the standard way we multiply integers. This is why I brought up the multiplication of irrationals, because we wouldn't be able to multiply those either. All of a sudden a huge chunk of math is gone. We wouldn't be able to use those results.

Also, 10*.999... = 9.999... in the Hyper-reals, the Sur-reals, or the Granular numbers. That operation works in all of those systems. Why? no contradictions.

There is no axiom that says Hyper-reals don't exist. Can you find it for me? The Hyper-reals don't exist in the real number system (standard analysis), because no operation of arithmetic can produce them. That's what the axioms are for in the real number system. They outline what operations we can perform without obtaining a contradiction.

A clarification. When I say that most people can't use the Hyper-real system, I am not saying they can't learn to use the rules, I am saying the Hyper-real system isn't useful for them. The real number system is used everywhere in the sciences and engineering. The Hyper-real, or any non-standard analysis based system hasn't shown any value yet for these people. That doesn't make it forever useless. There are things that might prove valuable in the future. But it doesn't resolve the .999... = 1 problem. It has its own issues there .viz indeterminacy.

If you take a class in non-standard analysis you will learn that 1 - .999... is literally labeled "undetermined." There is no single value that can be named as the answer. Placing x between two numbers is a characterization. y - x is an arithmetic operation. An arithmetic operation over two concrete numbers should produce a concrete value--or at least this is what mathematicians have wanted for a long time. It is why Root(2) was so jarring to the Greeks. Of course we can describe ranges of values for a variety of purposes, but the closure principle for completing the real number system was so highly prized for precisely this reason. An arithmetic operation, should provide a concrete arithmetic result--i.e. a single number.

On the use and acceptance of a proposition as an axiom: The mathematical domain is different from the logical domain, but axioms in both serve the purpose of outlining the rules to obtain results without contradiction. I say this because I want to use a simpler example. Math is more complicated than standard logic, so the proofs are longer.

In logic we have

A=A,

A cannot equal notA (at the same time and in the same respect--if we allow change), and

it is the case that either A or notA is true.

These are axioms. Why should we accept them? Because to deny their use (any of them), would require using the axioms to make the denial!

Try and say we shouldn't use them and you violate the first one right away, because what you say has to refer to something. That something has an identity (call it A). If it isn't A, then I can't begin to believe what you are saying because your reference has no identity. Everything said would have to be treated as an utterance without meaning, but even that has an identity. So A=A is inescapable.

The l.u.b. and Archimedian property aren't that basic, but they share a similar characteristic being an axiom. To accept them, they should not introduce a contradiction. There aren't any contradictions, and the only contradiction possible requires using either the l.u.b. axiom or the Archimidean axiom (or property--the two are corollaries of each other). To say I can't use l.u.b axiom would require that you demonstrate a contradiction, but the only way to do this is to use it (or its corollary, the Archimedian property), which would then be a contradiction, because I am not supposed be able to use it. This is a high standard for a proposition. This is why axioms aren't some arbitrary whim.

It was hard to understand that last paragraph because it involves relationships we don't usually deal with, but it is perfectly logical. The justification is there if you are willing to see it. I tried to explicate it more explicitly here.

Thought experiment:

Star Trek the Next Generation:

Captain: Data, we need a calculation for jumping into Hyper-space, can you perform it for me?

Data: Yes Captain.

Data: Captain.

Captain: Yes Data.

Data: I have performed part of the calculation, but the answer is .999... repeating. To complete the calculation for the coordinates, I need to enter this value, but I cannot enter an infinitely long digit, shall I use 1?

Captain: No. We are jumping into Hyper-space, repeat the calculation using the Hyper-Real system.

Data: Yes Captain.

Data: Captain, I have performed the calculation.

Captain: Good, give us the coordinates.

Data: I can't do that sir.

Captain: Why?

Data: When I performed the calculation using the Hyper-Real system I ended up with an indeterminate result.

Captain: How is that possible Data?

Data: The answer is a definite Hyper-Real number, but we don't know which one sir. I could try to go through all of them to check, but there are an infinite number of them, and there is no ordering among them, thus we would never find it. Shall we go back to the real number system and use 1? It is a perfectly valid use sir.

This statement isn't clear in my previous post:

"There aren't any contradictions, and the only contradiction possible requires using either the l.u.b. axiom or the Archimidean axiom (or property--the two are corollaries of each other)."

What I mean, here is that any denial of the axioms requires their use which itself is contradictory.

Trevor:

This is a strange response given I didn't say anything about "find[ing] a place for a zero." What you say simply doesn't address what I said. What I said is the two number have a different amount of nines after the decimal point because one is infinite and the other is infinite less one. That difference is excluded within the real number system, but that is due solely to the construction of the real number system.

In other words, that proof is no different than simply saying, "Non-zero infinitesimals do not exist." That may be an assumption used with a particular system, but it is non-responsive to a person who asks a question that basically says, "Why isn't there an infinitely small difference between those two numbers?"

I don't know why you say there "is no axiom that says Hyper-reals don't exist" rather than deal with what I actually said. What I said is an axiom of the real number system is non-zero infinitesimals do not exist. That axiom excludes the possibility of hyperreals, but it does much more than that. Asking me for an axiom with such a narrow focus seems bizarre when I referred to an axiom with a broader focus.

That aside, there is no single set of axioms agreed upon for the real number system (it can be constructed in many ways), but the non-existence of non-zero infintesimals a necessary result of the least upper bound property, which is axiomatic for the reals.

So when you say "most people can't use" something, you mean "most people won't use it"? That's a strange way of phrasing things. I don't think most people consider "can't" and "won't" to be synonyms. Besides, most people won't ever have a use for calculus. I guess we should refuse to discuss limits when people ask if .9 repeating equals 1? I'm pretty sure that's not how things work, but if you want to go that route, I'm cool with it.

You keep repeating this claim as though saying thesame thing often will make it true. That's not how logic works. SImply repeating your assertion does not lend credibility to it.

Nonsense. I can say, "I don't accept that axiom." That is a perfectly valid thing to say. It doesn't require I offer an alternative set of axioms or contradict anything. There is no positive burden of proof on a person who merely fails to accept what you say.

This makes no sense at all. First, I never said you

can'tuse any axiom. You're arguing against a strawman. What I said is a person can choose which axioms they wish to use, which a fundamental truth of mathematics (and all logic).Second, you've done nothing to establish why rejecting one of these properties would require using the other. That idea is incoherent and has no basis in logic. It's merely an assertion you keep repeating without attempting to justify in any way, fashion or form. And it's also one that's trivially easy to prove false as I can simply point to any number system which uses neither property. Giving we've actively talked about number systems which use neither property (hyperreals/surreals, and those are just the tip of the iceberg), I find it baffling you'd keep repeating this insane idea.

For the record, saying the "justification is there if you are willing to see it" is rude and will never convince anyone you're right. It's also nonsense as you've made absolutely no attempt to justify what you said or challenge what i said in response to it.

By the way Trevor, this is a strange analogy used to conclude:

You keep making an issue of the fact there are an infinite number of hyperreals between any real numbers like that somehow causes the system to be unusable, but that's nonsense. There being an infinite amount of values between two numbers is not a problem for anything. It isn't. If one needs a definite point, one wouldn't be use a hyperreal so they'd take the standard part of any hyperreal they have to find the value they need (which is effectively the same as taking a limit). Making a big deal about how hyperreals aren't usable in certain situations is like saying, "Real numbers don't work, how can I set my thermostat to sqrt(2)?" You can't. So what? Not all number systems are useful for all questions.

As a side note, I think it's interesting it hasn't been pointed out .999... isn't a hyperreal. One would never write that in the hyperreals as .999...999... isn't a hyperreal while .999.990 is. Similarly, I don't know why you only talk about the hyperreals when I already pointed out they are not the only system in which infinitesimals exist (specifically pointing to surreals as a further extension). This is a very strange conversation.

This is a strange response given I didn't say anything about "find[ing] a place for a zero." What you say simply doesn't address what I said. What I said is the two number have a different amount of nines after the decimal point because one is infinite and the other is infinite less one.

Ok, but the distinction is one without a difference. Either way you want it, the multiplication is valid. It is valid because you are complaining about an artifact that applies to integers. You can multiply using methods other than the integer algorithm. I again repeat, you would have to remove ALL multiplication involving irrationals, yet you seem to have no problem with those--or do you?

You also complain that these are the results of the axioms.

Of course! That's why they are so important. They aren't mere artifacts. They aren't dreamed up to "make" something happen the way we wish, they have to be consistent with the other axioms, and provide value for their use. In this case "completeness" of the arithmetic operations. We measure using arithmetic, this simply means that we can now measure without running into operations that provide values we don't know what to do with (like root(2)).

What I said is an axiom of the real number system is non-zero infinitesimals do not exist.

Again, there such is no axiom, it is a consequence (i.e. we can prove they don't exist). If you mean that the axioms of the real number system consequentially disallow for infinitesimals, then ok, but I would say your statement is unclear in that regard.

Me: A clarification. When I say that most people can't use the Hyper-real system, I am not saying they can't learn to use the rules, I am saying the Hyper-real system isn't useful for them. The real number system is used everywhere in the sciences and engineering. The Hyper-real, or any non-standard analysis based system hasn't shown any value yet for these people. That doesn't make it forever useless. There are things that might prove valuable in the future. But it doesn't resolve the .999... = 1 problem. It has its own issues there .viz indeterminacy.

You: So when you say "most people can't use" something, you mean "most people won't use it"? That's a strange way of phrasing things. I don't think most people consider "can't" and "won't" to be synonyms. Besides, most people won't ever have a use for calculus. I guess we should refuse to discuss limits when people ask if .9 repeating equals 1? I'm pretty sure that's not how things work, but if you want to go that route, I'm cool with it.

I am unsure of what you mean here. I'll be more specific with an example. I recently heard a talk by Professor Emeritus Dana Scott (of Carnegie Mellon University and Stanford). He was one of the pioneers in the development of a variety of axiomatic systems for study in mathematics and computer science. He is an Alan Turing award winner. In his talk he discussed a new axiomatic system that he thought was interesting. At the end of his talk he asked if anyone thought it was useful. He made the statement that over his career he had seen lots of developments in mathematics that provided systems to work in, but had no real use for anyone else other than mathematicians. He did not want to be doing that.

Incidentally, when we work in "algebras" we are identifying which axioms are appropriate to use for that particular "algebra," so it isn't that we always have a complete system to work with. What's different is that the axioms aren't dropped, they simply can't be used in that circumstance. Thus some questions that would be answerable in the real number system simply can't be answered at all--regardless of the axiomatic system adopted.

This is why I stress the value of the axiomatic system for the real numbers. Every science uses it! There are only very special, and limited areas, where anyone has found non-standard analysis (which includes Hyper-reals, Sur-reals, Granulars, etc.), useful outside of demonstrating some interesting proofs, which always get translated back to the standard reals. In other words, it hasn't been able to produce results of interest that aren't producible by the standard real axiomatic system.

You keep insisting that the result .999... = 1 is merely because someone made up some axiom to make it true. But think about what non-standard analysis does. It makes an arbitrary definition for 1/infinity. It has no justification for doing so, it simply likes that definition. But in order to make that definition work they have to drop the property of having a complete system for arithmetic. But now they at least can have their definition. There isn't any basis for doing this other than they like that definition. However, I can understand the interest. It might be something interesting that comes out of it, that may prove useful in other areas of mathematics. But it needs to be understood that it will never be able to replace standard analysis for the reals. Not because of some arbitrariness, but because of completeness! A complete system for arithmetic operations is extremely valuable.

If you were to change your rant to .999... = 1 doesn't have to be the case, I could understand. But you point to some proofs (some of which are correctly identified by you to be misleading if not outright incorrect), as if there is NO correct reason to believe it unless you accept arbitrariness in mathematics (i.e. you can choose whatever math you want to use--you can't if you want to use it). A correctly (yes completely logical!) given proof was provided to you by me, but you complain that they depend on axioms. YEP! they have too. That is what a proof is. The axioms are NOT arbitrary as you assert.

You: You keep making an issue of the fact there are an infinite number of hyperreals between any real numbers like that somehow causes the system to be unusable, but that's nonsense.

Why is it nonsense? If I need a specific coordinate, but I have an infinite number to choose from, how is that useful? Hyper-real numbers cannot interact with finite numbers.

I have no problem using Root(2) in measuring temperatures, or any other science. The density of the rational numbers theorem allows me to use that result beautifully.

You: For the record, saying the "justification is there if you are willing to see it" is rude and will never convince anyone you're right.

I don't know why it is rude. I thought you were honestly interested in finding the result of why .999... = 1. I was trying to appeal to your honest interest in inquiry. I was letting you know that there is nothing more needed. I am not trying to force you to believe anything. I am trying to show how this all works. You seem to demand more than is needed.

It's also nonsense as you've made absolutely no attempt to justify what you said or challenge what i said in response to it.

That isn't rude? I actually did justify it. I have tried multiple ways to provide a perspective that would allow an honest investigator to see why this works. Ask yourself why something becomes an axiom. Why would one statement be an axiom and another one not? When you see that these are simply chosen on a whim you will see the value of .999... = 1.

As a side note, I think it's interesting it hasn't been pointed out .999... isn't a hyperreal. One would never write that in the hyperreals as .999...999... isn't a hyperreal while .999.990 is.

Yep, and there is no way to get .999.990 using a standard arithmetic operation. You have to take .999... + Hyper-Real. In other words, you have to have the specific Hyper-real ahead of time, but there is no way to get it.

I will make the following statement to see if you would accept it.

For all scientists and engineers it is crucial that .999... = 1, however if you would like to look into the theoretical possibility of .999 not= 1, there are several systems you can work in that provide for that result.

I can live with that, since it is very clear about what is going on here.

A point on axioms.

You say: "I don't accept that axiom."

Which axiom? A=A? Hmmm. Let's see. "I" refers to whom? "axiom" refers to what? What is the difference between "I" and "axiom"? To answer these questions you have to believe that the references exist, and if they exist they have an identity. Whoops? A=A Says just that. "A" has an identity. To say you don't accept the axiom A=A, you had to make use of its property. To really not accept the axiom, "I don't accept that axiom" means the same as LDOIE*(#$JFG*(#L:JKFDKS. So you really aren't saying anything at all. You couldn't, but I think you actually meant what you said. I don't think you just typed a jumble of letters. So you can't honestly say you don't accept that axiom.

Trevor, it would help if you'd use simple HTML formatting available on most blogs, or even just quotation marks, when quoting things. Especially when making a long comment with multiple quotations, it's annoyingly difficult to follow what you mean if you don't set quotations off in some manner. Since you didn't bother to do something so simple, I'm not going to try to parse your entire comment. Instead, let me focus on a couple simple points:

The existence of non-zero infintesimals directly violates the completeness axiom, “If a non-empty set A has an upper bound, it has a least upper bound.” The fact it violates this axiom is demonstrated by you saying infinitesimals are disallowed by the least upper bound property. Given vyou say that property doesn't allow for infintesimals and that property is an axiom of the real number system, I am mystified as to why you say there is no axiom which forbids the existence of non-zero infintesimals.

Indeed. That is how all axioms works. I don't know why you feel the need to specify it for one system yet deny it is true for other systems. That is how logic works. First order principles are always chosen without any (internal) justification.

Yes, they are. Axioms are, by definition, arbitrary. Axioms are inherently arbitrary because they are first order principles, meaning there is nothing they can be based upon.

If you need a specific coordinate for some physical purpose, then what you need is not a hyperreal. Dismissing hyperreals as they are not useful for certain purposes is no more valid than dismiss real numbers because your thermostat only accepts integers.

If you truly don't understand why what you said was rude, I'll explain. Saying, "X is there, if you're willing to see it" to a person who has already said they don't agree with you about X is, via simple transformative properties, the same as saying, "X is there, but you choose not to see it."

1) X is there, if you are willing to see it.

2) You don't see X.

3) You must not be willing to see X.

4) X is there, but you are not willing to see it.

5) X is there, but you choose not to see it.

No, it isn't rude. There is no way to say what I say more politely. And while you claim I am wrong in what I say, you've again chosen not to make any effort to show how I am wrong. A simple truth of all axioms is they are arbitrary. There is no inherent basis for them. There cannot be as they are first principles.

You've claimed otherwise, repeatedly saying certain axioms are true because it's impossible to reject them. Despite being challenged on this claim, you've made no attempt to show how rejecting a first principle could possibly require accepting a (specific) different first principle.

Of course I wouldn't accept this. This issue is in no way crucial to most scientists or engineers. For most, this won't matter at all.

Trevor:

Prove this.

I'm serious. Offer a proof of this statement. You can't. Any proof you attempt to offer will necessarily require assuming A = A, meaning you will have assumed the question you are attempting prove, which is begging the question. It's nothing more than saying, "My framework is right because my framework is right."

Can I offer an alternative system in which A does not equal A? Perhaps not. That's irrelevant though. There is no negative burden of proof. A person failing to disprove A = A does not make A = A true. And certainly, person failing to offer an alternative framework to A = A does not make A = A true.

A = A because we say that is true. We say that is true because we find it useful to say it is true. That is all there is to it, and that's how all first principles work. By definition, it is impossible to

provea first principle. Your repeated claims of having proven first principles are utterly false.Hi Brandon.

Ok, I think I've narrowed down the problem.

You have two arguments against .999... = 1. One is a faulty argument based on how multiplication works (in both the real number system and non-standard number systems). The second is the belief that we can make any mathematical system we want by arbitrarily choosing some axioms.

I think if you were to make that clear up front (as I suggested), rather than simply stating an assertion that all 'proofs' are wrong, and then burying the equivocation down in the body of your blog, there wouldn't be this visceral reaction by some of the posts.

I don't know if you came to this 'arbitrary' belief on your own, or if you read it somewhere (it is certainly out there). However, any time a mathematician who makes such a claim is asked to explicate that belief, there is a retreat. They generally retreat to the idea that well, it's not really 'arbitrary' per se (in the sense that we can state whatever axioms we want), but rather it is interesting to explore what happens when we add or subtract axioms from well known axiomatic systems--or even it is interesting to build up an axiomatic system from known statements to see what is possible. No matter how its done, it's never 'arbitrary', or simply 'because we say so.'

But It's not irrelevant. If you can't make a system without A=A, then that would mean every system has to have it.

You are asking too much from proof. You use the term correctly when you recognize that axioms cannot be proven, but then you demand that I prove an axiom. Axioms are chosen for their usefulness. To be useful it has to be consistent with reality. This is why I can demonstrate that you can't build a system without A=A. It is so damn useful that to deny it renders every statement into jibberish.

Nope.

Yep. Which directly refutes your earlier statement.

Correct. Proofs are explicitly defined to render a given proposition consistent with the axioms.

I never made such a claim. I only showed that to deny the axiom, requires using its consequence in the denial. Your whole blog requires A=A, otherwise it would be simply jibberish.

I think the positions are clear. I am not demanding you change your mind. I thought you would be interested in knowing why .999... = 1. First, by showing that you CAN in fact demonstrate a correct proof (even if it is in the real number system), and second, that you don't have to go down the 'arbitary' rabbit hole. Invoking non-standard analysis doesn't really resolve anything, it just creates different difficulties that have to be dealt with.

Trevor:

No. Just no. There is nothing faulty about what I said regarding multiplication. And even if there were, that was never an argument I offered against that equality. I offered it as a response to a supposed proof for that equality, nothing more. It has always been inconsequential to my point, save in that it addresses something people sometimes say. I don't know why you present it as more than that.

Nonsense. I didn't bury anything in my post. My post isn't even 500 words, and I bring this issue up starting in the second paragraph of the post. It's the entire focus of the post. The entire message of this post is axioms are arbitrary, and as such, should not be presented as fact. It is not buried, at all.

Again, you are relying solely upon argument by assertion. Writing lots of words to say, "I'm right because I'm right" is a waste of everyone's time. If you have some basis for these claims, you're welcome to show them. However, vaguely alluding to things so you can try to pretend you're right without addressing any points contrary to your views is pointless.

This is simply not true. You're presupposing a system must exist at all, an arbitrary assumption that has no logical basis. If you truly believe what you say, then as I said before, prove it. Offer some logical construction which shows your claims are valid. You can't. It's impossible as first principles cannot, by definition, be proven to be true.

I said axioms cannot be proven to be true, and as such, they are arbitrary. You said I am wrong, that you can prove axioms are true by showing the rejection of one axiom requires the acceptance of another, supporting principle. Crying foul when I demand you provide the basis for your claim is silly.

I know fully well you cannot prove any axiom to be true. That's my point. I am showing why your claims that axioms can be proven to be true is false. If you stop claiming to be able to prove axioms true, then I'll stop demanding you prove axioms to be true. For instance, when you say:

You are claiming to prove an axiom true. It is silly to simultaneously claim to prove an axiom true while saying it is unreasonable for people to demand you prove that axiom to be true.

This post directly states the two numbers are equal to one another within the real number system so I cannot begin to imagine why you would think I'd be interested in you showing me a proof the two numbers are equal to one another within the real number system. The post literally contains a proof the two numbers are equal within the real number system. Why would I need you to provide me another?

I think everything is clear here for those who want to read it. You seem to claim that my arguments are assertions, when they aren't, you have retreated from your claim in the title (thankfully). But you still 'arbitrarily' assert that axioms can be made arbitrarily when they can't. Just because we can explore axiomatic systems by making choices about what we make them out of, doesn't mean we can have a functional axiomatic system--and without a functional axiomatic system, the statements are meaningless.

When you say "You're presupposing a system must exist at all" we know where this has come. No system has to exist, but then no consistent statements can be made. Everything you've said or will say is rendered meaningless--yet you write as if people should understand what you say, why? You have no basis anymore to make any meaning at all.

You also misunderstand proof. "You said I am wrong, that you can prove axioms are true by showing the rejection of one axiom requires the acceptance of another." I never said I could prove an axiom. All I did was demonstrate that you have to make use of the axiom in order to deny it--that isn't a proof. It is a performance issue. You cannot make an argument without using the axiom--nor can you make one without a system of axioms. You have essentially hoisted yourself on your own petard.

For anyone who is seriously interested in the question you have in your title, there are two choices. You can learn why .999...=1, or you can just accept that nothing has any meaning at all.

This is not why mathematicians study non-standard analysis. They need, and want consistent axiomatic systems. The interest is in how much of a system we need, what we can do without a complete system, etc. It is pretty clear after years of study, a complete axiomatic system is what we want. It is consistent, and lets us perform operations necessary for use in any science or engineering task.

Thanks for clearing up the issues you have with this question.

Trevor:

What are you talking about? All arguments are, by definition, assertions.

I don't know why you say this. NOthing I've said to you is any different than what I said in my post. The message of my post is .9 repeating does not equal 1, nor does it not equal 1. That's why the last paragraph begins, "If you feel 0.999... does not equal 1, you're right. If you feel it does equal one, you're right too." This is the same thing I've said to you.

This is rubbish. You're trying to claim to be able to prove an axiom must be accepted while saying it's unreasonable to demand you prove that axiom. You do so by claiming, "I'm not proving an axiom, I'm proving you can't do without the axiom!" That's not any sort of logical argument. At best, it's an argument of practicality.

If you want to claim something is true for practical purposes, you can. That's not what you've done though. You've repeatedly stated things are tre in an absolute sense, as in based in logic. But logic doesn't care what is practical so what you say is simply rubbish.

This is simply dishonest. You've accepted the existence of mathematical frameworks which expand the real number system and allow those two numbers not to be equal, yet you then turn around and create this false dichotomy? That's lying. YOu know fully well this is a false dichotomy, as you've demonstrated time and time again on this page. The fact many mathematicians use number systems you dismiss here proves you're full of it.

If you want to leave because you refuse to engage in an honest manner, you can. But the fact you still haven't done a thing to justify your claim rejecting the least-upper bound axiom requires accepting the Archimedean property shows you're not engaging honestly. That claim was baseless, and I demonstrated it was false by pointing to accepted mathematical frameworks which accept neither property. Ignoring challenges to your false claims and haughtily walking off won't convince anyone you are right.

I'd like to follow up my last comment with one that addresses this issue more directly. Trevor dismisses number systems other than the real for whatever reasons, but the hyperreal number system is an extension of the real number system. That is, the hyperreal number systems, by definition, contains the real number system. Any problem that could be solved within the real number system can, by definition, also be solved inthe hyperreal number system. The same is true for the surreal number system and the hyperreal number system.

The natural number system rejects the existence of numbers like -3 because they are negative. The integer system rejects numbers like 2.5 because they have decimals. The rational number system rejects the existence of numbers like the square root of 2 because the decimal expansion of them is non-repeating. The real number system rejects the existence of non-zero infinitesimals because they don't allow for the existence of infinitely small or large things. That is how our numerical systems are constructed. When one system rejects the existence of something, another system gets created which allows for the existence of it.

Do we use the real number system for all our problems? Of course not. If you count on your fingers how many times something happens, you're using the natural number system. You're not using calculus because you have no reason to. Most people don't. In the same way, you likely won't use the hyperreal number system. But that doesn't make the hyperreal number system less "real" or valid. Calculus doesn't go away because most people never use it. The hyperreal number system doesn't go away because most people never use it.

That said, ona historical note, hyperreals were actually accepted within the mathematical community for thousands of years. They were used by people like Isaac Newton, just without the formalized system we now have for them. The idea of infinitely small numbers existing and being useful for problems was even accepted by people like Cauchy, a founder of modern calculus, just 200 years ago. It was only in the late 1800s infintesimals were decreed not to exist, and that was largely due to a couple people like Cantor falsely claiming to have proven they cannot exist (because, he claimed, they are internally inconsistent).

For thousands of years infintesimals were accepted within mathematics. Approximately 150 years ago, they were formally rejected. Approximately 50 years ago, they were formally accepted again. Now, few mathematicians dispute that they can exist (though some still dispute whether or not they *should* exist).

Just some Clarifications.

Actually that is almost exactly backward. I don't dismiss these systems, I actually have produced work in them. I just recognize them for what they are. But importantly, the hyper-real system is NOT an extension of the real number system. Dropping an axiom literally makes that not the case. We view these systems from the perspective of the real system because the real system is so widely used and was developed first. It was only when a consistent way of putting the axioms together without the completeness axiom was shown to have interesting results, that these non-standard systems gained any interest. But importantly, they are not "better" than the real system, cannot provide anything that cannot be produced in the real number system, and are demonstrably less useful than the real number system for everyone outside of mathematics. But most importantly, the .999 not equaling 1 in these systems comes with a side effect never mentioned--you can't get these special numbers using arithmetic. It was thought that these special numbers could be what Newton was referring to when he developed the Calculus using infinitesimals. We now know that isn't the case. Newton's method would never have any need for them. However, the propositional statement that the numbers are so small that we can drop them remained true--why? Because you will never have to deal with a Hyper-real number (or any other number of Surreal or otherwise). The similarity with the real-number system is that we treat .999...= 1 because we can't produce a real number that goes in between them--ever! Hyper-real numbers are like ghosts, we can find a way to define them into a system, but we never have to worry about them because they are walled off.

Let's not anthropomorphize number systems. They don't 'reject' or 'accept' anything. The systems are categories for US--they allow us to analyze based on restrictions, or extensions of their domains. Peano was the first to lay down a system for the Natural Numbers. The problem mathematicians had, going back to the Pythagoreans, was that certain operations would produce values outside of the domain of what most people thought were 'numbers'. Every number domain, up until the real number system, had a problem with producing arithmetic values outside its domain. Hyper-real numbers are not produced this way. They are defined into a system (not, as you say, defined out of the other number systems). Robinson found a way to make that definition compatible with a set of axioms, but he had to drop completeness in order to do so.

Thanks to the OP for the OP itself and the vigilantly straightforward responses to the copious red herrings. I hope you have found contemplating and responding to them worth your time. 🙂

Trevor:

I find it incredibly difficult to believe your claims about your personal experience given you say something so mind-bogglingly stupid. Dropping an axiom for a system does nothing to prove that system is not an extension of another. What you say directly contradicts hundreds of mathematical references. Even Wikipedia's basic description of hyperreals contradicts you:

What you say shows a completely lack of understanding of what it means to extend one field into another. You've been doing things like this all along. You keep stating things as fact, without offering any basis for them, while they directly contradict basic and fundamental concepts of the fields in question (math/logic).

I'm going to stop here. I don't mind people having a lack of knowledge. Ignorance is not bad. However, a person who so consistently fabricates things in order to pretend to have knowledge they clearly do not possess is a person who simply shouldn't waste anyone else's time with their presence. I don't like doing this, but for people like you, there is a very simple solution. This site has a long-standing rule that any factual claim made by a person here (including myself) can be challenged with a demand of justification/proof. You state, as a fact:

This is a factual claim which I know to be false. Under this site's rules, I ask you to provide a justification/proof of what you say or else admit it is not a fact and you were wrong to present it as one. You are required to do so if you wish to comment further.

chronoblip, thanks. Some of the responses have been interesting, and some have even made me think about things differently than I might have otherwise. I'd say that means it has been worth my time, even if not every comment posted here has been.

I've actually been meaning to re-visit this post with a new one that discusses the issue in a more systematic manner. Whenever I see a new response to my post saying I'm wrong, I get more motivation to do so.

There is a difference between extending the real number system and extending the real numbers. The Hyper-real numbers do not extend the real number system (i.e. the axiomatic system that covers the domain of the real numbers). When one says they extend the real numbers, one is saying they are making room for them on the real number line.

You earlier gave the example of Natural Numbers excluding -3, Integers excluding rational numbers etc. There is a big difference among these domains (domains are not the same as the axiomatic systems in which they operate), and the Hyper-reals in the real domain. Each of the numbers excluded from the domains (negatives from the Natural Numbers, rational numbers from the Integers etc.) are produced by arithmetic operations. Hyper-reals are not. This is why a set of axioms was worked on over a long period of time in order to get a complete set that allowed operations in the axiomatic system that would not produce new numbers outside the domain.

Now, take Complex numbers. These too have to drop an axiom, however complex numbers are NOT placed on the real number line. They are in the complex plane. Hyper-reals have to have a special place carved out for them on the number line. So not only do they drop an axiom, they squeeze themselves on the real number line--despite not being computationally derived by the arithmetic operations.

One of the things that is important is that we keep the concepts being used distinguished. You have a rather quick temper and ask for demands that are unnecessary. All that was needed was a clarification.

Trevor, the requirement I explained you are under was you provide a basis for the factual claim(s) you made. You have chosen not to do so. Simply saying something is true does not provide a justification for claiming it is factually true. All you've done is make a new series of claims which depend upon a form of semantic hair-splitting that has no basis in anything other than your say-so. You've offered no evidence, no reference, no citation, nothing. That is not acceptable.

What you say is untrue. If you believe otherwise, provide some sort of evidentiary basis for what you say. If you continue to choose not to do so, you will be violating this site's rules. This is not difficult or complicated. If what you say is true, it should be quite easy to demonstrate.

Also, for anyone reading this, I want to stress it is complete and utter nonsense to say "extending the real number system" is so incredibly different from "extending the real numbers" as to justify what Trevor said. In fact, the two phrases mean basically the same thing. That's why it's trivially easy to find things being referred to an an extension of the real number system or of real numbers, interchangeably. A perfect example is this Wikipedia article on quaternions, an extension of the complex number system. The article begins:

The idea there is some super important distinction between an extension of a number system and an extension of the numbers of that system is laughable. Trevor would have you believe the two phrases mean basically the exact opposite things, and he'd have you believe it for no reason other than he tells you so. It's nonsense. An extension of the reals is an extension of the real numbers which is an extension of the real number system. The idea there is some magical semantic parsing which completely inverts the meaning of one's sentences that was never brought up until this point is just silly.

"Hah, hah, hah, ha, I shrunk the real number system by extending the real numbers!"

What I write has a specific meaning. When you say the real numbers, you are talking about the domain. The system includes the axioms over that domain. Mathematicians can use language that isn't quite so precise as long as everyone understands what is being said.

Unfortunately, you read something and then inject your own interpretation onto what you read. Then you demand a proof of something from this imprecise reading. You never ask for clarifications, and simply assail the author demanding proofs for things not stated, or for things that don't need proofs.

I am not here to be your tutor or your counselor. My original points have been made, and are correct. I thought you would ask questions if something was unclear, but that is not your modus operandi. I don't know how you ever learn anything.

For anyone who comes across this blog and wants to know why .999... = 1, the answers are here.

Trevor, I believe anyone who reads this exchange will not see what you hope they will see in it. Regardless, you have made it abundantly clear you refuse to follow a very simple rule of this site, and as such, you are now the recipient of a "soft ban." What this means is you are still allowed to post in specially designated posts (such as this one). You are also allowed to post in any other thread, so long as any comment you post consists of nothing but a link to a comment you have written in one of these specially designated posts.

I am fine with that.

1 does equal .99999999.. . . . . .

Change my mind plz

Evaluating the proof. The question is if two numbers are equal, x and y.

multiplying an odd number times an even number is always an even number.

https://en.m.wikipedia.org/wiki/Parity_(mathematics)

Odd numbers leave a remainder of 1 when divided by two.

http://mathworld.wolfram.com/OddNumber.html

0.9… divided by 3 equals 0.3…; division is done by digits.

The repeating decimal number 0.9… divided by 2 results in a remainder.

0.9… = x; ok, but the question is x = y, not what is x.

9.9… = 10x; note that 10 is an even number and the repeating decimal is odd.

10 times 0.9… equal what ?

making the claim that an odd number times 10 is an odd number is wrong.

⅓ is equal to 1 divided by 3 or 3 divided into 1, so the question is what number times three equals 1. Fact: 3 can’t fit into 1 but can fit into 10/10. The result and proof is in long division: 3 times 3/10= 9/10, not 10/10; and no matter how many times you divide 3 into 10, there will always be a remainder, so based on logic, 0.3… ; has less value than ⅓ therefore the self evident fact is three times a value less than ⅓ is a value less than 1.

The purpose of the decimal point is to separate the units from the fractional parts, example: 1.5 means 1 unit and half a unit, 0.5... ; 0.7 means no value of 1, just as 10 (10.0) has no value of 1, the repeating 9 has a value of 0 for the ones place, that means that 0 doesn't mean a value of 1. So some of you here believe that part of a unit has the same value as a whole unit ?

This is elementary math, taught with fraction where the fact is that only a 1 to 1 ratio of a fraction can have a value of 1, but the 9 followed by an infinitely repeating 9's over 10 followed by an infinite number of 0's is an odd to even ration, not a 1 to 1 ratio, not a value of one.

Algebra states that two numbers are equal when they subtract to 0 (x = y when x - y = 0) subtraction is done by comparing the digit to digit of the two numbers, resulting in a non zero number because once any set of digits do not subtract to 0, the resulting number can not be equal to 0.

Decimal values are defined by the value and position of the digits constructing the numbers. So adding digit 1 to 1 equals 2 is performed by processing the digits, and changing the position or pattern of the digits results in the addition of another digit to 1, so what is that other digit added to 1 that equals 2 ?

This and many more proofs, proves and confirms that these numbers are not equal. Math is a language to express logic, some of you simple lack the required logic to comprehend.

So the people that believe these numbers are equal, fail in basic arithmetic. They refuse to accept or ignore established and confirmed math rules and facts, in favor of their believe and opinions of others.