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.

I completely agree, just use common sense. The digit in the ones place is 1 in one of these numbers, and in another, it’s 0. 1 > 0. Also, a decimal place with no digit before it means the number is less than 1. .999 does not equal 1.

AG, thanks for your comment. Your thinking is what causes most people to ask the question. It just doesn’t seem to make sense that 0.anything could be equal to 1.0. 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.

By the way, you may be interested to now I’ve transferred domains. I copied all the old posts and comments over to:

http://www.hi-izuru.org/blog/

Because I prefer running things on my own website rather than using WordPress.com. I’m keeping this site up so people can find the new one (it’s discussed in the newest post), but otherwise, I’m not using it for anything.

Now that I think about it though, it might be worth going through and posting a comment on each post directing people to the version at the new site. That’d make sure visitors would know about the new domain.

These two 10 minute videos explain clearly why 0.999… does NOT equal 1

Why 0.999… does NOT equal 1 (Part 1: The Problem) https://www.youtube.com/watch?v=–HdatJwbQY

Why 0.999… does NOT equal 1 (Part 2: The Solution https://www.youtube.com/watch?v=xSlS2xE8rH8

Don’t forget to ‘like’ them.

Considering that there are proofs as to why 0.999… does not equal 1, I am questioning problem 30 of the 2014 Mathcounts school sprint round. To solve this problem, you need to use a concept similar to the one where you substitute 0.999… as x. Is this solution also invalid? I still have doubts about the claim that 0.999… is not equal to 1, but I also have doubts about the claim that 0.999… is equal to 1. Maybe we’ll see in the future which side can give a valid proof.

Sorry, I meant chapter not school

Just so you guys know, I’ve largely discontinued this site so I can move it over to a new domain. You can find the corresponding version of this post there at:

http://www.hi-izuru.org/wp_blog/2014/01/0-999-does-not-equal-1/

As you can see, it’s not a very different URL. What’s different is that site runs on a server I can control, rather than one WordPress controls. That gives me a lot more options, such as allowing me to install plugins.

Unfortunately, there’s no way for me to redirect users who access this site to that one (without paying money to WordPress to do it for me), so all I can do is try to alert readers of the change. The newest post here tells people about the change, and I’ve updated the site’s tagline to try to draw attention to it. I hope that’ll help people find the new site as I pretty much never check this site anymore, and all of my new material is posted at the new one.

0.999… is simply an endless series just like 1+2+4+8+… is an endless series.

The most famous proof that 0.999… equals 1 involves multiplying it by 10 and subtracting the original series, and this supposedly removes the endless part.

But we can apply the same logic to 1+2+4+8+…; by simply multiplying it by 2 and subtracting the starting series we can supposedly remove the endless part giving the result 1+2+4+8+… = -1.

This uses exactly the same logic. And so if 0.999… = 1 then 1+2+4+8+… = -1.

In both cases the logic is flawed because we are not lining up the 1st terms of the series when we do the subtraction. In both cases we are comparing the first (n+1) terms of the multiplied series with the first n terms of the starting series in order to create the illusion that the trailing terms all cancel out.

However, 1 is related to 0.999… in the same way that -1 is related to 1+2+4+8+…, it is the ‘fixed part’ of the expression for the sum to the n-th term (or the fixed part of the ‘partial sum expression’ if you prefer – but I don’t like the word ‘partial’ as it implies there may be a full sum).

By the way, just to clarify, 1/3 does not equal 0.333… because the algorithm to convert the fraction to a decimal has no defined end point. The endless terms are not zeros and so the algorithm must have some defined way of ending in order to produce a fixed value.

0.333… is an endless series that cannot equate to any fixed value because the series does not end; it has no ‘last term’. It has an endless supply of positive non-zero terms.

An endless series should be treated as an endless series, not a fixed value. We should not simply assert/define that a series must equal the fixed part of the expression for the n-th sum (known as the ‘limit’ for a series that is said to converge).

You’re treating 9.999…>10*0.999… Proof being; When you do that to any number in decimal series of any length; 0.444*10=4.444, 10*777777=7.777777, etc and put it through the Cantor number manipulation you get 1=0.9…n to whatever length the original decimal series is. When the starting number of places is 3 you get 0.999=1 when 7 you get 0.9999999=1 when 0.xxx… you get 0.999…=1. A correction factor illustrated in finite terms 10*0.999=9.999-0.009 results in the correct output from the Cantor number manipulation for any number of any decimal length.

Actually, OP is completely right, even this (0.999… doesn’t equal to one), was proved rigorously many years back, the clever point behind this famous ruined example must be explained to every student to avoid the nonsense mathematics, one must ask himself why the professionals unusual defence for this ruined example, it is not a silly issue but very important question that has a very obvious answer, (the term: 0.999… with infinite repeated digits of 9’s) is completely a fectious number, or on other words “non existing number” on the real line number, but the consequences of this failure example are so huge in maths, this immediately would imply the same, i.e “nonexistence of all transcendental numbers or all algebraic numbers (that are not constructible), along with the infinite decimal representation of any constructible number”

This would eventually raise an urgent alarm to repair the mathematics to save it from infinitely fectious numbers that are impossible to exist on our real line number or else it is just as interesting games like chess

To clarify my idea, you may kindly have a look on my answers on Quora at this link:

https://www.quora.com/Philosophy-of-Mathematics/When-and-Who-was-the-first-to-break-the-mathematics/answer/Bassam-Karzeddin-1

The… At the end of these numbers indicate that they go on forever. That is to say there is no blast nine, and therefore no zero that is added on when it is multiplied by 10. Likewise, 0.00…1 makes no sense since there is no last 0, and therefore they can’t be followed by 1.

Lol there is no 0 on the end 0.9999….. Never ends

the 9s in 0.9999……………… are actually infinity, any finite number of them wont clear the gap, but we cant no wheather an actual infinity number of them clear or don’t clear the gap from 1. the problem is rather with actual infinity. do not treat 0.99999………asa potential infinity decimals! its sctual infinity number of decimals.

lets say that .99999…. is n

.9999…=1

now multiply by 10

10n=10

then subtract 9

n=1

Of course if you change the axioms of maths you get different results for things but if you want to meaningfully say x = y or x != y you have to say in which number system you mean, if you do not specify it is presumed that you mean the real number system so 0.999… = 1. Your same argument can be applied to any mathematical proof of anything.

0.999…=1 bear true even in number systems which do include infinitesimal numbers. That is because the definition of periodic decimals yields to be true, and it has nothing to do with what number system you use, but with the the definition of infinite sums.

Consider a sequence {0.9, 0.09, 0.009, …}, where the n+1th term is the nth term multiplied by 0.1. We can add all the terms of this sequence, in the form of a sum. Notice that this sequence is inherently infinite, hence we are summing an infinite number of terms. This summation is called a series.

An infinite series is equal to the limit as n approaches Infinity of the expression for the partial sums. The definition of a limit already implicitly and secretly includes infinitesimals, so this definition holds true regardless of the number system that you use. The n-th partial sum of the sequence, S(n) = 0.9 + 0.9*10^1 + … + 0.9*10^(n-1) + 0.9*10^n. Mathematicians have proved that sums of this form are equal to a(1-r^n/1-r), where is the initial term of the sequence and r is the common ratio that produces the next term. As n approaches infinity, this expression becomes a/(1-r) so long as |r|<1. For 0.999…, this condition is true since 0.1 < 1. Now, 0.9/(1-0.1) = 0.9/0.9 = 1. Therefore, 0.999… = 1.

This holds true in all number systems which obey the basic axioms of mathematics, including number systems with infinitesimals incorporated into them.

The argument that the elipsis cannot represent infinity is moot and invalid, one can choose to represent Infinity however one wants as there isn't a rule that states otherwise. I can choose to represent 1 as 1.00000…… or 0.99…… they're the same number in the end.

Multiplying an infinite series of nine’s by 10 simply move the decimal point one unit to the right. There is no need for a zero, as the nines have no end

The problem with 0.999….=1. goes deeper than just a matter of choice of number system. Treating 10×0.999… as equal to 9.999… by “moving the decimal” hides the crucial fact that “moving the decimal” is valid if, and only if, there are the same number of digits on both sides of the equals sign as in 10×0.444=4.44. If you have it like this 10×0.444=4.444 with the number of decimal place fours the same on both sides of the equals, like the people who do 10×0.999…=1 do , it’s the wrong answer obviously, and you run that number through the above formula you get 0.999=1 and 0.9999=1 or any number of decimal places you start the formula with you get 0.999… to that number of places =1. If we do it correctly then, because of the common misunderstanding, or dichotomy of treatment of infinity, you have to use a nomenclature that specifies that the number of decimal digits when infinite follows the same rules that any other number with a defined number of digits follows. Just as in the rule followed here 10×0.444=4.44. here the number of fours is plainly visible. But it’s not visible in 0.999… The question is what nomenclature should be used. if I did this 10×0.444=4.444-0.004 then we have both the right answer and a method to treat 10×0.999…That is 10×0.999…=9.999…-0.000…009. Use that equation in the above formula and you get 0.999…=0.999… not 0.999…=1.