# Comprehending 1=0.99(9)

On first look it is weird when they tell you that 1 equals 0.999(9) where 9 is repeating.
Or, which is the same thing – that 3 * 0.333(3) , equals both 0.999(9) and 1.

I know I searched once on web for the explanation of why is it so – an explanation I could easily grasp/comprehend,  as intuitively it seems there is something wrong – our intuition tells us – “As much nines are repeating there, they won’t add-up to 1”. All I found is proofs that it is so, but from the philosophical point (well from common-sense point even) we can’t be happy (me at least) just with symbolic proofs, we need to comprehend the equality. So here goes the easy explanation…

When we have integer, let’s say four, we can say that we have a quantum of four things, a quantum which can be divided to four ones. Say you have four apples. You can divide them to four people – each person will get one apple.
Decimal numbers can be comprehended in similar way, just that if we add decimal fractions, we need to imagine that even the ones are further dividable – that they are made from parts. We need to imagine each one as made from ten parts, and each of those ten parts as made from even smaller ten parts, and so on…

In such way 1 would mean the item, 0.1 would mean one of the ten parts of the item (1), 0.01 would mean one of the ten part of the 0.1 part, and so on…

Now, let’s try to divide one item to three people. Because the item consist of ten parts, we will give three parts to each person, and we will be left with one of those ten parts that we need to divide:

1/3= 0.3 + 0.1/3

This 0.1 is again consisting of ten parts and we will likewise divide it between those three people – three parts to each, but we will be left again with one (now ten times smaller) part to divide further…

1/3= 0.3 + 0.03 + 0.01/3 , and dividing further…
1/3 = 0.3 + 0.03 + 0.003 + 0.001/3 , and so on..

Or if we sum the parts which are already divided we get

1/3 = 0.3 + 0.1/3
1/3 = 0.33 + 0.01/3
1/3 = 0.333 + 0.001/3 etc…

We see that this adding of cipher 3 to infinity, is not consequence of the part which is already divided, but of the part (left-over) which is left to be divided. As many threes are generated there will be still a part to be divided which would generate new threes. We can now say the following… the infinity in the never-ending repeating cipher 3, is not about the infinity of how much ciphers 3 are generated, but about the impossibility to finish the dividing process.

So, it is the wrong comprehension of what 0.333(3) means, that is causing the intuition against 1 being equal to 0.333(3) * 3. If we imagine 0.33(3) vaguely as some never ending string of cipher 3 – a decimal number in decimal notation that just has infinity of ciphers, our intuition tells you that however far you multiply with 3, you will only get 0.999(9), and however far you go, it won’t be 1.
But comprehending 0.33(3) in that way, is ignoring the reason of that infinity, for example in the given process – ignoring the left over which is still to be divided that causes us to add new and new ciphers 3, and which will never end. So, we need to see repeating cipher as a new notation added to the decimal system, which tell us about that impossibility to finish the description in the normal decimal notation, and not a shortcut for normal decimal notation, in which we could write the cipher 3 infinite number of times.

By the way we can divide 1 in similar way to get 0.99(9) directly…
We divide 10 of the parts (0.1) to 10 people, so that we divide 9 or those parts to 10 people, and leave one more to divide; and that we repeat that:
1/10=0.9/10+0.1/10=0.9/10 + 0.09/10 + 0.01/10, etc…
We will get 1 = 0.99(9), again this repeating 9 signifying the impossibility to finish the process.

* It isn’t really necessary for each part consist of 10 parts. What we need is just that each part can be divided to 10 parts, and if it can, we can treat it as consisting of 10 parts. For example if we don’t have quantum, but a continuous magnitude, it can be divided in any number of parts we want, hence to 10 also.

## 21 thoughts on “Comprehending 1=0.99(9)”

1. This is a good way to look at it. We tend to assume 0.9999… and 1.000… differ as 9 and 10 (or 99 and 100, or 999 and 1000) do; when actually a better way to think of it is that 0.999…, because it is infinitely repeating, can’t differ from 1 by any finite amount — any finite number + 0.9999…. will be greater than 1. And if number A and number B don’t differ by any finite amount, they’re equal.

This particular line of reasoning only works, though, if we don’t have infinitesimals (numbers greater than zero but smaller than any finite number). My understanding is that in nonstandard analysis these can arise, even though they don’t in standard calculus; but I don’t know enough about nonstandard analysis to say much about them, or how they relate to this problem. And, in any case, I doubt that any of us have difficulty with 0.9999…=1 because we have developed a consistent system of infinitesimal numbers.

2. Hi Brandon, thanks for the comment,
What you say helps to comprehend why the opposite can’t be the case, i.e. why 0.99(9) can’t differ from 1. But without comprehending why 1 = 0.99(9) positively, common sense is left with a situation which seems like a paradox, as those negative proofs say – “1 and 0.99(9) can’t be different”, but the intuition says “but they are”.
I fancy that the post gives a positive account of the identity – to remove that “but they are” intuitive part.

3. I think you’re right that it probably does make up for a lack in the ‘negative’ approach to the question.

4. I think the real issue at hand when people have trouble with infinite sums is the very cavalier attitude that most people take towards the concepts of “number” and “infinity.” Just because something is true in the finite case does not mean that it holds as well in the infinite case, or in other words, infinity means much, much more than “a very, very large number.” For instance:

(1) If we are summing a finite series, we can permute the numbers in the series any way we wish and get the same result. In general, this is not true in the infinite case.

(2) The statistical concept of the expected value of a random event is based on an infinite sum. Statisticians (and gamblers) who are too quick to assume that this matches up with the finite case will lose a lot of money playing the St. Petersburg game. That game works something like this: “You’re gonna flip a coin. If it’s tails, you lose. If not, keep on flipping. If it’s tails, you get \$1; if not, keep on flipping. If it’s tails, you get \$2; if not, keep on flipping. If it’s tails, you get \$4, and so on.” The expectated value of this game to the gambler is infinite, so it would seem a good deal to pay \$5 a game to play. Try this, however, and you’ll rapidly lose all your money.

The fact is, in standard analysis the real numbers are defined as either the Cauchy sequences or Dedekind cuts of the equivalence classes of pairs of integers under the cross-product relation, endowed with a cross-product-based ordering. In other words, there’s really nothing intuitive about them. They are a mathematical construct designed to have certain properties, and beyond these properties they behave in manners which one might find bizarre.

To put it in more philosophical terms, a car and a horse can be made to serve the same purpose, but a man whose intuition tells him that he can buy a pair of cars and let nature take its course is, unfortunately, mistaken.

5. Yeah…actually, .999 repeating != 1.

Let’s assume for a moment that .999… is = 1.

Therefore, 0 = 1/infinity (if you don’t accept this premise, just think about it, 1/infinity = .0000000001… just as 1 = .999…)

Consequently: 0 = 1/infinity = 2(1/infinity) = 3(1/infinity) = infinity(1/infinity) = 1.

So if .999.. = 1, then 0 = 1.

——
Have a good philosophical article? Post it on the the philosophy wiki (http://sophiasdialectic.com).

6. Hi Ben,

I’m not sure that you are addressing anything that was said in the post, so let me point to it:
When you divide 1 on 3 parts in decimal system, you will get new decimals 3 over and over, and you will *never* be able to finish the process. So, we have those two things a)the decimal 3 will be repeating in every iteration and b)the process can’t be ended.

So, this repeating doesn’t signify some aggregation of “lots of lots” of ciphers 3, but the ones that are written, and the “power” which is generating the new ones.

As for your proof (1/infinity=2/infinity=…infinity/infinity), you make the mistake of thinking of infinity as an aggregate. It assumes that you can get to infinity by starting from 1, and then adding +1, +1, +1.
But it is impossible to get to infinity by adding +1. Let me put it like this: infinity-1=infinity. Namely, there is no number from which you can get to infinity, by adding 1.

7. Simple math says:

Okay guys, yes i agree that .999 = 1
But with the use of simple math this can be proved wrong just as easily

Infinity x 1 = that same number

But…

Infinity x .999 = A different result.
For example
4 x .999= 3.996

Hmmm… so if your telling me .999=1 then because of rounding any number is actually equal to any other number!? Whhooaaa i just blew my own mind.

8. Nobody said that .999=1
The post and the comments are about .9999… where 9 is repeating infinite number of times.
I wonder if I should close comments on this post. :-/

9. David says:

Also, infinity * 0.999 = infinity * 1. go to the wikipedia article on infinity, it will state the properties there.

10. Kyle Joecken says:

@Ben,

Your proof has two issues with it. First, you say that 1/infinity=.00000000001… . This is not true; in fact, 1/infinity=0. The issue people have with this fact is the same as the issue people take with the fact that 1=.999… — that of a lack of understanding of the infinite process. Infinite processes are very touchy things; something that holds true at every single finite step need not hold at the “infinite” step — whatever that means.

The second problem with your proof is the line

0 = 1/infinity = 2(1/infinity) = 3(1/infinity) = infinity(1/infinity) = 1.

The last two equalities are not true. In fact, infinity/infinity is what’s called an “indeterminate form,” meaning that it is not feasible to assign it any value whatsoever. Consider the following problem. Choose any number x, and multiply it by three. Divide the result by your original number x. You will get an answer of 3. In other words, we have that 3x/x=3, a fact that you will no doubt concede. Thus, you might think it natural to assign “3*infinity/infinity” the value 3, and you’d be correct. But then 3*infinity=infinity, so what you are really saying is that infinity/infinity=3. Of course, there is nothing special about 3 at all; you could repeat the process with any finite number in place of 3, including 1 or even 0. So in this sense it is “natural” to assign any value you want to infinity/infinity. But this doesn’t make any sense — any symbol we write down and want to do computation with must have a precise meaning. It can’t mean a bunch of different things at the same time.

Like I said, infinite processes are tricky things.

In fact, we have that n(1/infinity)=0 for any finite value of n, but infinity/infinity is indeterminate (has no meaning or intrinsic value; is left undetermined). Other indeterminate forms include 0*infinity, 0/0, 0^0, infinity^0, and 0^infinity (where “^” means “to the power of”).

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The real issue with the decimal expansion system is that the representation of a number is not unique. A ‘number’ does not change based on its representation. The concept of ‘5’ is something that we all understand; we all know what it means to have ‘5’ of something. The fact the the Romans called it ‘V’ and the Japanese call it ‘rocking chair with a roof’ (or ‘go’) have nothing to do with what it is.

Our decimal system is an extension of the usual base-10 representation of natural numbers (positive integers) that is meant to represent values between the integers. This is done in the most natural, innocent of ways. We know the number 365 does not mean 3+6+5 (as an alien might think upon learning the digits), but 3*100+6*10+5*1. It is a shortcut, so that we do not have to come up with an infinite set of symbols; instead, we write arbirarily long “words” of a finite set of symbols (namely, 0123456789). What we gain by doing this is that we can represent arbitrarily large integers without having to create a new symbol each time.

In the same vein, then, we attempt to go the other way by letting numbers of the form 36.576 mean 3*10+6*1+5*(1/10)+7*(1/100)+6*(1/1000). You may be suprised to notice that using numbers of finite length is not sufficient to represent all of the numbers we want to. Obviously numbers such as sqrt(2) (the square root of 2) and pi have issues (the latter is actually not simple to prove), but even such “normal” (rational) numbers such as 1/3 an 1/7 cannot be represented finitely by such sums. It is not because these values don’t exist; rather it is the fault of our choice of a system of representation. Choosing another base (meaning choosing another number to serve the role of 10) does not help; while 1/3=.3=3*(1/9) in base-9, we now have the new issue that 1/2 cannot be respresented by a finite “word.”

The only thing we can do is allow these “words” to be infinite. Fortunately, we can make a precise definition of what these infinite words are; the infinite sums that we get “converge” in the nicest way possible (what we call “absolute convergence”). Even more fortunately, we now have that every real value can be assigned a number (a decimal representation). This includes 1/3, sqrt(2) and pi (both rational and “irrational” numbers). Unfortunately, we get the “problem” that is the root of this discussion, namely that we can assign multiple “numbers” to the same value. 1 (or 1.0000000…) is not the only exception either. In fact, ANY number with a finite decimal representation (except 0) can be represented with an infinite representation that ends in an infinite string of 9’s. This is because

1/(10^n)=9/(10^n+1)+9/(10^n+2)+9/(10^n+3)+… for every positive integer n.

Now, as it turns out, the values that have this exact problem are the only values that have multiple representations, and moreover these two are the only possibilities. Thus, the natural thing to do is to choose the finite representation in each case, as the whole point of this system is for book-keeping purposes, anyway. We do NOT choose the finite representation because the infinite one doesn’t work; you can in fact use strings that end in infinite 9’s in any of the usual (+-*/) calculations, and as long as you are careful you will still arrive at the correct answer. But since we’re only book-keeping, why bother?

The next natural question to ask is, “why don’t we choose a system where each number has one and only one representation?” As far as I know, a natural such system does not exist. In fact, it would not surprise me if it could be proven that so such system existed. Remember, we aren’t just writing numbers down, we want to calculate with them as well. Finding a system of representation in which it is easy to perform arithmetic calculations is not easy.

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Yes, I know this post is way too long. But hey, if can sit around for hours every day talking about math, you guys can read a long comment, can’tcha?

11. Floris van Dorp says:

And I was just thinking and playing with my calculator. Dividing through 9. When you do that, you will see (which is quite obvious) that the number you will divide through 9 will repeat itself, endlessly (and this of course is the issue, as I saw in a glimpse). 7 divided by 9 = 0.777777777
16 divided by 9 = 1.77777777 (1.6 + 0.16 + 0.016 + etc.). Until you use a number which you can divide by 9. Lets take 45 and divide it by 9, we can say that’s 4.99999999 (4.5 + 0.45 + 0.45 + etc.) or just say it’s 5.
Is it a flaw in the calculator’s ‘mind’, I don’t know, but what I do know is that there is more than meets the eye, espacially in the world of numbers. I will end this message with the beautiful number 6174.

12. Guenther says:

Actually, 0.9 periodic = 1 only when you assume the Dedekind Axioms (or equivalent) for real numbers.

Other axioms will lead to other results, viz. 0.9 periodic != 1 (you just have to introduce infinitesimals)

For a good and accessible introduction as well as a cognitive science view on this subject see:

Lakoff and Nunez: Where Mathematics comes from: How the embodied mind brings mathematics into being. 2001

13. Closing the comments on this post, after I just deleted dozen of silly ones made yesterday. :)
Sorry for that you posters of silly comments. I would leave them (even personal attacks), but they fill up the Recent Comments widget on sidebar, so people can’t see that there was serious discussion at the other posts recently.

EDIT:11 May 2008. Reopening and hoping for the best :)

14. Math Rocks says:

I am very convinced that 0.999 = 1, because of how many different kinds of math prove it. Algebra proves it, Calculus proves it, even simple 6th grade math proves it. http://en.wikipedia.org/wiki/0.999
But I still would like to prove it with my own equations, once I learn the required information.

15. Then 0 = 1.
Thanks, I will try to apply at myself.

16. I am very convinced that 0.999 = 1.
Very interesting note!

17. Kid at heart says:

im just lost in my mind
all of you seem to say different things
all i see is 1/3=0.333…
*3=0.999…
and whatever you do 0.999… won’t give the exact value as one
i think there is something wrong with our system of numbers or just multiply and dividing.

1. Simon says:

Well, this is really tough one :-)
_ _ _
but IMO: 1/3 = 0.3 ; (1/3) * 3 = 1 so I think it’s onlz proof that 0.3 * 3 = 1 not 0.9 :-)
also n/∞ != 0 it’s only limitly closing to 0
_
for that matter I still cannot accept that 0.9 = 1
It’s just kinda close :D

So that is how I see it, but I’m not an expert.

18. my mind will explode!!!!!!! .999…= 1 PERIOD.

19. you ve really forgotten the notion of limit;it ‘s something that a 17 year child could think. There exists a quantity R(n) (here, R(n)1. As we let n go to infinity, R(n) becomes smaller and smaller, something that we express mathematically “R(n) goes to 0”. That ‘s all. lim(1/n) is 0, as n goes to infinity; 1/infinity has no mathemaitcal meaning because infinity is not a number you can divide. Only some “clever” physicians use these things; but, to be fair, they think that when we say infinity we know that it s about limits and only. I think you misunderstood.

20. Hello, interesting article, but very small. Where can I learn more on this subject.