Why do people say that 1 and 0.999... are equal? Aren't they two different numbers?

Greg Bernhardt

Why do people say 1 and 0.999... are equal? Aren't they two different numbers?

No, they really are the same number, though this is often very counterintuitive to many beginning students. Here are some non-rigourous proofs that 1=0.999...:

Proof #1

For any two unequal numbers, there is always another number in between them. (That is intuitively obvious, and can be pictured on a number line, which will be familiar to many people.) Therefore, if 0.999.... and 1 are different, there must be another number in between them. But there is no way to write a number that is greater that 0.999... and less than 1 in decimal notation.

Proof #2

First, we put
[tex]\ x = 0.999...[/tex]
Multiplying by 10 gives us
[tex]\ 10x = 9.999...[/tex]
But then
[tex]\ 10x-x = 9.999...-0.999...,[/tex]
thus
[tex]\ 9x = 9[/tex]
Hence we get that
[tex]\ x = 1[/tex]

Proof #3

If you accept that 0.9999… is a number, then how much less than 1 is it? It's larger that 0.9999, so it's less than 0.0001 less than 1. But it's also larger that 0.9999999, so it's less than 0.0000001 less than 1. So the difference between 0.9999… and 1 is less than 0.00001, 0.000000001, or 0.any number of 0s followed by 1 … so the difference must be zero. If you accept that two numbers whose difference is 0 must be the same, then that proves that 0.9999… = 1. If you don't accept that, and you think that two different numbers can have a difference of zero, then you're in an 'extended number system' which has more numbers than we normally use.

Proof #4

First, we have that
[tex]\ 1/3 = 0.333...[/tex]
If we multiply things by 3, then we get
[tex]\ 1=3\times (1/3) = 3\times (0.333...) = 0.999...[/tex]

All of these proofs are correct, but they are not rigourous. For example, how do we know that [itex]3 \times (0.333...) = 0.999...[/itex]? This is not that obvious if we think about it. A more rigourous proof is given in the post following this one.

Some further questions you might have:

But 1 cannot equal 0.999..., as every number can only have one representation!
Well, the thing is that this is just a misconception that is simply not true. Numbers can have many representations. For example,
[tex]\frac{1}{3}=\frac{2}{6}=\frac{3}{9}=0.333... ,[/tex]
but somehow, many people don't have any problems with this thing. The same thing happens to 1=0.999... really, it's just another way to write the same number. Does this make our number system ugly? I understand that you might think that, but that's just something we need to accept. Not having that 1=0.999... would make our number system much uglier!

The way I see it, is that 0.999... gets closer and closer to 1, but never quite reaches 1.
This reasoning appears a lot and apparently, many people see 0.999... as some kind of process that gets close to 1. But this is not quite what mathematicians mean with 0.999...
Mathematicians say that 0.999... is a number, just like 2 and 3. So phrases like "it gets close to 1, but never reaches 1" are meaningless. It's the same as saying 1 gets closer and closer to 2, but never quite reaches 2. This sentence makes no sense, and the same thing happens with 0.999...

Can we define number systems such that 1=0.999... does not hold?
Of course! But these number systems are not as useful, because they don't conform to our intuition about numbers and limits.

In Proof #2, you say 10x=9.999... But this 9.999... has one fewer nine than 0.999...
Another popular argument. This time, the confusion arises from not grasping infinity. 0.999... has an infinite number of nines. If we somehow remove a nine from this sequence, then we would still have an infinite number of nines. So there are an equal number of nines in 0.999... and 9.999...

The same thing happens here: consider two sets of numbers, A and B, where
[tex]A=\{0,1,2,3,4,...\}[/tex]
and
[tex]B=\{1,2,3,4,...\}[/tex]
Both sets are infinite. And actually, both sets have an equal number of elements. But A doesn't contain 0, so it has one fewer element than B? Yes, but this reasoning only applies to finite sets. For infinite sets, it's quite possible to have one element less and still have an equal number of elements. Indeed, consider the following correspondence:

[tex]0\leftrightarrow 1,~1\leftrightarrow 2,~2\leftrightarrow 3,~3\leftrightarrow 4,...,~n\leftrightarrow n+1,...[/tex]
So for an element n in A, there exists a unique element in B that corresponds to n, namely n+1. This means, by definition actually, that both sets have the same number of elements.

Maybe we should just abandon our base 10 number system and move to a number system where every number does have a unique representation.
Tempting, but sadly this is not possible. The problem arises in every base! For example, in base 2, we have [itex]1=0.111...[/itex]. There is no way around it.


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