(This page is entirely factually accurate. It is neither a joke nor a satire nor a collection of fallacious proofs. All these proofs are genuine and the results are true. Thanks.)
Preliminary note
ALL numbers are infinite decimal expansions.
For example, whenever you write "1" you are using this as a convenient shorthand for "1.0000..." in the same way that "1.0000..." is a convenient shorthand for a 1, a decimal point, and infinitely many zeros.
Similarly, "13556" is short for "13556.0000...", "1/3" is short for "0.3333...", and "pi" is short for "3.14159265...". Shorter forms are merely useful notation because it's tiresome/impossible to write out infinitely many decimal digits whenever you want to write a number.
Believe it or not, this has always been true since the moment you started doing mathematics. This is also part of the fundamental bedrock of mathematics and not something which you can argue with or debate about.
NOW READ ON:
Quickest proof
For any two different real numbers, you can pick a third number which is between them.
So, if 0.9999... and 1.0000... were different numbers, then it would be possible to find a number which was between them.
But it's impossible to write out the decimal expansion of a number between 0.9999... and 1.0000... .
Therefore, they cannot be different numbers.
Therefore, they are the same number.
"That argument doesn't work in the integers. There aren't any integers between 3 and 4, but they aren't equal."
So? Different sets of numbers have different properties. We're looking at real numbers, not integers. There are plenty of real numbers between 3 and 4.
Quickest mathematical proof
If the difference between two numbers is zero, then they are equal. For example, 5 - 5 = 0 because 5 = 5.
The difference between 1.0000... and 0.9999... is:
1.0000... - 0.9999... = 0.0000...
= 0
Therefore, they are equal.
"At that first step, you're already assuming 1 = 0.9999...!"
No I'm not, I'm just doing a simple subtraction. Work it out yourself if you like.
"But 0.0000... should have a 1 at the end!"
No, it shouldn't. "0.0000...1" is meaningless. The "..." means the zeros go on forever. "Forever" means "without end". There IS no end for the final 1 to go on.
A third proof
Let
x = 0.9999...
Multiply both sides by ten:
10x = 9.9999...
Subtract x from both sides:
10x - x = 9.9999... - 0.9999...
9x = 9.0000...
Divide by nine:
x = 1.0000...
"But on line two, 9.9999... should be 9.9999...0, because you multiplied it by ten!"
"9.9999...0" is meaningless. The "..." means the nines go on forever. "Forever" means "without end". There IS no end for the final 0 to go on. See the Preliminary note.
Add and divide and conquer
Verify this yourself using long division:
1 + 0.9999... 1.9999...
--------------- = -----------
2 2
= 0.9999...
Multiply both sides by 2:
1 + 0.9999... = 2 * 0.9999...
= 0.9999... + 0.9999...
Now subtract 0.9999... from both sides:
1 = 0.9999...
Geometric series argument
0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...
n=∞
= Σ 0.9 * 0.1n
n=0
n=∞
= Σ a * rn where a=0.9 and r=0.1
n=0
= a / (1-r)
= 0.9 / (1-0.1)
= 0.9 / 0.9
= 1
This uses the formula S = a / (1-r) for the sum of a geometric series with initial term a and ratio r, proof of which is left to the reader.
The real proof
Most of the reason why people don't understand why point nine recurring is equal to one is because they don't fully understand what a decimal representation actually means. Take a look at the definition of 0.9999... and things become abundantly clear:
0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...
= 9·0.1 + 9·0.01 + 9·0.001 + 9·0.0001 + ...
= 9·10-1 + 9·10-2 + 9·10-3 + 9·10-4 + ...
n=∞
= Σ 9·10-n
n=1
n=N
:= lim Σ 9·10-n
N→∞ n=1
= lim ( 9·10-1 + 9·10-2 + ... + 9·10-N )
N→∞
= lim ( 9·0.1 + 9·0.01 + ... + 9·0.000...0001 )
N→∞ \________/
N digits
= lim ( 0.9 + 0.09 + ... + 0.000...0009 )
N→∞ \________/
N digits
= lim ( 0.999...999 )
N→∞ \_______/
N nines
= lim ( 1 - 0.000...0001 )
N→∞ \________/
N digits
= lim ( 1 - 10-N )
N→∞
= lim 1 - lim 10-N
N→∞ N→∞
= 1 - 0
= 1
"I didn't understand that proof."
Try an earlier one.
A lengthy, rigorous proof by contradiction
If your lines of reasoning are correct, but the conclusion you arrive at is definitely wrong, there must be something wrong with your assumptions.
Clearly
0.9999... ≤ 1.
Assume
0.9999... ≠ 1 (*).
Then
0.9999... < 1,
so there must be some positive number P so that
0.9999... + P = 1.
But for ANY positive P,
0.9999... + P > 1,
which is a contradiction, and definitely wrong. Therefore we are forced to conclude that the assumption (*) was incorrect, that is:
0.9999... = 1
By popular demand
1/3 = 0.3333...
1 = 3/3
= 3 * 1/3
= 3 * 0.3333...
= 0.9999...
Proof that 1/3 = 0.3333... is left to the reader.
You might find this convincing
Use long division to find 9 / 9, but, instead of 90/9 = 10, put 81 remainder 9 each time, and see what happens.
0.9999...
-----------
9 | 9.0000...
8.1
---
90
81
--
90
81
--
9...
Limit argument
A sequence can only have one limit.
Observe that the limit of the sequence
0.9 0.99 0.999 0.9999 0.99999 ...
is
0.9999...
That is, the sequence gets closer and closer to 0.9999..., in fact, infinitely close.
But the sequence also gets closer and closer to 1.0000..., in fact, infinitely close. So 1.0000... is a limit of this sequence too.
But a sequence can only have one limit, so 0.9999... and 1.0000... must be the same.
Lastly, consider this
There are NO proofs that 0.9999... and 1 are different numbers.
Anywhere.
Common counter-arguments and my responses
"0.9999... and 1 are obviously different numbers."
Not good enough. Intuition counts for nothing. In mathematics, proof is everything, and "obvious" means "a proof springs immediately to mind". Please PROVE that 0.9999... and 1 are not equal. Without proof, no hunch, feeling, or intuition is worth anything.
"1 and 0.9999... are written differently, therefore they are different numbers."
There are many ways of writing ANY number. You could write 1/1, or 2/2, or 9/9, or 2-1, or 1.0, or 1.00, or 1.0000... or any number of other expressions, and all of them ultimately have the same meaning, "one".
"0.9999... is a concept, not a number."
All numbers are concepts. Some numbers, like 1, have stronger links to reality than others, but we are looking at mathematics here, not the real world. If you're going to throw away numbers which can't concretely exist, then you're throwing away pi, e, i, zero, and, frankly, almost all of mathematics.
"There is a rounding error. 0.9999... and 1 are approximately equal."
Do you see any rounding or approximation going on around here? That only happens when you stop counting after a certain number of decimal digits. But I have kept and counted every single one of the infinitely many decimal digits in my proofs. No rounding, no error.
"0.9999... gets closer and closer to 1, but never reaches it."
Closer and closer? How can it be getting closer and closer? It's one number! Try the limit argument, above.
"0.9999... is a decimal representation of infinity, not a number."
Well, how come it's DEFINITELY bigger than 0.5 and smaller than 2? Just because something has infinitely many pieces doesn't mean it's infinite. Zeno figured this out 2500 years ago.
"Humans can't comprehend infinity, and not being able to comprehend infinity means you can't do mathematics with it."
This statement is wrong on many levels. When you say humans cannot comprehend infinity, you are quite likely projecting your own inability to comprehend infinity onto everybody else in the world, among them many thousands of mathematicians who are perfectly capable of sitting down and dealing with infinite values in a sane and rational fashion. It may impossible to literally conceive of infinite values - whatever formal definition such conception could possibly have - but that does not and never will stop mathematicians from dealing with them. Mathematics is all about rules. Mathematicians have discovered that there are rules for dealing with infinity. These rules are perfectly consistent with our rules for dealing with other numbers, and given a little time you could probably learn to apply them yourself.
In case the connection isn't clear, what is true of infinite values is equally true of infinite decimal expansions. There are rules and procedures and they work and give meaningful results. See "The Real Proof" above for a relatively tame glimpse of this, which is actually a vast region of mathematics known as "analysis", naturally based on rock-solid fundamental axioms.
"My mate/my dad/my mathematics teacher/Professor Stephen Hawking told me that 0.9999... and 1 were different numbers."
They were wrong. In science, credentials are as worthless as intuition (above). Proof is everything.
"But they proved it, too!"
The proof was fallacious. Send it to me and I'll show you why.
"I still don't believe it, and I'm entitled to my own opinion."
Mathematics is unlike regular science in that we can actually prove things, permanently, for real, instead of just finding increasing amounts of evidence supporting our hypotheses. That's why we have what we call "theorems" instead of theories. That point nine recurring equals one is just such a theorem (although it's so easy that it's barely worth the name). You aren't in a position to argue or debate about it. It's a fact. Your opinion is wrong.
You are entitled to be wrong, I suppose. However, if you are unable to admit that you are wrong when you have been proven wrong, then you have no business taking part in a mathematical discussion or pretending that you know anything about mathematics. Therefore, I must request that you please distance yourself from any future discussions that you may encounter on this topic, so that those who do want to learn, rather than just promote an ideology, can do so.