Point Nine Recurring Equals One

To end the constant stream of disbelief coming from almost everyone I know here is a complete collection of proofs and logically and mathematically correct arguments that the number 0.999... (naught point nine recurring) is equal to 1 (one). This is not a joke nor are there any logical contradictions or any fallacious proofs. It is all true. (Big thanks to this very nice page from Things of Interest)

Proof 1: The Quickest

If two numbers are different then one can find a third number which is between the two.

Therefore if 0.999... and 1 are two separate numbers a third number exists which is between the two. But there is NO number between them. It is impossible to find one. Therefore they aren't different but are the same.

Proof 2: Quick Mathematical Proof

If two numbers are equal then their difference is zero

The difference between 1 and 0.999... is:

1.000000000... - 0.999999999... = 0.000000000...

Therefore they are equal.

Common counter-argument "But the 0.000000... should have a 1 at the end."
No it shouldn't. There is no end to and infinite number of zeros so there is no end and therefore no place for the "final" 1.

Proof 3:

Consider:

Multiply both sides by 10:


Subtract c from both sides:




Dividing by nine gives:


Common counter-argument: "When you multiplied by 10 you should have added a 0 to the end making it 9.999999...0!"
The recurring decimal goes on forever with no end. Saying that a 0 should be put on the end of an endless decimal is meaningless. See previous common counter-argument.

Proof 4: Geometric Series

Consider this:


That is, 0.999... is the sum of an infinite number of fractions of 9 over powers of ten:


This is a geometric series with first term a = 9/10 and common ratio r = 1/10 . Therefore using the formula for the sum to infinity of a geometric series we can conclude that:

Maybe these different (and all logically and mathematically correct) proofs will convince those who didn't already know or accept this truth.

0.999... = 1 Q.E.D