# 1 = 0.999? How can you prove that 1 = 0.999999999999999999....?

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### 5 Answers

(1) You might also consider that 1/3+1/3+1/3=1 or .333333...+.33333...+.33333...=1.

Note that we **are** saying that .99999...=1 exactly, not approximately.

You will encounter other proofs using other ideas as you go through school: the sum of an infinite series, the limit of an infinite sequence, etc...

If you disagree that .99999999....=1, you have some company in the math world, though not much. Look up **infinitesimals** and you will find a fascinating history of a school of thought that differs from mainstream mathematics.

Try this:

0.9999... /= 1

x=0.9999...

10x = 9.9999...

10x-x=9.0...

10x = 9+x = 9.999...

From your line:

10x - x = 9.9999... - 0.9999...

the next line 9x=9 does not follow. It should be

10x = 9.999....

1 is .99 rounded. The more nines you get, the closer to one you will be. However, 1 will never exactly each 0.99999999, no matter how many nines you add. You will get closer and closer until the differnece is so minimal that you decide to call it 1.

It look like a infinite decimal

so

999999999999999.../1000000000000000...

as

S Infinity=a/1-r

Well... but can you also see it this way?

0.9999... = 1

x = 0.9999...

10x = 9.9999...

10x - x = 9.9999... - 0.9999... (becuase x = 0.9999...)

9x = 9

x=1