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

Expert Answers
embizze eNotes educator| Certified Educator

(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.

maadhav19 eNotes educator| Certified Educator

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....

 

 

litteacher8 eNotes educator| Certified Educator

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.

wanasyraf | Student

It look like a infinite decimal

so

999999999999999.../1000000000000000...

as

S Infinity=a/1-r

 

 

soccerfreak9 | Student

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