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

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

(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's profile pic

maadhav19 | College Teacher | (Level 2) Assistant Educator

Posted on

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's profile pic

litteacher8 | High School Teacher | (Level 3) Distinguished Educator

Posted on

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's profile pic

wanasyraf | Student, Grade 9 | (Level 3) eNoter

Posted on

It look like a infinite decimal

so

999999999999999.../1000000000000000...

as

S Infinity=a/1-r

 

 

soccerfreak9's profile pic

soccerfreak9 | Student, Grade 9 | (Level 2) eNoter

Posted on

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

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