Why is 0.99999..=1?

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Answer 1 · 2016-10-20T23:15:24

One of the enigmas of Maths.....

Let $" "x =0.99999.....$

$:.10x =9.9999999...$

Subtract these two to get:

$9x= 9" "$ all the decimals subtract to 0

$x = 9/9$

$x =1$

$:. 0.999999.... = 1$

Answer 2 · 2016-10-20T23:18:05

See explanation.,,

Consider:

$(10-1)*0.99999... = 9.99999... - 0.99999... = 9$

Divide both ends by $(10-1)$ to find:

$0.99999... = 9/(10-1) = 9/9 = 1$

So the expressions $0.99999...$ and $1.00000...$ are both decimal representations of the number $1$ .

Answer 3 · 2016-10-20T23:19:12

Series explanation

$0.9999 . . . = 9/10+9/10^2+9/10^3 + 9/10^4 + * * *$

This is a geometric series with $a=9/10$ and $r = 1/10$ ,

so the sum is $a/(1-r) = (9/10)/(1-1/10) = (9/10)/(9/10) = 1$