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#