Why 10/9 = 1.111… ; 9×1.111… = 9.999… ; but 9×10/9 = 10. Why is 10 equal to 9.999…?
In: Mathematics
9.99999… = 9 + 0.9 + 0.09 + 0.009 + …
= 9 * (1 + 0.1 + 0.01 + 0.001 + …)
= 9 * ((0.1)^0 + (0.1)^1 + (0.1)^2 + (0.1)^3 + …)
= 9 * sum of ((0.1)^k) with k from 0 to infinity
= 9 * (1-0)/(1-0.1) ***
= 9 * 1/0.9
= 9/0.9
= 90/9
= 10
the part marked *** is from the known formula :
sum of (r^k) with k from 0 to n = (1 – r^(n+1)) / (1-r) (if r is not 1)
Particularly, if |r|<1 (which is the case with r=0.1), we know that r^n tends toward 0 when n tends to infinity. So this formula towards infinity becomes
sum of (r^k) with k from 0 to infinity = (1-0) / (1-r) if |r|<1
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