Mystery of Infinity-1

Calculus Level 3

We all know the famous identity:

$0.9999\ldots{ }\ldots{ }\ldots = 1$ .

Here the numbers on the both sides are in base $10$ .

Is the identity true in some other base $>10$ ?

No Yes

1 solution

Muhammad Rasel Parvej
Dec 2, 2016

In base $b\geq2$ , the left side can be written as

$9 \times \frac{1}{b} + 9 \times \frac{1}{b^2} + 9 \times \frac{1}{b^3} + \ldots{ } \ldots{ } \ldots{ }$

= $9 ( \frac{1}{b} + \frac{1}{b^2} + \frac{1}{b^3} + \ldots{ } \ldots{ } \ldots{ } )$

= $9 \times \frac{\frac{1}{b}}{1-\frac{1}{b}}$ (as $b>1$ )

= $\frac{9}{b-1}$ , which is equal to $1$ only if b=10.

Hence, No is the answer.

Mystery of Infinity-1

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