Question

2. Prove that if a and b are positive integers such that a is a factor of b, then any finite (terminating) mantissa of base a can be converted to a finite (terminating) mantissa of base b. Note: Express the number in summation form

Answer 1

Consider any finite mantissa of , let it be $.a_1a_2\ldots a_k$ where each . Let be a factor of , hence for some $c \geq 1$ .

The mantissa can be expressed as a summation
$\frac{a_1}{a} + \frac{a_2}{a^2} + \ldots + \frac{a_k}{a^k}$ . Substitute to get
$\frac{ca_1}{b} + \frac{c^2a_2}{b^2} + \ldots + \frac{c^ka_k}{b^k}$ . As this summation only uses powers of in the denominator upto a finite , the mantissa is represented in base finitely as well.

Comment in case of any doubts.