How are these two sets equivalent

Question

Given S = [0,1] and another set as R = {$$sum_{1}^{infty} frac{a_i}{10^i} $$ } where $$a_1 in {0,1,2,...,9}$$. Also the set R - {$$sum_{1}^{infty} frac{a_i}{2^i} $$ } where $$a_1 in {0,1}$$. My professor said these sets are same. How ?

zkutch · Answer

Hint: these are representations of real numbers with radix 2 and radix 10, where radix is base of positional numeral system. For fractions are used negative degrees.
For example:
$$(0.678)_{10} = (0.1010110110010001011010000111001010110000 cdots)_2$$
$$(0.998)_{10} = (0.1111111101111100111011011001000101101000 cdots)_2$$

How are these two sets equivalent

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