Infinite binary having infinite decimal \frac{1}{2} = 0.5 = (0.1)_2

luminarc24lry

luminarc24lry

Answered question

2022-04-22

Infinite binary having infinite decimal
12=0.5=(0.1)2

Answer & Explanation

eslasadanv3

eslasadanv3

Beginner2022-04-23Added 20 answers

Step 1
We have
12=0.5=(0.1)2
13=0.333=(0.010101)2
15=0.2=(0.001100110011)2
So it can happen that
both expansions are finite
both are infinite
the decimal is finite and the binary is infinite.
What cannot happen is that the decimal is infinite and the binary is finite. May be that's what you are trying to prove.
In other words we have the implication:
Decimal infinite decimal finite.
To prove this observe that if x=(0.b1b2bn)2 then
x=m2n=m,5n10n
which has a finite decimal expansion.

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