0.124 0.124five \: base \: two ​

Question

0.124

`0.124five \: base \: two`
​

Answer 1

Let x be the number equal to 0.124₅, so that

x = 1•5⁻¹ + 2•5⁻² + 4•5⁻³

Multiply x by 5³ to convert it to a whole number:

5³x = 1•5² + 2•5¹ + 4•5⁰

Convert everything to base 2:

5³ = 125 = 64 + 32 + 16 + 8 + 4 + 1 = 1111101₂

1•5² + 2•5¹ + 4•5⁰ = 25 + 10 + 4 = 39 = 32 + 4 + 1 = 100111₂

Then

1111101₂ x = 100111₂

so that

x = 100111₂ / 1111101₂

I've carried out the first several steps in the division in the attached table, with more bits added to the result. The number in base 2 does not terminate. You end up with approximately

0.010011111101111100111...₂