Let <em>x</em> be the number equal to 0.124₅, so that
<em>x</em> = 1•5⁻¹ + 2•5⁻² + 4•5⁻³
Multiply <em>x</em> by 5³ to convert it to a whole number:
5³<em>x</em> = 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 <em>x</em> = 100111₂
so that
<em>x</em> = 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
