0.625₁₀ → binary — number base conversion
coreterminating fraction
Answer
0.101
Why this example is worth doing
Fractions convert by repeated multiplication by two, taking the integer part of each product as the next bit — the mirror of the repeated-division method for integers, and a genuinely different procedure that many courses skip. This value terminates after three bits because its denominator is a power of two. The next example shows what happens when it is not, and the pair is the point.
Try your own input in the Binary ⇄ decimal ⇄ hex ⇄ octal. Convert between any two bases, integers and fractions, with the positional working.
How the answer is reached
Base 10 to base 2
Convert 0.625 from base 10 to base 2. The sign is carried across unchanged; only the magnitude is converted.
| digit | value | weight | weight (base 10) | digit x weight |
|---|---|---|---|---|
| 0 | 0 | 10^0 | 1 | 0 |
| 6 | 6 | 10^-1 | 0.1 | 0.6 |
| 2 | 2 | 10^-2 | 0.01 | 0.02 |
| 5 | 5 | 10^-3 | 0.001 | 0.005 |
| total | 0.625 |
Integer part: divide by 2 and read the remainders bottom to top.
| n | divide by | quotient | remainder |
|---|---|---|---|
| 0 | 0 ÷ 2 | 0 | 0 |
Fraction part: multiply by 2 and read the digits top to bottom.
| fraction | multiply by | product | digit |
|---|---|---|---|
| 0.625 | 0.625 × 2 | 1.25 | 1 |
| 0.25 | 0.25 × 2 | 0.5 | 0 |
| 0.5 | 0.5 × 2 | 1 | 1 |
0.625 (base 10)0.101 (base 2)