Knowledge base · From Zero to EngineerFrom Zero to Engineer
Converting a decimal number to another number system
Dividing by the base with remainders converts the integer part, and multiplying by the base converts the part after the point. Worked examples for the binary, octal and duodecimal systems.
The previous lessons went one way: from another system to decimal. Now we go back. Divide the integer part by the base until nothing is left, then read the remainders from the bottom up. Multiply the part after the point by the base and take the whole part of each result - those are the digits after the point. Both ladders are mechanical: nothing to guess and no place values to remember.
Conversion from the decimal system to another number system
Binary form
| remainder | ||
| 245 | : 2 | 1 |
| 122 | : 2 | 0 |
| 61 | : 2 | 1 |
| 30 | : 2 | 0 |
| 15 | : 2 | 1 |
| 7 | : 2 | 1 |
| 3 | : 2 | 1 |
| 1 | : 2 | 1 |
| 0 | ↑ |
Octal form
| remainder | ||
| 524 | : 8 | 4 |
| 65 | : 8 | 1 |
| 8 | : 8 | 0 |
| 1 | : 8 | 1 |
| 0 | ↑ |
Duodecimal form
| remainder | ||
| 897 | : 12 | 9 |
| 74 | : 12 | 2 |
| 6 | : 12 | 6 |
| 0 | ↑ |
Converting a fraction in the decimal system to octal and duodecimal form
| 0.526 | · 8 | |
| 4 | .208 | · 8 |
| 1 | .664 | · 8 |
| 5 | .312 | · 8 |
| 2 | .496 |
| 0.306 | · 12 | |
| 3 | .672 | · 12 |
| 8 | .064 | · 12 |
| 0 | .768 | · 12 |
| 9 | .216 |
Convert to the octal system
| remainder | ||
| 492 | : 8 | 4 |
| 61 | : 8 | 5 |
| 7 | : 8 | 7 |
| 0 | ↑ |
| 0.731 | · 8 | |
| 5 | .848 | · 8 |
| 6 | .784 | · 8 |
| 6 | .272 | · 8 |
| 2 | .176 |
Convert to the duodecimal system
| remainder | ||
| 384 | : 12 | 0 |
| 32 | : 12 | 8 |
| 2 | : 12 | 2 |
| 0 | ↑ |
| 0.426 | · 12 | |
| 5 | .112 | · 12 |
| 1 | .344 | · 12 |
| 4 | .128 | · 12 |
| 1 | .536 | · 12 |
| 6 | .432 |
The fifth digit decides how the fourth is rounded: 6 is half of twelve, so 0.51416 is written as 0.5142.
PhiBoard is a free online whiteboard for maths
This lesson is part of it. On the board itself you can draw a number line, a graph or a figure, write a formula and solve a problem together with a teacher or a friend - live, in the browser. Our app is completely free.
Check whether you really know it
Theory is half the way. Work through a few exercises on the same material - each with a step-by-step solution, no account and no fees.
Frequently asked questions
How do you convert a decimal number to binary?
Divide it by 2 until the quotient is zero, writing down the remainders. Read from the bottom up they give the answer: dividing 245 by 2 repeatedly gives remainders 1, 0, 1, 0, 1, 1, 1, 1, that is 11110101 in binary.
Why are the remainders read from the bottom up?
Because the first remainder belongs to the smallest place, the units, and each next one to a place higher up. The last remainder is therefore the leftmost digit of the answer.
How do you convert the part after the point?
Multiply it by the base of the system. The whole part of the result is the first digit after the point, and the fractional part goes into the next multiplication. For 0.526 and base 8 the successive products are 4.208, 1.664, 5.312 and 2.496, that is 0.4152 in octal.
When do you stop multiplying the fractional part?
When the fractional part becomes zero, or when you have as many digits as you need. The expansion is often non-terminating, so you usually compute one extra digit and round the last one.