Number Systems
Place value, bases, and how to convert between them.
A number system is a way of writing quantities using a fixed set of symbols. In a positional system the position of a digit decides its value, so the same symbol means different amounts depending on where it sits. Base ten is the familiar case, but nothing about mathematics requires ten.
Place value
In base ten, 4,207 means 4×103 + 2×102 + 0×101 + 7×100. The zero is doing real work: it holds a column open. Systems without a zero placeholder, such as Roman numerals, cannot use position this way and need far more symbols.
Common bases
| Base | Name | Digits | Used for |
|---|---|---|---|
| 2 | Binary | 0 1 | Digital logic and computing |
| 8 | Octal | 0-7 | Compact binary, file permissions |
| 10 | Decimal | 0-9 | Everyday arithmetic |
| 16 | Hexadecimal | 0-9 A-F | Memory addresses, colour codes |
| 60 | Sexagesimal | - | Time and angle measure |
Converting to base ten
Multiply each digit by its place value and add. Binary 1011 is 1×8 + 0×4 + 1×2 + 1×1 = 11.
Converting from base ten
Divide repeatedly by the target base and collect remainders bottom to top. For 11 into binary: 11÷2 = 5 r 1, 5÷2 = 2 r 1, 2÷2 = 1 r 0, 1÷2 = 0 r 1. Reading upward gives 1011.