Numeral Systems
Positional number systems and radix explained
About Numeral Systems
A numeral system (or number system) is a way of representing numbers. In a positional system, each digit's value depends on its position: the rightmost digit is multiplied by the base to the power 0, the next by the base to the power 1, and so on. The base (or radix) is how many distinct digits are used—from 0 up to base minus one.
For example, in decimal (base 10) the number 42 means 4×10 + 2×1. In binary (base 2), 42 is written as 101010, meaning 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 0×1. Zero is essential in positional systems as a placeholder so that digit positions keep their meaning.
Key Features
- Decimal (base 10) is the everyday system; binary (base 2) and hexadecimal (base 16) are standard in computing.
- Digits range from 0 to base−1; bases greater than 10 use letters (e.g. A–F in hex) for values 10–15.
- Place value: each position represents a power of the base (ones, base¹, base², …).
- Negative bases (e.g. negabinary, base −2) are possible and have theoretical uses.
Further Reading
Numeral Systems Table
Common positional numeral systems. The example column shows the number 42 in each system (see 42 on NumDic for all representations).
| Base | Name | Digits | Example (42) | Common uses |
|---|---|---|---|---|
| 2 | Binary | 0–1 | 101010 | Computing, digital logic |
| 3 | Base-3 | 0–2 | 1120 | Ternary logic |
| 4 | Base-4 | 0–3 | 222 | Quaternary |
| 5 | Base-5 | 0–4 | 132 | Quinary |
| 6 | Base-6 | 0–5 | 110 | Senary |
| 7 | Septimal | 0–6 | 60 | Base-7 |
| 8 | Octal | 0–7 | 52 | Computing (historical) |
| 10 | Decimal | 0–9 | 42 | Everyday use worldwide |
| 12 | Duodecimal | 0–9, A–B | 36 | Time, dozen counting |
| 16 | Hexadecimal | 0–9, A–F | 2a | Computing, programming |
| 20 | Vigesimal | 0–9, A–J | 22 | Maya, some natural languages |
| 24 | Base-24 | 0–9, A–N | 1i | Time (hours) |
| 36 | Base-36 | 0–9, A–Z | 16 | Compact encoding |
| −2 | Negabinary | 0–1 | 1111110 | Theoretical, negative base |