Radix (base)

Also called: number base, base of a number system

The number of distinct digits a positional number system uses, and the factor between neighboring place weights: 10 for decimal, 2 for binary.

The radix, or base, of a positional number system tells you two things at once:

how many digit symbols there are: base b uses the digits 0 to b − 1
how much bigger each place is than the one to its right: a factor of b

So in base b the place weights, from the right, are b⁰ = 1, b¹, b², b³, and so on. Decimal (base 10) has weights 1, 10, 100, 1000. Binary (base 2) has 1, 2, 4, 8. Hexadecimal (base 16) has 1, 16, 256, 4096.

A base never has a digit equal to itself. When a column fills up, it resets to 0 and carries 1 into the next column. That's why ten is written 10 in decimal, two is 10 in binary and sixteen is 10 in hex. In every base, 10 means "one of the base".

When a number could be read in more than one base, mark it with a subscript: 101₂, 101₁₀ and 101₁₆ are three different values (5, 101 and 257). See base notation for the other common markers, such as 0x.

Worked example

Example

Reading 231 in different bases

The same three digits are worth different amounts, because the weights change with the base.

1.
Base 10: 2 × 100 + 3 × 10 + 1 = 231.
2.
Base 8: 2 × 64 + 3 × 8 + 1 = 128 + 24 + 1 = 153.
3.
Base 16: 2 × 256 + 3 × 16 + 1 = 512 + 48 + 1 = 561.
4.
Base 2: not allowed. Binary only has the digits 0 and 1, so 231₂ isn't a valid number.

Common mistakes

Saying base 8 has a digit 8. Base b runs from 0 to b − 1, so octal stops at 7 and binary stops at 1.
Numbering positions from 1. The rightmost position is position 0, with weight b⁰ = 1.
Writing the base in its own base. Every base is 10 in itself, so subscripts are always written in decimal: ₁₆, ₈, ₂.

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