Radix (Number Base) — Definition, Formula & Examples

A radix is the number of unique digits (including zero) used in a positional number system. For example, the decimal system has a radix of 10 because it uses the digits 0 through 9.

In a positional numeral system, the radix (or base) $b$ is a positive integer greater than 1 that defines both the set of permissible digit values $\{0, 1, \ldots, b-1\}$ and the factor by which each successive place value increases from right to left.

Key Formula

N = d_n \cdot b^n + d_{n-1} \cdot b^{n-1} + \cdots + d_1 \cdot b^1 + d_0 \cdot b^0

Where:

$N$ = The value of the number
$b$ = The radix (base) of the number system
$d_i$ = The digit in position i, where each digit satisfies 0 ≤ d_i < b
$n$ = The position of the leftmost digit (counting from 0 on the right)

How It Works

Each digit in a number occupies a position, and that position's value depends on the radix. The rightmost digit has a place value of

b^0 = 1

, the next has

b^1

, then

b^2

, and so on. To find the total value of a number, multiply each digit by its place value and add the results. Common radixes include 2 (binary, used by computers), 8 (octal), 10 (decimal, everyday counting), and 16 (hexadecimal, used in programming).

Worked Example

Problem: Convert the binary number 1101 (base 2) to decimal (base 10).

Identify the radix: The radix is 2, so each place value is a power of 2.

Expand by place value: Write each digit multiplied by its place value, from left to right.

1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0

Calculate: Evaluate each term and add them together.

8 + 4 + 0 + 1 = 13

Answer: The binary number

1101_2

equals

13_{10}

in decimal.

Why It Matters

Computers store and process all data in base 2, so understanding radix is essential for computer science courses. Hexadecimal (base 16) is widely used to represent colors in web design and memory addresses in programming.

Common Mistakes

Mistake: Using digits that are too large for the base — for example, writing the digit 5 in a base-5 number.

Correction: In base

b

, the largest allowed digit is

b - 1

. Base 5 only uses the digits 0, 1, 2, 3, and 4.