Introduction to Number Bases

What is a Number Base?

A number base (or radix) tells us how many digits are used to represent numbers in a counting system. The number system we use every day is base 10 (denary/decimal), which uses ten digits: 0–9. Other bases use different numbers of digits.

Common Number Bases

• Base 10 (Denary): uses digits 0–9. This is our everyday counting system.
• Base 2 (Binary): uses only digits 0 and 1. Computers use binary internally because electronic switches have two states: on (1) and off (0).
• Base 5 (Quinary): uses digits 0–4.
• Base 8 (Octal): uses digits 0–7.

Place Value in Different Bases

In base 10, each digit's place value is a power of 10 (units, tens, hundreds...). In any base b, each digit's place value is a power of b. For example, in binary (base 2), place values from right to left are 1, 2, 4, 8, 16, and so on (powers of 2).

Converting Base 2 to Base 10

To convert a binary number to denary, multiply each digit by its place value (power of 2) and add the results. For example, 1011₂ = (1×8) + (0×4) + (1×2) + (1×1) = 8 + 0 + 2 + 1 = 11 in base 10.

Converting Base 10 to Base 2

To convert a denary number to binary, repeatedly divide the number by 2 and record the remainders, then read the remainders from bottom to top. For example, to convert 13 to binary: 13÷2=6 r1, 6÷2=3 r0, 3÷2=1 r1, 1÷2=0 r1. Reading remainders bottom to top gives 1101₂.

Why Number Bases Matter

Number bases are fundamental to computer science because all data inside a computer — text, images, sound — is ultimately stored and processed as binary numbers. Understanding number bases helps explain how computers represent and process information.