Decimal, Binary, and Hexadecimal

The base tells us how many different digits are available, including zero. Decimal has ten digits, binary has two and hexadecimal has sixteen.

These are all positional number systems: a digit's position affects its contribution to the number. For whole numbers, the rightmost position has a place value of 1. Each step to the left multiplies the place value by the base.

Decimal: place values increase by ten

Decimal, also called denary, is the familiar base-10 system used for everyday counting and calculations. Its digits are 0–9.

Reading from right to left, its place values are 1, 10, 100, 1000 and so on. These are powers of ten: $10^0$, $10^1$, $10^2$ and $10^3$. The power records how many times we multiply 1 by the base; $10^0 = 1$.

For example, the digits in 3268 occupy these positions:

The number means three thousands, two hundreds, six tens and eight ones:

Each digit contributes its value multiplied by its place value. We then add the contributions together.

Binary: place values double

Binary is base 2, so its only digits are 0 and 1. Each binary digit is called a bit.

The positional rule stays the same, but now each place value is twice the one to its right:

A 1 contributes that position's place value; a 0 contributes nothing. In the binary number 1100, the 8 and 4 positions contain ones, while the 2 and 1 positions contain zeros:

So binary 1100 represents decimal 12. Having only two possible digits does not restrict binary to two numerical values: additional positions let us represent larger numbers.

Hexadecimal: sixteen possible digits

Hexadecimal is base 16. It uses 0–9 for the first ten digit values, then A–F for the remaining six:

The letters act as single digits. For example, A represents the value ten, rather than a separate variable.

Hexadecimal place values increase by a factor of sixteen. The rightmost position is the ones position, $16^0$, and the next is the sixteens position, $16^1$.

For example, hexadecimal 13 means one sixteen and three ones:

It therefore represents decimal 19, not decimal thirteen.

The same digits can mean different values

The written number 10 shows why the base matters. It means one group of the base and zero ones:

• Decimal 10 represents ten.
• Binary 10 represents two.
• Hexadecimal 10 represents sixteen.

In each system, once all the single digits have been used, counting continues by using another position. Decimal goes from 9 to 10, binary from 1 to 10, and hexadecimal from F to 10. The underlying quantity has not changed its nature: only the way we write it has changed.