The hexadecimal number system, why hexadecimal is used as a shorthand for binary, and converting between hexadecimal, binary and denary.

What this topic is asking

WJEC wants you to know the hexadecimal (base 16) number system, why it is a useful shorthand for binary, and how to convert between hexadecimal, binary and denary. This is part of the Data representation and data types content in Unit 1 of WJEC GCSE Computer Science (3500).

The hexadecimal number system

Why hexadecimal is used

It is important to remember that the computer never actually stores hexadecimal; it always stores binary. Hexadecimal is only a human-friendly way of writing binary down.

Converting hexadecimal to binary

The neat trick is that each hex digit is exactly four bits, so you convert one digit at a time.

Converting binary to hexadecimal

To go the other way, split the binary number into groups of four bits starting from the right (add leading zeros to the left group if needed), then convert each group of four bits into its hex digit. For example, $11010110$ splits into $1101$ and $0110$; $1101 = 13 = \text{D}$ and $0110 = 6$, so the answer is $\text{D6}$. If a binary number has a number of bits that is not a multiple of four, pad the left-hand group with zeros: $110110$ becomes $0011\,0110 = \text{36}$. Grouping from the right is essential, because grouping from the left would change the place values and give the wrong answer.

Converting hexadecimal to denary

Multiply each hex digit by its column value ($16$ for the left digit, $1$ for the right digit) and add the results, remembering to convert any letter to its value first ($\text{A} = 10$ up to $\text{F} = 15$). For a single hex digit the value is just the digit itself, so $\text{C} = 12$. For two digits such as $\text{7E}$, the calculation is $7 \times 16 + 14 = 112 + 14 = 126$. A quick way to check a two-digit answer is that the largest possible value, $\text{FF}$, is $15 \times 16 + 15 = 255$, so any valid two-digit hex number must come out between $0$ and $255$.

Try this

Q1. Convert the binary number $11101010$ to hexadecimal. [2 marks]

• Cue. Split into $1110$ and $1010$; $1110 = \text{E}$ and $1010 = \text{A}$, so the answer is $\text{EA}$.

Q2. Convert the hexadecimal number $\text{4C}$ to denary. [2 marks]

• Cue. $4 \times 16 + 12 = 64 + 12 = 76$.