Units and data representation - OCRHexadecimal

In computer science, different number bases are used:

• denary is base 10, which has ten units (0-9)
• binary is base 2 , which has two units (0-1)

Hexadecimal, also known as hex, is the third commonly used number system. It has 16 units (0-9) and the letters A, B, C, D, E and F.

Hex is useful because large numbers can be represented using fewer digits. For example, colour values and MAC addresses are often represented in hex.

Additionally, hex is easier to understand than binary. Programmers often use hex to represent binary values as they are simpler to write and check than when using binary.

Converting hexadecimal to denary

Whereas denary place values are powers of 10, and binary place values are powers of 2, hex place values are powers of 16.

Each place value can be represented by the units 0 through to F.

To convert hex to denary, simply take each place value that has a unit in it, and add them together.

Example - hex number 7C

Result: (7 × 16) + (C × 1) = (7 × 16) + (12 × 1) = (112) + (12) = 124

Converting between binary and hexadecimal numbers

Denary to hexadecimal

The OCR specification requires you to be able to convert from denary to two-digit hex. To convert:

• If the denary number is bigger than 16, divide it by 16. Take the hex equivalent of this result - this represents the first digit. Take the hex equivalent of the remainder - this represents the second digit.
• If the denary number is smaller than 16, take the hex equivalent of the denary number.

Example - convert denary 22 to hex

16 goes into 22 once with 6 left over, so 22 ÷ 16 = 1 remainder 6

1 = hex 1

6 = hex 6

Result: 16

Example - convert 138 to hex

138 ÷ 16 = 8 remainder 10

8 = hex 8

10 = hex A

Result: 8A

Converting between binary and hexadecimal

The simplest way to convert from binary to hex, and vice versa, is to convert to denary first.

Binary to hexadecimal

1. Start at the rightmost digit and break the binary number up into groups of four digits. These are known as nibbles. If there are less than four digits, use just that number of digits for that group.
2. Next, convert each group of four digits into denary.
3. Convert each denary value into its hex equivalent.
4. Put the hex digits together.

Example - 1101 to hex

1101 = denary 13

13 = hex D

Result: D

Example - 11000011 to hex

Break into groups of four - 1100 0011

1100 = denary 12 0011 = denary 3

12 = hex C 3 = hex 3

Result: C3

Example - 110011 to hex

Break into groups of four - 0011 0011. In this example, extra 0s are added at the highest values to create two groups of four bits.

0011 = denary 3 0011 = denary 3

3 = hex 3 3 = hex 3

Result: 33

Hexadecimal to binary

1. Split the hex number into individual values.
2. Convert each hex value into its denary equivalent.
3. Next, convert each denary digit into binary, making sure you write four digits for each value.
4. Combine all four digits to make one binary number.

Example - hex 28 to binary

2 = denary 2 8 = denary 8

2 = binary 0010 8 = binary 1000

Result: 00101000

Example - hex FC to binary

F = denary 15 C = denary 12

15 = binary 1111 12 = binary 1100

Result: 11111100