Binary, denary and hexadecimal

Where this is in your specification

Spec points: AQA 3.3.1 to 3.3.4, OCR 1.2.1 and 1.2.4, Edexcel Topic 2

Binary to denary and back

Add the place values with a 1 under them: 1011 0010 = 128 + 32 + 16 + 2 = 178.

To go from denary to binary, take the biggest place value that fits, write a 1 there, subtract it, and repeat with what is left. Write 0 in every column you skip.

Hexadecimal

Hex uses the digits 0 to 9 and then A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. Its place values are 1, 16, 256.

• Hex to denary: 3C = 3 × 16 + 12 = 60.
• Denary to hex: 200 ÷ 16 = 12 remainder 8, so C8.
• Binary to hex: split into nibbles from the right and convert each one. 1011 0010 = B2.
• Hex to binary: write each hex digit as four bits. 7E = 0111 1110.

Hex is used because it is shorter than binary, easier for people to read and copy without errors, and converts to binary directly. The computer still stores binary. You will see hex in colour codes, memory addresses and MAC addresses.

Binary addition and overflow

Work from the right-hand column. The rules: 0 + 0 gives 0, 0 + 1 gives 1, 1 + 1 gives 0 with 1 carried, and 1 + 1 + 1 gives 1 with 1 carried. If the answer needs more bits than the register has, the extra carry is lost: that is an overflow error.

Shifts

A left shift of one place multiplies an unsigned binary number by 2; a right shift divides by 2 (losing any remainder). Shifting left by 2 places multiplies by 4. Bits shifted off the end are lost, which can make the result wrong.

Units of data

Worked examples

Common mistakes

• Reading binary from the wrong end. The 1s column is on the right.
• Writing 10 to 15 as two digits in hex instead of A to F.
• Splitting into nibbles from the left when the number of bits is not a multiple of four. Pad with 0s on the left first.
• Saying computers use hex. They store binary; hex is for people.