Number representations: binary, denary and hexadecimal

OCR J277 is one of the few GCSE subjects with guaranteed calculation questions. Binary/hex conversions and binary arithmetic appear on virtually every Paper 1. Mastering these is important — they are easy marks if practised.

Denary (base 10)

The number system we use every day. Uses digits 0–9. Each position is a power of 10.

Binary (base 2)

Uses only digits 0 and 1 (bits). Each position is a power of 2:

Bit position	128	64	32	16	8	4	2	1
(2^n)	2⁷	2⁶	2⁵	2⁴	2³	2²	2¹	2⁰

Binary to denary

Add up the values of all positions where the bit is 1. Example: 10110101 = 128+32+16+4+1 = 181.

Denary to binary

Repeatedly divide by 2, record the remainder, read remainders bottom-up. Example: 181 ÷ 2 = 90 r1; 90÷2=45 r0; 45÷2=22 r1; 22÷2=11 r0; 11÷2=5 r1; 5÷2=2 r1; 2÷2=1 r0; 1÷2=0 r1. Read up: 10110101 ✓

Binary addition

Rules: 0+0=0; 0+1=1; 1+1=10 (write 0, carry 1); 1+1+1=11 (write 1, carry 1).

Example: 01101010

• 00110111 = 10100001

Overflow occurs if the result is too large to fit in the number of bits. E.g. adding two 8-bit numbers that produce a 9-bit result — the 9th bit is lost, giving a wrong answer.

Binary shifts

Logical left shift (multiply)

Shift all bits left by n positions; fill vacated right positions with 0; bits shifted off the left are lost.

• Left shift by 1 = multiply by 2¹ = 2.
• Left shift by 2 = multiply by 2² = 4. Example: 00000110 (6) left shift 1 → 00001100 (12).

Logical right shift (divide)

Shift all bits right by n positions; fill vacated left positions with 0; bits shifted off the right are lost.

• Right shift by 1 = divide by 2 (any remainder is discarded). Example: 00001100 (12) right shift 1 → 00000110 (6).

⚠ Right shift discards the remainder — 00001101 (13) right shift 1 → 00000110 (6), not 6.5.

Hexadecimal (base 16)

Uses digits 0–9 and A–F (A=10, B=11, C=12, D=13, E=14, F=15). Each hex digit represents exactly 4 bits (a nibble).

Why use hexadecimal?

• More compact than binary (4 bits → 1 hex digit).
• Used in: MAC addresses, IP addresses, colour codes (HTML), memory addresses, error codes.
• Human-readable shorthand for binary.

Binary to hex

Split binary into groups of 4 (from the right); convert each nibble to its hex digit. Example: 10110101 → 1011 | 0101 → B | 5 → B5.

Hex to binary

Replace each hex digit with its 4-bit binary equivalent. Example: 3F → 3=0011, F=1111 → 00111111.

Hex to denary

Multiply each digit by its positional value (powers of 16). Example: B5 = (11 × 16) + (5 × 1) = 176 + 5 = 181.

Common OCR exam mistakes

1. Forgetting to pad binary to 8 bits when converting to hex (must group into nibbles of exactly 4 bits).
2. Writing A=10 but then treating hex incorrectly in arithmetic — always convert to denary for arithmetic, then back.
3. Overflow: forgetting that the carry bit is lost — the question often asks what the result is AND whether overflow occurred.
4. Right shift: forgetting that any remainder/fractional result is discarded (it's integer division).