Binary Numbers and Denary Conversion

Binary is a base-2 number system that uses only two digits: 0 and 1.

Every number system has a base that defines how many unique digits exist before the count carries into the next place. Denary (everyday numbers) is base 10 — it uses digits 0 through 9 and carries at 10. Binary uses only 0 and 1, carrying at 2.

System	Base	Digits used	Denary value 10 written in this system
Denary	10	0–9	10
Binary	2	0, 1	1010
Hexadecimal	16	0–9, A–F	A

Computers represent all data in binary because transistors — the fundamental switches inside every processor and memory chip — have exactly two stable states: off (0) and on (1). Any more complex signal would require analogue electronics, which are far less reliable at speed and scale.

A single binary digit is a bit. Eight bits form one byte. Every piece of data a computer processes — text, images, sound, video, programs — is ultimately stored and processed as sequences of 0s and 1s.

AQA 8525 uses 8-bit (one byte) binary numbers throughout. All examples in this lesson are 8-bit; the maximum value is 255.

Each bit in a binary number represents a power of 2, starting from $2^0=1$ at the rightmost position and doubling with each step to the left.

For an 8-bit binary number, the place values are:

128	64	32	16	8	4	2	1
$2^7$	$2^6$	$2^5$	$2^4$	$2^3$	$2^2$	$2^1$	$2^0$

Worked example — read the value of 10110100:

128	64	32	16	8	4	2	1
1	0	1	1	0	1	0	0

Bits set to 1: 128, 32, 16, 4

Value = 128 + 32 + 16 + 4 = 180

An 8-bit byte holds values from 0 to 255 — a range of $2^8=256$ distinct values. The maximum 8-bit value 11111111 equals $128+64+32+16+8+4+2+1=255$.

To convert binary to denary: write the place value table, mark which bits are 1, then add those place values together.

Worked example 1 — convert 01101001 to denary:

128	64	32	16	8	4	2	1
0	1	1	0	1	0	0	1

Value = 64 + 32 + 8 + 1 = 105

Worked example 2 — convert 11001010 to denary:

128	64	32	16	8	4	2	1
1	1	0	0	1	0	1	0

Value = 128 + 64 + 8 + 2 = 202

Always write the full 8-bit place value table before reading any binary number. Pad short values with leading zeros: 01101001, not 1101001.

Use the division-by-2 method: repeatedly divide by 2 and record the remainder each time. Read the remainders from bottom to top for the binary result.

Steps:

1. Divide the number by 2; record the remainder (0 or 1)
2. Divide the quotient by 2; record the remainder
3. Repeat until the quotient reaches 0
4. Read remainders bottom to top
5. Pad to 8 bits with leading zeros if needed

Worked example — convert 109 to binary:

Division	Quotient	Remainder
109 ÷ 2	54	1
54 ÷ 2	27	0
27 ÷ 2	13	1
13 ÷ 2	6	1
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1

Remainders bottom to top: 1101101 → padded to 8 bits: 01101101

Check: 64 + 32 + 8 + 4 + 1 = 109 ✓

An alternative to repeated division is the subtraction method: work left to right through the place values, subtracting each one that fits into the remaining total.

Steps:

1. Start with the largest place value (128)
2. If it fits (remaining value ≥ place value), write 1 and subtract it
3. If it does not fit, write 0 and move right
4. Repeat for each place value until the total reaches 0

Worked example — convert 178 to binary:

Place value	128	64	32	16	8	4	2	1
Fits?	178 ≥ 128 ✓	50 < 64 ✗	50 ≥ 32 ✓	18 ≥ 16 ✓	2 < 8 ✗	2 < 4 ✗	2 ≥ 2 ✓	0 < 1 ✗
Bit	1	0	1	1	0	0	1	0
Remaining	50	50	18	2	2	2	0	0

Result: 10110010

Check: 128 + 32 + 16 + 2 = 178 ✓

Both methods produce identical results. The division method is systematic for any number; the subtraction method is often faster in exam conditions once you know the place values.

AQA 8525 requires you to explain why hexadecimal (hex) appears frequently in computer science. The core reason: each hex digit corresponds to exactly 4 binary bits (one nibble).

Binary nibble	Hex digit	Denary
0000	0	0
0001	1	1
0101	5	5
1001	9	9
1010	A	10
1011	B	11
1100	C	12
1101	D	13
1110	E	14
1111	F	15

An 8-bit byte such as 10110100 (180 in denary) converts to B4 in hex — two characters instead of eight. This compactness makes hex practical wherever binary values are written down by humans: memory addresses, RGB colour codes (#FF8800), MAC addresses, error codes.

Hex is not a different form of storage — computers store everything in binary. Hex is a shorthand notation for humans to read and write binary values more efficiently.

The full worked examples for binary↔hex and denary↔hex conversion are in the related lesson on Hexadecimal Representation.

1. Reading the division table in the wrong direction

The division-by-2 method produces remainders that must be read bottom to top. Reading top to bottom gives the bit-reversed value — an entirely different number.

2. Forgetting to pad to 8 bits

1101101 is 7 bits. AQA 8525 uses 8-bit numbers. Always pad with a leading zero: 01101101. Omitting the leading zero is one of the most common arithmetic errors in binary conversion questions.

3. Using the wrong place values

The 8 place values are 128, 64, 32, 16, 8, 4, 2, 1 — not 256, 128, 64, 32, 16, 8, 4, 2. The least significant bit (rightmost) is always 1 ($2^0$). Writing the table before reading any number prevents this error.

4. Confusing bit and byte

A bit is a single binary digit (0 or 1). A byte is 8 bits. These are not interchangeable. An 8-bit binary number is 1 byte — not 8 bytes.

5. Stopping the division method early

The division-by-2 method must continue until the quotient reaches exactly 0. Students who stop when the quotient is 1 miss the final remainder (which is always 1), losing the most significant bit of the result.