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.