Number Systems and Bases

AQA requires knowledge of the following number sets:

Symbol	Name	Description	Examples
$\mathbb{N}$	Natural numbers	Non-negative whole numbers (including 0)	0, 1, 2, 3, …
$\mathbb{Z}$	Integers	All whole numbers (positive, negative, zero)	…, −2, −1, 0, 1, 2, …
$\mathbb{Q}$	Rational numbers	Expressible as a fraction $p/q$ (integers and fractions)	1/2, 3, −4, 0.75
Irrational	Irrational numbers	Cannot be expressed as a fraction; non-terminating, non-repeating decimal	$\sqrt{2}$, $\pi$, $e$
$\mathbb{R}$	Real numbers	All rational and irrational numbers combined	any point on the number line
Ordinal	Ordinal numbers	Describe position in an ordered sequence	1st, 2nd, 3rd, …

Key relationships: $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$. Every natural number is an integer; every integer is rational (e.g. $-3=-3/1$); every rational number is real.

Purpose in computing:

• Natural numbers are used for counting (array indices, loop counters)
• Real numbers are used for measurement (temperatures, coordinates, probabilities)

A number base defines how many distinct digits are available and the place value of each digit position.

Base	Name	Digits	Prefix/suffix
10	Decimal	0–9	(default)
2	Binary	0, 1	0b or subscript ₂
16	Hexadecimal	0–9, A–F	0x or subscript ₁₆

Place values:

Decimal (base 10): each column is a power of 10. $542=5\times 10^2+4\times 10^1+2\times 10^0$

Binary (base 2): each column is a power of 2. $1011_2=1\times 2^3+0\times 2^2+1\times 2^1+1\times 2^0=8+0+2+1=11_{10}$

Hexadecimal (base 16): digits A=10, B=11, C=12, D=13, E=14, F=15. ${\text{2F}}_{16}=2\times 16^1+15\times 16^0=32+15=47_{10}$

To convert binary to decimal, multiply each bit by its place value (power of 2) and sum the results.

Worked example — convert $10110101_2$ to decimal:

Bit position	7	6	5	4	3	2	1	0
Place value	128	64	32	16	8	4	2	1
Bit	1	0	1	1	0	1	0	1
Contribution	128	0	32	16	0	4	0	1

$128+32+16+4+1=181_{10}$

Method: repeated division by 2. Divide by 2 repeatedly; collect remainders from bottom to top.

Worked example — convert $53_{10}$ to binary:

Division	Quotient	Remainder
53 ÷ 2	26	1 ← LSB
26 ÷ 2	13	0
13 ÷ 2	6	1
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1 ← MSB

Read remainders from bottom to top: $53_{10}=110101_2$

Verify: $32+16+4+1=53$ ✓

Each hexadecimal digit represents exactly 4 binary bits (a nibble). This makes hex a compact shorthand for long binary strings.

Hex digit table:

Hex	Decimal	Binary
0–9	0–9	0000–1001
A	10	1010
B	11	1011
C	12	1100
D	13	1101
E	14	1110
F	15	1111

Binary to hex: group bits in fours from the right; convert each group.

$\underset{A}{\underbrace{1010}}\underset{E}{\underbrace{1110}}{}_2={\text{AE}}_{16}$

Hex to binary: expand each hex digit to 4 bits.

${\text{3C}}_{16}=\underset{3}{\underbrace{0011}}\underset{C}{\underbrace{1100}}{}_2=00111100_2$

Why hex is used: binary strings are hard to read and write. 0xAE3F is far less error-prone than 1010111000111111. Hex is used in memory addresses, colour codes (e.g. #FF5733), machine code, and network addresses.

Unit	Size	Approximate size
Bit	1 bit (0 or 1)	—
Byte	8 bits	Can store $2^8=256$ values
Kibibyte (KiB)	$2^{10}$ = 1 024 bytes	~1 000 bytes
Mebibyte (MiB)	$2^{20}$ = 1 048 576 bytes	~1 000 000 bytes
Gibibyte (GiB)	$2^{30}$ bytes	~1 000 000 000 bytes
Tebibyte (TiB)	$2^{40}$ bytes	~1 000 000 000 000 bytes

Binary vs decimal prefixes:

Binary	Size	Decimal	Size
1 KiB	$2^{10}$ = 1 024 bytes	1 KB	$10^3$ = 1 000 bytes
1 MiB	$2^{20}$ bytes	1 MB	$10^6$ bytes
1 GiB	$2^{30}$ bytes	1 GB	$10^9$ bytes

Storage manufacturers use decimal prefixes (so a "1 TB" drive holds $10^{12}$ bytes). Operating systems typically report in binary units (GiB). This is why a "1 TB" drive appears as ~931 GiB in Windows.

How many values can $n$ bits represent? $2^n$ distinct values. $n$ bits can represent integers from 0 to $2^n-1$ (unsigned).

1. Confusing rational and irrational numbers

$\sqrt{4}=2$ is rational (it equals 2). $\sqrt{2}$ is irrational. Not all square roots are irrational — only non-perfect-square roots.

2. Reading binary remainders in the wrong order

In the repeated-division method, the first remainder is the least significant bit (rightmost). Read remainders from bottom to top to get the correct binary value.

3. Forgetting to pad hex digits to 4 bits

When converting hex to binary, each digit must become exactly 4 bits. $3_{16}$ becomes 0011, not 11. Missing leading zeros produces an incorrect binary value.

4. Confusing binary (KiB) and decimal (KB) units

1 KiB = 1 024 bytes (binary). 1 KB = 1 000 bytes (decimal). These are different. AQA expects students to distinguish them and know which context uses which convention.