Intermediate

Number Systems and Bases

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·A-Level Computer Science·AQA 7517·6 min
4.5.1 Number systems·4.5.2 Number bases·4.5.3 Units of information

Types of Numbers

AQA requires knowledge of the following number sets:

SymbolNameDescriptionExamples
Natural numbersNon-negative whole numbers (including 0)0, 1, 2, 3, …
IntegersAll whole numbers (positive, negative, zero)…, −2, −1, 0, 1, 2, …
Rational numbersExpressible as a fraction (integers and fractions)1/2, 3, −4, 0.75
IrrationalIrrational numbersCannot be expressed as a fraction; non-terminating, non-repeating decimal, ,
Real numbersAll rational and irrational numbers combinedany point on the number line
OrdinalOrdinal numbersDescribe position in an ordered sequence1st, 2nd, 3rd, …

Key relationships: . Every natural number is an integer; every integer is rational (e.g. ); 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)

Number Bases

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

BaseNameDigitsPrefix/suffix
10Decimal0–9(default)
2Binary0, 10b or subscript ₂
16Hexadecimal0–9, A–F0x or subscript ₁₆

Place values:

Decimal (base 10): each column is a power of 10.

Binary (base 2): each column is a power of 2.

Hexadecimal (base 16): digits A=10, B=11, C=12, D=13, E=14, F=15.

Binary to Decimal Conversion

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

Worked example — convert to decimal:

Bit position76543210
Place value1286432168421
Bit10110101
Contribution128032160401

Decimal to Binary Conversion

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

Worked example — convert to binary:

DivisionQuotientRemainder
53 ÷ 2261 ← LSB
26 ÷ 2130
13 ÷ 261
6 ÷ 230
3 ÷ 211
1 ÷ 201 ← MSB

Read remainders from bottom to top:

Verify:

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Hexadecimal: Shorthand for Binary

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

Hex digit table:

HexDecimalBinary
0–90–90000–1001
A101010
B111011
C121100
D131101
E141110
F151111

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

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

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.

Units of Information

UnitSizeApproximate size
Bit1 bit (0 or 1)
Byte8 bitsCan store values
Kibibyte (KiB) = 1 024 bytes~1 000 bytes
Mebibyte (MiB) = 1 048 576 bytes~1 000 000 bytes
Gibibyte (GiB) bytes~1 000 000 000 bytes
Tebibyte (TiB) bytes~1 000 000 000 000 bytes

Binary vs decimal prefixes:

BinarySizeDecimalSize
1 KiB = 1 024 bytes1 KB = 1 000 bytes
1 MiB bytes1 MB bytes
1 GiB bytes1 GB bytes

Storage manufacturers use decimal prefixes (so a "1 TB" drive holds 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 bits represent? distinct values. bits can represent integers from 0 to (unsigned).

Common Exam Mistakes

1. Confusing rational and irrational numbers

is rational (it equals 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. 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.

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