Module 5.2-5.3 Numbering systems and data conversion

Knowledge levels

Sub-module	A	B3	B1	B2 / B2L
5.2 Numbering systems	—	—	1	2
5.3 Data conversion	—	—	1	2

Positional number systems

Each digit is multiplied by a power of the base (radix).

System	Base	Digits	Example
Decimal	10	0-9	253 = 2×10² + 5×10 + 3
Binary	2	0, 1	1011₂ = 8 + 0 + 2 + 1 = 11
Octal	8	0-7	17₈ = 1×8 + 7 = 15
Hexadecimal	16	0-9, A-F	2F₁₆ = 2×16 + 15 = 47

Binary place values: 128, 64, 32, 16, 8, 4, 2, 1. A bit is one binary digit; 4 bits = a nibble; 8 bits = a byte. The left-most bit is the most significant bit (MSB), the right-most the least significant bit (LSB).

Conversions

Decimal to binary (repeated division by 2)

Convert 156:

• 156/2 = 78 r0; 78/2 = 39 r0; 39/2 = 19 r1; 19/2 = 9 r1; 9/2 = 4 r1; 4/2 = 2 r0; 2/2 = 1 r0; 1/2 = 0 r1
• Read upwards: 10011100₂

Binary to octal and hexadecimal

• Octal: group bits in threes from the right. 10 011 100 → 2 3 4 → 234₈
• Hex: group bits in fours. 1001 1100 → 9 C → 9C₁₆

Check: 9 × 16 + 12 = 156 ✓; 2×64 + 3×8 + 4 = 156 ✓.

Hex to decimal

3A7₁₆ = 3 × 256 + 10 × 16 + 7 = 768 + 160 + 7 = 935

Decimal fractions to binary

Multiply the fraction by 2 repeatedly and take the integer parts: 0.625 → 1.25 (1), 0.5 (0), 1.0 (1) → 0.101₂.