Sample Notes: Number Systems

O Level and IGCSE Computer Science

Chapter 1.1 – Number Systems

Understanding Binary Representation

• All data in a computer system—text, numbers, images, sound—is stored and processed as binary (base-2).
• Binary is used because computers are made of logic circuits, which operate using two states:
  • 0 (off, low voltage)
  • 1 (on, high voltage)
• Logic gates, registers, RAM, and all other components understand only binary inputs and outputs.

Number Systems Overview

Conversions Between Number Systems

1. Denary to Binary (positive values only)

• Use division-by-2 method or place value (128, 64, 32, …, 1)
• Example:
  13 → 00001101
  (8 + 4 + 1 = 13)

2. Binary to Denary

• Add place values for all 1s.
• Example:
  00011010 → 16 + 8 + 2 = 26

3. Denary to Hexadecimal

• Use division-by-16 or convert via binary.
• Example:
  255 ÷ 16 = 15 remainder 15 → FF

4. Hexadecimal to Denary

• Multiply each digit with corresponding power of 16.
• Example:
  1A = 1×16 + 10 = 26

5. Binary to Hexadecimal

• Break binary into 4-bit groups from the right.
• Convert each group:
  • 1010 1111 → A F → AF

6. Hexadecimal to Binary

• Convert each digit to 4-bit binary:
  • 2C → 2 = 0010, C = 1100 → 00101100

Why Hexadecimal is Used

• Easier for humans to read and remember than binary:
  • 8-bit binary: 11110000
  • Equivalent hex: F0
• Compact representation
• Widely used in:
  • Memory addresses (e.g., 0x1A3F)
  • Color codes in web design (e.g., #FF5733)
  • Machine code instructions

Binary Addition (8-bit only)

Rules

• 0 + 0 = 0
• 1 + 0 = 1
• 1 + 1 = 10 (0 with carry 1)
• 1 + 1 + 1 = 11 (1 with carry 1)

Example

  01101010  
+ 10101101  
= 100101111 (9 bits → overflow occurs)

Overflow in Binary Addition

• Overflow occurs when result exceeds available bits.
• In 8-bit: max value is 11111111 = 255
• Any result > 255 leads to overflow error
• System may store only last 8 bits, leading to incorrect value
• Example:
  • 200 + 100 → 300 in denary → needs 9 bits → overflow

Logical Binary Shifts

1. Logical Left Shift

• All bits move left; 0 enters from right
• Multiply number by 2 each time
• Example: 00001101 (13) → left shift → 00011010 (26)

2. Logical Right Shift

• All bits move right; 0 enters from left
• Divide number by 2 each time
• Example: 00001100 (12) → right shift → 00000110 (6)

Effect of Multiple Shifts

• Left shift n times → multiply by 2ⁿ
• Right shift n times → divide by 2ⁿ (integer division)

Two’s Complement Representation (8-bit)

Purpose

• Represents both positive and negative integers in binary.

Steps to Represent Negative Numbers:

1. Write positive binary of absolute value.
2. Invert all bits (1s → 0s, 0s → 1s)
3. Add 1

Example: -18

• +18 = 00010010
• Invert = 11101101
• Add 1 → 11101110 → This is -18 in two’s complement.

Two’s Complement Ranges

• 8-bit:
  • Positive: 00000000 (0) to 01111111 (+127)
  • Negative: 11111111 (-1) to 10000000 (-128)

Converting Two’s Complement to Denary

For Positive (starts with 0):

• Same as normal binary.

For Negative (starts with 1):

1. Invert all bits
2. Add 1
3. Convert to denary and place negative sign

Example:

11101110 → Invert: 00010001
→ Add 1: 00010010 = 18
→ Final: -18