Sample Cheat Sheets: Number Systems

IGCSE / O Level Computer Science Cheat Sheet
Topic: 1.1 Number Systems

🔢 Number Systems Overview

🔁 Conversions

✅ Denary ↔ Binary

To Convert Denary to Binary:

• Repeatedly divide by 2, write remainders in reverse order.

Example: 13 → Binary
13 ÷ 2 = 6 R1
6 ÷ 2 = 3 R0
3 ÷ 2 = 1 R1
1 ÷ 2 = 0 R1
→ Binary = 1101

To Convert Binary to Denary:

• Multiply each bit by 2^position (from right), then add.

Example: 1101 → Denary
1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8 + 4 + 0 + 1 = 13

✅ Denary ↔ Hexadecimal

To Convert Denary to Hex:

• Repeatedly divide by 16 and use hex digits.

Example: 255 → Hex
255 ÷ 16 = 15 R15 → FF

To Convert Hex to Denary:

• Multiply each digit by 16^position (from right).

Example: 7F → Denary
7×16¹ + 15×16⁰ = 112 + 15 = 127

✅ Binary ↔ Hexadecimal

Group binary in 4s from right → convert to hex.

Example: 10111111 → Hex
1011 1111 → B F → BF

Hex BF → Binary
B = 1011, F = 1111 → 10111111

🔠 Why Hex is Used

• Shorter than binary
• Easier for humans to read/debug
• Often used in color codes, memory addresses, MAC addresses

➕ Binary Addition (8-bit)

1. Add bits like normal numbers.
2. Carry over 1 if sum ≥ 2.

Example:

  11001101  
+ 01010111  
-----------
 100100100  → 9 bits (overflow!)

⚠️ Overflow

• Occurs when result > max value for bit size
• 8-bit max = 255 (11111111)
• If result requires 9 bits → overflow

🔁 Binary Shifts

• Bits shifted out are lost
• Zeros fill the opposite end

🔁 Two’s Complement (8-bit Signed Binary)

• MSB (Most Significant Bit) = sign bit (0 = +, 1 = –)

Convert Denary to 2’s Complement:

1. Convert magnitude to binary
2. Pad to 8 bits
3. If negative:
   • Invert bits
   • Add 1

Example: –18

1. 18 = 00010010
2. Invert → 11101101
3. Add 1 → 11101110

Convert Two’s Complement to Denary:

• If MSB = 0 → convert normally
• If MSB = 1 → invert + add 1 → then add (–)

Example: 11101110

1. Invert → 00010001
2. Add 1 → 00010010 = 18
3. Result = –18