Number Systems (Copy)

Binary Representation and Data Processing

• Why computers use binary
  • Computers are built on electronic circuits that operate with two distinct voltage levels:
    • High voltage → Represented as 1
    • Low voltage → Represented as 0
  • All forms of data (numbers, text, images, audio, video) are stored and processed as a sequence of binary digits (bits).
  • Logic gates (AND, OR, NOT, etc.) process binary data inside the Central Processing Unit (CPU).
  • Data is stored in registers, which are small storage locations inside the CPU.
• Data conversion to binary
  • Any form of data (e.g., a letter ‘A’ in text) is converted to binary before processing:
    • Example: ASCII code for ‘A’ = 65 in denary → 01000001 in binary.
  • Storing in binary makes it easier for a computer to perform calculations and apply logical operations.

Number Systems Overview

Converting Between Number Systems

1. Denary ↔ Binary

• Denary to binary (Repeated division by 2 method):
  1. Divide the denary number by 2.
  2. Record the remainder (0 or 1).
  3. Continue dividing the quotient by 2 until you reach 0.
  4. The binary number is the remainders read from bottom to top.

  Example: Convert 25₁₀ to binary

  25 ÷ 2 = 12 remainder 1  
  12 ÷ 2 = 6  remainder 0  
  6 ÷ 2  = 3  remainder 0  
  3 ÷ 2  = 1  remainder 1  
  1 ÷ 2  = 0  remainder 1  
  Binary = 11001₂
  

• Binary to denary (Positional value method):
  • Multiply each bit by 2ⁿ, where n is the position from right (starting at 0), then sum.

  Example: Convert 101101₂ to denary:

  (1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)  
  = 32 + 0 + 8 + 4 + 0 + 1  
  = 45₁₀
  

2. Denary ↔ Hexadecimal

• Denary to hexadecimal (Repeated division by 16):
  Example: Convert 255₁₀ to hexadecimal:

  255 ÷ 16 = 15 remainder 15  
  15 in hex = F  
  Read from bottom to top: FF₁₆
  

• Hexadecimal to denary (Positional value method):
  Example: Convert 2F₁₆ to denary:

  (2 × 16¹) + (15 × 16⁰)  
  = 32 + 15  
  = 47₁₀
  

3. Binary ↔ Hexadecimal

• Binary to hexadecimal:
  • Group binary digits into 4-bit groups from right to left.
  • Convert each group to its hexadecimal equivalent.

  Example: 10101111₂ → Group as 1010 1111

  1010 = A  
  1111 = F  
  Result: AF₁₆
  

• Hexadecimal to binary:
  • Convert each hexadecimal digit into a 4-bit binary value.

  Example: 3C₁₆ →

  3 = 0011  
  C = 1100  
  Result: 00111100₂
  

Why Hexadecimal is Beneficial

• Shorter representation than binary (e.g., 11111111₂ is FF₁₆).
• Easier for humans to read and write.
• Common uses in computing:
  • Memory addresses (e.g., 0x3F2A)
  • HTML colour codes (e.g., #FF5733)
  • Machine code representation for debugging.
• Reduces human error when working with long binary strings.

Binary Addition and Overflow

Binary addition rules:

Example: Add 11001010₂ and 01110111₂

   11001010  
+  01110111  
-------------
  100000001

• The result is 9 bits in length → overflow has occurred.

Overflow concept:

• Happens when the result of a calculation is too large for the allocated number of bits.
• 8-bit register: Maximum value = 255₁₀ (11111111₂).
• If adding results in a number > 255, an overflow occurs.
• Overflow can cause incorrect calculations or wrap-around errors in computing.

Logical Binary Shifts

Logical left shift:

• Each bit moves one position to the left.
• A 0 is added to the rightmost bit.
• Leftmost bit is lost.
• Effect: Multiplies the number by 2 for each shift.

Example: 00101100₂ (44₁₀) left shift by 1 → 01011000₂ (88₁₀).

Logical right shift:

• Each bit moves one position to the right.
• A 0 is added to the leftmost bit.
• Rightmost bit is lost.
• Effect: Divides the number by 2 for each shift.

Example: 00101100₂ (44₁₀) right shift by 1 → 00010110₂ (22₁₀).

Two’s Complement Representation

Why use two’s complement:

• Allows binary to represent both positive and negative integers.
• Simplifies binary arithmetic (same addition/subtraction rules for all integers).

8-bit two’s complement range:

• Minimum value: −128 (10000000₂)
• Maximum value: +127 (01111111₂)

Converting positive denary to two’s complement binary:

1. Convert denary to binary normally (7 bits for magnitude, plus sign bit 0).
2. Example: +18 → 00010010₂.

Converting negative denary to two’s complement binary:

1. Convert absolute value to binary.
2. Invert all bits (1 → 0, 0 → 1).
3. Add 1 to the result.

Example: −18 in 8-bit:

+18 = 00010010  
Invert = 11101101  
Add 1 = 11101110

Final: 11101110₂.

Converting from two’s complement binary to denary:

• If MSB (most significant bit) is 0 → positive number → convert normally.
• If MSB is 1 → negative number:
  1. Invert all bits.
  2. Add 1.
  3. Convert to denary → add minus sign.

Example: 11101110₂:

Invert: 00010001  
Add 1: 00010010 (18)  
Result: −18