Digital Logic Basics: Number Systems, Complements, and Binary Arithmetic

1. Introduction to Digital Logic

1.1 Digital vs. Analog Systems

• Digital Systems: Represent data as discrete values (0s and 1s). Used in computers, digital clocks, and microcontrollers.
• Analog Systems: Represent data as continuous signals (e.g., voltage levels). Used in thermometers, audio signals.
• Advantages of Digital Systems:
  • Noise immunity (less error-prone).
  • Easier storage and processing.
  • Scalability (e.g., binary logic scales to complex circuits).

1.2 Binary Number System

• Base-2 System: Uses digits 0 and 1.
• Weighted Positional Notation: $$(b_n{b}_{n-1}\dots b_0{)}_2=b_n\times 2^n+b_{n-1}\times 2^{n-1}+\cdots +b_0\times 2^0$$
• Example: Convert $1011_2$ to decimal: $$1\times 2^3+0\times 2^2+1\times 2^1+1\times 2^0=8+0+2+1=11_{10}$$

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2. Number System Conversions

2.1 Decimal to Binary (Fractional and Integer Parts)

Integer Part (Division by 2)

1. Divide the number by 2, record the remainder.
2. Repeat until the quotient is 0.
3. Read remainders in reverse order.

Example: Convert $13_{10}$ to binary:

13 ÷ 2 = 6 (remainder 1)
6 ÷ 2 = 3 (remainder 0)
3 ÷ 2 = 1 (remainder 1)
1 ÷ 2 = 0 (remainder 1)

Result: $1101_2$

Fractional Part (Multiplication by 2)

1. Multiply the fraction by 2, record the integer part.
2. Repeat until the fraction becomes 0 or reaches desired precision.

Example: Convert $0.625_{10}$ to binary:

0.625 × 2 = 1.25 → 1
0.25 × 2 = 0.5 → 0
0.5 × 2 = 1.0 → 1

Result: $0.101_2$

Combined Example: Convert $13.625_{10}$ to binary:

• Integer part: $1101_2$
• Fractional part: $.101_2$ Final Result: $1101.101_2$

2.2 Binary to Decimal

Use weighted summation: $$(1101.101{)}_2=1\times 2^3+1\times 2^2+0\times 2^1+1\times 2^0+1\times 2^{-1}+0\times 2^{-2}+1\times 2^{-3}$$ $$=8+4+0+1+0.5+0+0.125=13.625_{10}$$

2.3 Octal (Base-8) and Hexadecimal (Base-16) Systems

Example: Convert $101101101_2$ to octal and hexadecimal.

• Octal: Group: 1 011 011 001 → 1 3 3 1 → $1331_8$
• Hexadecimal: Group: 1011 0110 0100 → B 6 4 → $B64_{16}$

Decimal to Octal/Hex:

1. Convert decimal to binary.
2. Group into 3/4 bits and convert.

Example: Convert $75_{10}$ to hexadecimal.

• Binary: $1001011_2$ → Group: 0010 0101 → 2 5 → $25_{16}$

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3. Signed Number Representations

3.1 Signed Magnitude

• Format: Most significant bit (MSB) = 0 (positive), 1 (negative).
• Example: Represent $-6_{10}$ in 8-bit signed magnitude: $$00000110\text{ (positive 6)}\text{or}10000110\text{ (negative 6)}$$
• Drawback: Two representations for zero ($+0$ and $-0$).

3.2 1’s Complement

• Negative Numbers: Invert all bits of the positive number.
• Example: $-6_{10}$ in 8-bit 1’s complement:
  • Positive 6: $00000110$
  • Negative 6: $11111001$
• Addition/Subtraction:
  • For $A+B$, if $B$ is negative, add $A$ and $1$'s complement of $B$, then add end-around carry.
  • Example: $5+(-3)$ in 4-bit 1’s complement:
    • $5=0101$, $-3=1100$
    • Sum: $0101+1100=0001$ (end-around carry: $1$ → $0010_2=2_{10}$)
    • Error: Incorrect due to limited bits (overflow).

3.3 2’s Complement

• Negative Numbers: Add 1 to the 1’s complement.
• Example: $-6_{10}$ in 8-bit 2’s complement:
  • Positive 6: $00000110$
  • 1’s complement: $11111001$
  • 2’s complement: $11111010$
• Advantages:
  • Single representation for zero.
  • Simplified arithmetic (no end-around carry for subtraction).
• Example: $5-3$ in 4-bit 2’s complement:
  • $5=0101$, $-3=1101$
  • Sum: $0101+1101=0010_2=2_{10}$ (correct).

Comparison Table:

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4. Arithmetic Operations in Binary

4.1 Binary Addition and Subtraction

• Addition Rules:

  0 + 0 = 0
  0 + 1 = 1
  1 + 0 = 1
  1 + 1 = 0 (carry 1)
  

• Subtraction Rules:
  • Use 2’s complement for negative numbers.
  • Example: $7-5$ in 4-bit:
    • $7=0111$, $-5=1011$
    • Sum: $0111+1011=0010_2=2_{10}$ (correct).

4.2 Multiplication and Division

• Multiplication: Shift-and-add method. Example: $3\times 5$ in binary:

  0011 (3)
  × 0101 (5)
  -----
  0000 (0)
  0011 (3, shifted left 1)
  0000 (0, shifted left 2)
  

• 0011 (3, shifted left 3)

  01111 (15)

• Division: Repeated subtraction. Example: $15÷3$:
  • $15-3=12$, $12-3=9$, $9-3=6$, $6-3=3$, $3-3=0$ → 5 times.

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5. Complements in Digital Logic

5.1 9’s and 10’s Complement (BCD)

• Used in Binary-Coded Decimal (BCD) arithmetic.
• 9’s Complement: Subtract each digit from 9. Example: $9$'s complement of $123$: $876$
• 10’s Complement: Add 1 to 9’s complement. Example: $10$'s complement of $123$: $876+1=877$

Application: Simplifies subtraction in BCD systems. Example: $123-45$ using 10’s complement:

1. Find 10’s complement of $45$: $955$.
2. Add to $123$: $123+955=1078$.
3. Discard the overflow digit: $078$ → $78$.

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6. Exam Tips

Common Pitfalls

1. Forgetting End-Around Carry in 1’s Complement: Always add the carry to the LSB.
2. Incorrect Grouping in Octal/Hex Conversion: Group from the right (LSB first).
3. Sign Bit Errors: Ensure the MSB is correctly set for negative numbers in signed representations.
4. Overflow in 2’s Complement: If the result exceeds the bit limit, it wraps around (e.g., $4+(-4)=0$ in 4-bit).

High-Scoring Strategies

• Show All Steps: For conversions, write intermediate binary groupings.
• Use Tables: For signed arithmetic, list the binary representations clearly.
• Practice Complements: Master 1’s and 2’s complement for subtraction problems.
• Diagrams for BCD: Draw the logic circuit for 9’s/10’s complement generators if asked.
• Time Management: Allocate 10–15 minutes for this unit in exams (typically 10–15 marks).

Past Exam Patterns

• Direct Conversions: Expect 2–3 decimal-to-binary/octal/hex questions (5–10 marks).
• Arithmetic: 1’s/2’s complement subtraction or addition (5–8 marks).
• Complements in BCD: Design circuits for 9’s/10’s complement (8–10 marks).
• Theoretical Questions: Compare signed representations or explain overflow (3–5 marks).

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Practice Problems (TU-Style)

1. Convert $255.75_{10}$ to:
   • Binary: $11111111.11_2$
   • Octal: $377.6_8$
   • Hexadecimal: $FF.C_{16}$
2. Perform $(-15)+8$ in 8-bit 2’s complement.
   • $-15=11110001$, $8=00001000$
   • Sum: $11111001$ → $-7_{10}$ (correct).
3. Design a circuit to generate the 9’s complement of a 4-bit BCD input.
   • Use 4 NOT gates (one per bit) and 4 XOR gates with a constant 1 for each digit.
4. Explain why 2’s complement is preferred over 1’s complement for arithmetic operations.
   • Answer: Eliminates the need for end-around carry and provides a unique zero representation, simplifying hardware design.