Two’s Complement

Overview

Two’s complement is the standard method for representing signed integers in binary. It eliminates the dual-zero problem of sign-magnitude and simplifies arithmetic circuits.

Why Two’s Complement?

How Two’s Complement Works

For an n-bit number:

• Positive numbers: Same as unsigned binary
• Negative numbers: Invert all bits and add 1 (or equivalently: 2^n - |number|)

Converting to Negative

+5  = 00000101
Invert:  11111010
Add 1:   11111011  = -5

Converting from Negative

-5  = 11111011
Invert:  00000100
Add 1:   00000101  = +5

Range

For n-bit two’s complement:

• Minimum: -2^(n-1)
• Maximum: 2^(n-1) - 1

Quick Tricks

Sign Detection

MSB (Most Significant Bit) = sign bit:

• 0 → positive (or zero)
• 1 → negative

Negation Shortcut

Starting from the right, copy all bits up to and including the first 1, then invert the rest:

-5 in 8-bit:
+5  = 00000101
                ↑ first 1 from right
Copy: 00000101  (up to first 1)
Invert rest: 11111011
Result: 11111011 = -5 ✓

Sign Extension

Extend a negative number to more bits by copying the sign bit:

-5 in 4 bits:  1011
-5 in 8 bits:  11111011
-5 in 16 bits: 1111111111111011

Two’s Complement Arithmetic

Addition

Just add normally (ignore overflow):

  00000101  (+5)
+ 11111011  (-5)
----------
1 00000000  (0, carry out is discarded)

Subtraction

Subtract by adding the negation:

7 - 5 = 7 + (-5)

  00000111  (+7)
+ 11111011  (-5)
----------
1 00000010  (+2, carry discarded)

Overflow Detection

Overflow occurs when:

• Adding two positives → negative
• Adding two negatives → positive

flowchart TD
    A[Add A + B] --> B{Same sign?}
    B -->|No| C[No overflow possible]
    B -->|Yes| D{Result sign different?}
    D -->|No| E[No overflow]
    D -->|Yes| F[OVERFLOW!]

Example (8-bit):

  01111111  (+127)
+ 00000001  (+1)
----------
  10000000  (-128) ← OVERFLOW! (positive + positive = negative)

Two’s Complement vs Others

Interview Questions

1. Q: Why is two’s complement preferred over sign-magnitude? A: 1) Single representation of zero. 2) Addition/subtraction work without special cases. 3) One extra negative number (-128 in 8-bit). 4) Simpler hardware (no special subtraction circuit needed).

2. Q: What is the two’s complement of 5 in 8 bits? A: +5 = 00000101. Invert: 11111010. Add 1: 11111011. So -5 = 11111011.

3. Q: How do you detect overflow in two’s complement addition? A: Overflow occurs when adding two numbers of the same sign produces a result of different sign. Check: if both operands have the same MSB but the result has a different MSB, overflow occurred.

4. Q: What is the range of 8-bit two’s complement? A: -128 to +127. The asymmetry comes from zero taking one of the positive representations (00000000). So there’s one more negative number than positive.

5. Q: How does sign extension work? A: To extend a two’s complement number to more bits, copy the sign bit (MSB) into all new high-order bits. This preserves the value: -5 in 4 bits (1011) becomes -5 in 8 bits (11111011).

6. Q: What’s the two’s complement of 0? A: 00000000. Invert: 11111111. Add 1: 100000000 (9 bits). Discard carry: 00000000. Zero is its own negation.

Common Mistakes

• Forgetting that negation = invert + 1 (not just invert)
• Not understanding that the range is asymmetric (-128 to +127)
• Confusing sign extension (copy MSB) with zero extension (add zeros)
• Not detecting overflow correctly (must check same-sign inputs)
• Thinking -128 can be negated (it can’t — it’s the most negative value)

Summary

Two’s complement is the standard signed integer representation. Negative numbers are formed by inverting bits and adding 1. It simplifies arithmetic (addition/subtraction use the same circuit), has a single zero, and provides one extra negative value. Overflow detection checks for same-sign inputs producing different-sign results.

Cross-References

• Number Systems Overview
• Binary — Unsigned binary
• Hexadecimal — Compact notation
• IEEE 754 — Floating point representation

Cross References

• Binary
• ALU
• IEEE 754