Two's Complement Signed Integers

Two's complement is the universal method used by digital computers to represent both positive and negative integers in binary.

Learning Objectives

• Understand the concept and need for two's complement signed representation.
• Convert between denary integers and 8-bit two's complement binary across the range $-128$ to $+127$.
• Explain the engineering advantages of two's complement over sign-and-magnitude representation.

Core Syllabus Theory

1. Why Two's Complement is Needed

Unsigned binary representation can only store positive integers (e.g. 8 bits represent $0$ to $255$). Computer processors require a reliable system to represent negative numbers as well.

An older approach called sign-and-magnitude used the leftmost bit as a sign indicator ($0 = +, 1 = -$). However, sign-and-magnitude has two major design flaws:

1. Two representations for zero: $+0$ ($00000000$) and $-0$ ($10000000$), wasting code space and complicating logic equality comparisons.
2. Complex hardware: Standard binary addition hardware fails when adding negative numbers; dedicated subtraction circuitry is required.

Two's complement solves both problems:

• There is only one representation of zero ($00000000$).
• Subtraction can be performed using standard addition circuitry: $A - B = A + (-B)$.

2. Positional Place Values for 8-Bit Two's Complement

In an 8-bit two's complement register, the Most Significant Bit (MSB) carries a negative weight of $-128$. All other positions carry positive binary weights:

• Positive integers (MSB = 0): Range from $00000000_2 \ (0_{10})$ to $01111111_2 \ (+127_{10})$.
• Negative integers (MSB = 1): Range from $11111111_2 \ (-1_{10})$ down to $10000000_2 \ (-128_{10})$.
• Total range: $-128$ to $+127$ ($256$ distinct values).

3. Converting Negative Denary to 8-Bit Two's Complement

To convert a negative denary number (e.g. $-38$) to 8-bit two's complement, follow the Invert and Add 1 algorithm:

1. Step 1: Write the positive binary number in full 8 bits:

1. Step 2: Invert all bits ($0 \to 1$ and $1 \to 0$) to obtain One's Complement:

1. Step 3: Add 1 to the inverted binary string:

4. Converting Two's Complement Binary to Denary

Method A: Direct Positional Place Value Addition

Evaluate $11110100_2$:

Method B: Invert and Add 1 (Magnitude Retrieval)

1. Invert $11110100 \to 00001011$.
2. Add $1$: $00001011 + 1 = 00001100_2 = 12_{10}$.
3. Attach the negative sign: $-12_{10}$.

Exam Tips & Common Pitfalls

Note: Always Write 8 Full Bits: When converting positive numbers, never omit leading zeros before inverting. Inverting $100110$ is completely wrong; you must invert the full 8-bit representation $00100110$. Checking Negative One: $-1$ in two's complement is always represented by all 1s ($11111111_2$).

Key Terminology Glossary

• Two's Complement: A method of encoding signed integers where the MSB has a negative place value.
• One's Complement: The intermediate binary value obtained by inverting all bits of a binary number.
• Sign Bit: The leftmost bit (MSB), where 0 indicates positive and 1 indicates negative.

Written practice and self-review

Explain two distinct reasons why computer processors use two's complement representation rather than sign-and-magnitude representation to store signed integers in hardware.

Write an explanation before checking the theory. Check that each claim answers the question, uses the correct concept and explains why it applies. Discuss uncertain points with a teacher. The structured exercise below checks specific facts; it does not automatically mark this explanation.