7.0 Arithmetic Operations in Binary

Tags: #◇Computer Science #◇CCICOMP #◇Problem Solving #◇M2

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◇Overview

Arithmetic operations (addition, subtraction, multiplication, and division) form the core of computer processing. In order for arithmetic operations to be performed by a computer, all operands must be converted to the same base (binary). When working with signed integer data, the arithmetic logic varies depending on the specific representation scheme used (e.g., 1's complement vs. 2's complement). Specifically, how a final carry-out bit is handled distinguishes 1's complement addition from 2's complement addition. Furthermore, subtraction is universally handled by negating the subtrahend (complementing it) and then performing addition.

◇Important Formulas

Basic Binary Arithmetic Rules Binary Addition:

$0 + 0 = 0$
$0 + 1 = 1$
$1 + 0 = 1$
$1 + 1 = 0 \text{ carry } 1$ Binary Subtraction:
$0 - 0 = 0$
$1 - 0 = 1$
$1 - 1 = 0$
$0 - 1 = 1 \text{ borrow } 1$ (equivalent to borrowing $2$ from the next higher significance column)

Signed Integer Addition & Subtraction 1's Complement Addition: Add the two binary strings. If there is a final carry-out bit beyond the most significant bit, add that $1$ back into the intermediate result (end-around carry). 2's Complement Addition: Add the two binary strings. If there is a final carry-out bit, simply discard it. Subtraction ($A - B$): Negate the subtrahend ($B$) to its complement equivalent ($-B$) using the appropriate complement rules, and then perform addition: $A + (-B)$.

◇Worked Examples

1. Basic Binary Addition & Subtraction Addition: $1100_2 + 1010_2$

$1100$
$+ 1010$
$= 10110_2$ (with a carry on the final bit) Subtraction: $1100_2 - 1010_2$
$1100$
$- 1010$
$= 0010_2$

2. 1's Complement Addition

Add $+6$ and $-2$ using $8$-bit 1's complement strings. $+6 = 00000110_2$ $-2 = 11111101_2$ Add them: $00000110_2 + 11111101_2 = 1 00000011_2$ (carry out is $1$) End-around carry (add $1$ to result): $00000011_2 + 1_2 = 00000100_2$ Result: $00000100_2 = +4_{10}$

3. 2's Complement Addition

Add $-6$ and $-2$ using $8$-bit 2's complement strings. $-6 = 11111010_2$ $-2 = 11111110_2$ Add them: $11111010_2 + 11111110_2 = 1 11111000_2$ (carry out is $1$) Discard the final carry out: $11111000_2$ Result: $11111000_2 = -8_{10}$

4. 2's Complement Subtraction

Perform $(+2) - (+6)$ using $8$-bit 2's complement strings. Equivalent to $(+2) + (-6)$. $+2 = 00000010_2$ $+6 = 00000110_2 \implies -6 = 11111010_2$ (Invert bits and add $1$) Add them: $00000010_2 + 11111010_2 = 11111100_2$ Result is negative. Invert and add $1$ to check value: $00000011_2 + 1 = 00000100_2 = 4$. So, $-4_{10}$.

◇References & Navigation

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• Related Notes: ◇Binary Addition, Subtraction & Carry Lookahead Adders, ◇Complements