Adders, multiplier, and ALU

01

Binary numbers

An unsigned n-bit value ranges from 0 to 2ⁿ−1. Two's complement ranges from −2ⁿ⁻¹ to 2ⁿ⁻¹−1. To form −x, invert positive x and add 1.

02

Half-adder

A half-adder adds A and B: Sum=A XOR B and Carry=A AND B. It has no carry-in.

Build it yourself: Half-adder

03

Full-adder

A full-adder adds A, B, and Cin: Sum=A XOR B XOR Cin; Cout=(A AND B) OR (Cin AND (A XOR B)). It can be built from two half-adders and an OR gate.

Build it yourself: Full-adder

04

Ripple-carry adder

In a multi-bit adder, each lower Cout feeds the next Cin. Carry ripple forms a long critical path; delay grows roughly linearly with width.

Build it yourself: 2-bit ripple-carry adder

05

Binary multiplier

Each partial product is an AND of one A bit and one B bit. Shifted partial products are added. Multiplying two n-bit unsigned values may require 2n bits.

Build it yourself: 2×2 multiplier

06

ALU and subtraction

An ALU computes logic and arithmetic candidates and a MUX selects one. Subtraction reuses the adder: A−B=A+NOT(B)+1.

Build it yourself: 1-bit ALU · 1-bit comparator