Binary Addition CSC 103 September 17, 2007

Recap: Binary Numbers Physical representation Abstract representation Transistor Concept of “on” and “off” for physical manufacturing of computers  T/F… Abstract representation Logic: NOT, AND, OR Truth tables ANY Boolean expression can be built with transistors – wired as AND, OR or NOT

Recap: Transistors = 0 = 1 = 0 or = 1 = 1 or = 0 = 1

Logic Functions: NOT The ‘NOT’ function A A’ 0 1 1 0

Recap Logic Gate: AND Function (=1) 1 (=1) 1

Logic Gate: OR Function 1 1 1 1

Onto Addition and the Adder Circuit...

Binary Addition Add Add these numbers 1000111 1011010 0100110 0111001 0 + 0 = 0 + 1 = 1 + 0 = 1 + 1 = Add these numbers c: 1000111 1011010 0100110 0111001 s:

Binary Addition: Half Adder We need a circuit to add two bits Either bit can be ‘0’ or ‘1’ The function in the truth table is Sum = A’B + AB’  Exclusive-OR function Carry = AB

The Half-Adder and Exclusive OR Gate A’B + AB’ = Exclusive OR Typically abbreviated to XOR Simulator uses EOR A B’ A’ B A B A B | S C 0 0 | 0 0 0 1 | 1 0 1 0 | 1 0 1 1 | 0 1

Recap Logic Gates: Symbols AB, AB A+B A, A’ XOR

Summary: The Half-Adder and Exclusive OR Gate Typically abbreviated to XOR Simulator uses EOR A B

Binary Addition: Half Adder

Half-Adder  Full-Adder

The Full Adder A full adder is a circuit with three inputs (including a ‘carry-in’) and two outputs (the sum and carry-out) What is the third input? Exercise: Add 111+ 101 (carry) 1 1 1 ( ‘A’ ) 1 0 1 ( ‘B’ ) (sum) For adding two numbers, we need three inputs

The Full Adder Cascade two half-adders to get a full adder A B Cin

HW: Cascade 2 Full Adders for a 2-Bit Adder A2A1  1 1 + B2B1  + 1 0 B2 A2 B1 A1 Full Adder Full Adder Cout2 Cin2 = Cout1 Cin1 S2 S1

Summary Binary addition Concept of ‘sum’ and ‘carry’ Half adder and full adder circuits Cascading circuits to make larger ones