Binary Addition

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Binary Addition

• CSC 103
• September 17, 2007

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Recap Binary Numbers

• Physical 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

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Recap Transistors
0
1
0 or 1
1 or 0
1
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Logic Functions NOT

• The NOT function
• A A
• 0 1
• 1 0

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Recap Logic Gate AND Function
(1)
1
(1)
0
0
0
1
0
0
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Logic Gate OR Function
1
0
1
1
0
1
0
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• Onto Addition and
• the Adder Circuit...

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Binary Addition

• Add
• 0 0
• 0 1
• 1 0
• 1 1
• Add these numbers
• c
• 1000111 1011010
• 0100110 0111001
• s

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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 AB AB ? Exclusive-OR function
• Carry AB

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The Half-Adder and Exclusive OR Gate

• AB AB Exclusive OR
• Typically abbreviated to XOR
• Simulator uses EOR

A B 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
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Recap Logic Gates Symbols
AB, A?B AB
A, A
XOR
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Summary The Half-Adder and Exclusive OR Gate

• Exclusive OR
• Typically abbreviated to XOR
• Simulator uses EOR

A B
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Binary Addition Half Adder
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Half-Adder ? Full-Adder
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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

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The Full Adder

• Cascade two half-adders to get a full adder

A B Cin
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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
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Summary

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