Binary Addition Circuits in Computing
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Programming for GCSE
Topic 9.2: Circuits for
Adding
T
eaching
L
ondon
C
omputing
William Marsh
School of Electronic Engineering and Computer Science
Queen Mary University of London
Aims
•
Show how computers are built from logic
gates
•
Circuit for Adding
•
… two inputs
•
… three inputs – one column
•
… many columns
Key Idea:
•
Represent numbers as binary digits
•
Digits as logic levels
H
ALF
A
DDER
Half Adder
•
Simplest circuit for binary addition
•
input: two bits A and B
•
output: sum S and carry C
•
Sums
?
circuit
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 carry 1
Half Adder – Truth Table
•
Binary addition – truth table
•
input: two bits A and B
•
output: sum S and carry C
Quiz: Determine the
formula for S and C
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0
A B C S
Half Adder – Formula
•
Simplest circuit for binary addition
•
input: two bits A and B
•
output: sum S and carry C
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0
A B C S
Quiz: draw the circuit
Half Adder – Circuit
•
Simplest circuit for binary addition
•
input: two bits A and B
•
output: sum S and carry C
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0
A B C S
F
ULL
A
DDER
One Columns of a Binary Addition
Full Adder – One Column
•
Each digit (column) of
binary add has 3 inputs
•
A, B and carry C
in
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1
A B C
in
C
out
S
A
Full-+
C
in
B
S
C
out
A
B
S
C
out
C
in
Full-Adder from 2 Half
Adders
•
Step 1: add A + B
•
Step 2: add carry to result
•
Step 3: carry
A
B
S’
C’
Half-+
S
Half-+
C
in
C
out
C’’
Ripple Adder
•
Add each bit, carry from previous bit
A
Full-+
C
in
B
S
C
out
Ripple Adder
•
Add each bit, carry from previous bit
A
0
Full-+
C
in
= 0
B
0
S
0
Full-+
Full-+
C
out
S
1
S
2
A
1
B
1
A
2
B
2
S
YLLABUS
Syllabus – Binary
•
GCSE (OCR)
•
Logic circuits: and, or , not
•
Truth tables
•
Writing boolean expressions
•
AS/A2 (AQA)
•
(AS) More boolean algebra
•
(AS) More gates
Joined up view?
How to make sense
of logic unless used
e.g. adder circuit.
binary
truth table
circuit
Summary
•
Show how logic circuits build a computer
•
Binary digits become logic inputs
•
Circuits operate on numbers
•
Adder stages
•
One column, no carry
•
One column, with carry
•
Many columns
Explore the concepts of half adders, full adders, and binary addition circuits used in computer systems. Learn how binary digits are represented as logic levels to perform addition operations. Dive into the working principles of half adders, truth tables, formulas, and circuit diagrams for binary addition. Gain insights into the role of logic gates in building computer systems from basics to complex operations.
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Presentation Transcript
TeachingLondon Computing Programming for GCSE Topic 9.2: Circuits for Adding William Marsh School of Electronic Engineering and Computer Science Queen Mary University of London
Aims Show how computers are built from logic gates Circuit for Adding two inputs three inputs one column many columns Key Idea: Represent numbers as binary digits Digits as logic levels
Half Adder Simplest circuit for binary addition input: two bits A and B output: sum S and carry C Sums ? circuit A S Half-+ B C 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 carry 1 Example: 0 ? Half-+ 1 ?
Half Adder Truth Table Binary addition truth table input: two bits A and B output: sum S and carry C A B C S 0 0 0 0 Quiz: Determine the formula for S and C 0 1 0 1 1 0 0 1 1 1 1 0
Half Adder Formula Simplest circuit for binary addition input: two bits A and B output: sum S and carry C A B C S 0 0 0 0 = + B . B . S A A 0 1 0 1 = B . C A 1 0 0 1 Quiz: draw the circuit 1 1 1 0
Half Adder Circuit Simplest circuit for binary addition input: two bits A and B output: sum S and carry C C A B C S 0 0 0 0 A 0 1 0 1 S 1 0 0 1 B 1 1 1 0
FULL ADDER One Columns of a Binary Addition
Full Adder One Column Each digit (column) of binary add has 3 inputs A, B and carry Cin A B Cin Cout S 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 CoutCin A B S Cin S A B Full-+ Cout
Full-Adder from 2 Half Adders Step 1: add A + B Step 2: add carry to result Step 3: carry S S A Half-+ Half-+ C C B Cout Cin
Ripple Adder Add each bit, carry from previous bit Cin S A B Full-+ Cout
Ripple Adder Add each bit, carry from previous bit Cin = 0 S0 A0 B0 Full-+ S1 A1 B1 Full-+ S2 A2 B2 Full-+ Cout
Syllabus Binary GCSE (OCR) Logic circuits: and, or , not Truth tables Writing boolean expressions Joined up view? How to make sense of logic unless used e.g. adder circuit. binary truth table circuit AS/A2 (AQA) (AS) More boolean algebra (AS) More gates
Summary Show how logic circuits build a computer Binary digits become logic inputs Circuits operate on numbers Adder stages One column, no carry One column, with carry Many columns
