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Chapter 6Digital Arithmetic :
Operations and Circuits
IntroductionTo understand how digital
machines (i.e., computers) perform the basic arithmetic operations.
Chapter 6 Digital Arithmetic: Operations & CircuitsINC222 Logic Theory and Digital Circuit Design
Binary Addition
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Representing Signed Numbers Sign-Magnitude System
INC222 Logic Theory and Digital Circuit Design
2’s-Complement System
Chapter 6 Digital Arithmetic: Operations & Circuits
Representing Signed Numbers Using 2’s Complement 2’s-Complement System
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
1’s-Complement Form
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
2’s-Complement Form (Negation)
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Special Case in 2’s-Complement Representation
INC222 Logic Theory and Digital Circuit Design
Range of values :
N : Number of magnitude bits
There are a total of 2N +1 different values, including zero
Chapter 6 Digital Arithmetic: Operations & Circuits
INC222 Logic Theory and Digital Circuit Design
Decimal Value Signed Binary Using 2’s Complement+ 7 = 23 - 1 0111+ 6 0110+ 5 0101+ 4 0100+ 3 0011+ 2 0010+ 1 0001
0 0000- 1 1111- 2 1110- 3 1101- 4 1100- 5 1011- 6 1010- 7 1001- 8 = -23 1000
Chapter 6 Digital Arithmetic: Operations & Circuits
Addition in the 2’s-Complement System Case Ⅰ: Two Positive Numbers
INC222 Logic Theory and Digital Circuit Design
sign bits
+13
Chapter 6 Digital Arithmetic: Operations & Circuits
Addition in the 2’s-Complement System Case Ⅱ: Positive Number and Smaller
Negative Number
INC222 Logic Theory and Digital Circuit Design
sign bits
This carry is disregarded
+5
Chapter 6 Digital Arithmetic: Operations & Circuits
Addition in the 2’s-Complement System Case Ⅲ: Positive Number and Larger
Negative Number
INC222 Logic Theory and Digital Circuit Design
sign bits
-5
Chapter 6 Digital Arithmetic: Operations & Circuits
Addition in the 2’s-Complement System Case Ⅳ: Two Negative Numbers
INC222 Logic Theory and Digital Circuit Design
sign bits
-13
This carry is disregarded
Chapter 6 Digital Arithmetic: Operations & Circuits
Addition in the 2’s-Complement System Case Ⅴ: Equal and Opposite Numbers
INC222 Logic Theory and Digital Circuit Design
sign bits
0
This carry is disregarded
Chapter 6 Digital Arithmetic: Operations & Circuits
Subtraction in the 2’s-Complement System When subtracting one binary number (the
subtrahend) from another binary number (the minuend), use the following procedure:
Negate the subtrahend.
Add this to the minuend.
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Arithmetic Overflow
INC222 Logic Theory and Digital Circuit Design
incorrect magnitudeincorrect sign
Chapter 6 Digital Arithmetic: Operations & Circuits
Multiplication of Binary Numbers
INC222 Logic Theory and Digital Circuit Design
1 0 0 11 0 1 11 0 0 1
1 0 0 10 0 0 0
1 0 0 11 1 0 0 0 1 1
multiplicand = 910
multiplier = 1110
partial products
final product = 9910
Chapter 6 Digital Arithmetic: Operations & Circuits
Multiplication of Binary Numbers
INC222 Logic Theory and Digital Circuit Design
1 0 0 1
1 0 1 1
1 0 0 11 0 0 1
0 0 0 0
1 0 0 11 1 0 0 0 1 1
multiplicand = 910
multiplier = 1110
final product = 9910
1 1 0 1 1
0 1 1 0 1 14th partial product shifted left
sum of 1st 3 partial products
3rd partial product shifted left
sum of 1st 2 partial products
2nd partial product shifted left
1st partial product
Chapter 6 Digital Arithmetic: Operations & Circuits
Multiplication in the 2’s-Complement System
If +A x +B = +(A x B)
If -A x –B = +(A x B)
If +A x –B or –A x +B = -(A x B)
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Binary Division
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
BCD Addition Sum Equals 9 or Less
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
BCD Addition Sum Greater Than 9
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
INC222 Logic Theory and Digital Circuit Design
8 2
Chapter 6 Digital Arithmetic: Operations & Circuits
INC222 Logic Theory and Digital Circuit Design
9 7
Chapter 6 Digital Arithmetic: Operations & Circuits
Hexadecimal Arithmetic Hex Addition
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Hexadecimal Arithmetic Hex Subtraction
2’s Complement of a hex number
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Ex. 2’s complement of 3A5 Add to the minuend
INC222 Logic Theory and Digital Circuit Design
Disregard carry
Chapter 6 Digital Arithmetic: Operations & Circuits
Hex Representation of Signed Numbers
INC222 Logic Theory and Digital Circuit Design
Hex Address Stored Binary Data Hex Value Decimal Value
4000 00111010 3A +58
4001 11100101 E5 -2７
4002 01010111 57 +87
4003 10000000 80 -128
Chapter 6 Digital Arithmetic: Operations & Circuits
Arithmetic Circuit
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Parallel Binary Adder
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
5-bits parallel adder circuit
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Design of a Full Adder
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Full Adder
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Complete Parallel Adder with Registers
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Integrated-Circuit Parallel Adder
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Cascading Parallel Adders
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
2’s-Complement System
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Parallel adder used to subtract
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Combined Addition & Subtraction
Addition : ADD = 1 & SUB = 0note: C0 = 0
Subtraction : ADD = 0 & SUB = 1note: C0 = 1
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
BCD Adder
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Correction invalid code
INC222 Logic Theory and Digital Circuit Design
S4S3S2S1S0 can range anywhere from 00000 to 10010
Chapter 6 Digital Arithmetic: Operations & Circuits
INC222 Logic Theory and Digital Circuit Design
01010 to 10010 are the invalid codesS4 S3 S2 S1 S0 Z
(0-8) 00 1 0 0 1 (9) 00 1 0 1 0 (10) 10 1 0 1 1 (11) 10 1 1 0 0 (12) 10 1 1 0 1 (13) 10 1 1 1 0 (14) 10 1 1 1 1 (15) 11 0 0 0 0 (16) 11 0 0 0 1 (17) 11 0 0 1 0 (18) 11 0 0 1 1 (19) x
(20-31) x
Chapter 6 Digital Arithmetic: Operations & Circuits
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Cascading BCD Adders
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
ALU Integrate Circuits
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
Expanding the ALU
INC222 Logic Theory and Digital Circuit Design Chapter 6 Digital Arithmetic: Operations & Circuits
