Scribd
Upload
200 views29 pages

Computer Organization and Assembly Language: Lecture 5 & 6 Computer Arithmetic Integer Representation, Integer Arithmetic

This document discusses computer arithmetic and integer representation. It covers: 1) How integers are represented in binary using sign-magnitude and two's complement representations. Two's complement is commonly used as it allows for simple addition, subtraction, and negation operations. 2) Integer arithmetic operations like addition, subtraction, multiplication and division in two's complement. Addition and subtraction are performed using basic binary addition and subtraction with checks for overflow. Multiplication and division are more complex and involve calculating partial products and remainders. 3) Key aspects of two's complement representation like the valid value ranges for different bit lengths, converting between lengths, and handling overflow during arithmetic operations.

Uploaded by

Darwin Vargas
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
200 views29 pages

Computer Organization and Assembly Language: Lecture 5 & 6 Computer Arithmetic Integer Representation, Integer Arithmetic

This document discusses computer arithmetic and integer representation. It covers: 1) How integers are represented in binary using sign-magnitude and two's complement representations. Two's complement is commonly used as it allows for simple addition, subtraction, and negation operations. 2) Integer arithmetic operations like addition, subtraction, multiplication and division in two's complement. Addition and subtraction are performed using basic binary addition and subtraction with checks for overflow. Multiplication and division are more complex and involve calculating partial products and remainders. 3) Key aspects of two's complement representation like the valid value ranges for different bit lengths, converting between lengths, and handling overflow during arithmetic operations.

Uploaded by

Darwin Vargas
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 29

COMPUTER ORGANIZATION

AND ASSEMBLY LANGUAGE

Lecture 5 & 6
Computer Arithmetic
Integer Representation, Integer Arithmetic
Course Instructor: Engr. Aisha Danish
Arithmetic & Logic Unit
 Does the calculations
 Everything else in the computer is there to service
this unit
 Handles integers
 May handle floating point (real) numbers
 May be separate FPU (maths co-processor)
 May be on chip separate FPU (486DX +)
ALU Inputs and Outputs
Integer Representation
 Only have 0 & 1 to represent everything
 Positive numbers stored in binary
 e.g. 41=00101001
 No minus sign
 No period
 Sign-Magnitude
 Two’s compliment
Sign-Magnitude
 The simplest form of representation that employs a
sign bit is the sign-magnitude representation
 In an n-bit word, the rightmost n-1 bits hold the
magnitude of the integer
 +18 = 00010010
 -18 = 10010010
Sign-Magnitude
 Problems
 Need to consider both sign and magnitude in arithmetic
 Two representations of zero (+0 and -0)

 Inconvenient because it is slightly more difficult to test


for 0 (an operation performed frequently on
computers) than if there were a single representation
 Sign-magnitude representation is rarely used in
implementing the integer portion of the ALU
 Instead, the most common scheme is twos complement
representation
Two’s Compliment
 Like sign magnitude, twos complement
representation uses the most significant bit as a sign
bit, making it easy to test whether an integer is
positive or negative
 It differs from the use of the sign-magnitude
representation in the way that the other bits are
interpreted
Characteristics of Twos Complement
Representation and Arithmetic
Two’s Compliment
 +3 = 00000011
 +2 = 00000010
 +1 = 00000001
 +0 = 00000000
 -1 = 11111111
 -2 = 11111110
 -3 = 11111101
Alternative Representations for 4-Bit
Integers
Benefits
 One representation of zero
 Arithmetic works easily
 Negating is fairly easy
3 = 00000011
 Boolean complement gives 11111100
 Add 1 to LSB 11111101
Negation Special Case 1
 0= 00000000
 Bitwise not 11111111
 Add 1 to LSB +1
 Result 1 00000000
 Overflow is ignored, so:
 -0=0
Negation Special Case 2
 128 = 10000000
 bitwise not 01111111
 Add 1 to LSB +1
 Result 10000000
 So:
 -128 = 128
 Monitor MSB (sign bit)
 It should change during negation
Range of Numbers
 8 bit 2s compliment
 +127 = 01111111 = 27 -1
 -128 = 10000000 = -27
 16 bit 2s compliment
 +32767 = 011111111 11111111 = 2 15 - 1
 -32768 = 100000000 00000000 = -215
Conversion Between Lengths
 Positive number pack with leading zeros
 +18 = 00010010
 +18 = 00000000 00010010
 Negative numbers pack with leading ones
 -18 = 10010010
 -18 = 11111111 10010010
 i.e. pack with MSB (sign bit)
Addition and Subtraction
 Normal binary addition
 Monitor sign bit for overflow

 Take twos compliment of substahend and add to


minuend
 i.e. a - b = a + (-b)

 So we only need addition and complement circuits


Overflow
 For unsigned integers, overflow occurs when there is
a carry out of the msb.
 1000 (8) +1001 (9) ----------- 1 0001 (1)
 For 2's complement integers, overflow occurs when
the signs of the addends are the same, and the sign
of the result is different
 0011 (3) + 0110 (6) ---------- 1001 (-7)
 (note that a correct answer would be 9, but 9
cannot be represented in 4-bit 2's complement)
Hardware for Addition and Subtraction
Problems
 Represent the following decimal numbers in both
binary sign/magnitude and twos complement using
16 bits:
I. +512
II. -29
 Represent the following twos complement values in
decimal:
I. 1101011
II. 0101101.
Problem
 Assume numbers are represented in 8-bit twos
complement representation. Show the calculation of
the following:
I. 6+13
II. -6+13
III. 6-13
IV. -6-13
Problem
 Find the following differences using twos complement
arithmetic:
Multiplication
 Complex
 Work out partial product for each digit
 Take care with place value (column)
 Add partial products
Multiplication Example
 1011 Multiplicand (11 dec)
 x 1101 Multiplier (13 dec)
 1011 Partial products
 0000 Note: if multiplier bit is 1 copy
 1011 multiplicand (place value)
 1011 otherwise zero
 10001111 Product (143 dec)
 Note: need double length result
Unsigned Binary Multiplication
Execution of Example
Flowchart for Unsigned Binary
Multiplication
Division
 More complex than multiplication
 Negative numbers are really bad!
 Based on long division
Division of Unsigned Binary Integers

00001101 Quotient

Divisor 1011 10010011 Dividend


1011
001110
Partial
1011
Remainders
001111
1011
Remainder
100
Flowchart for Unsigned Binary Division

You might also like