Basic Concepts

Computer Organization and Assembly Language

Lec#2

Computer Science Department

National University of Computer and Emerging Sciences Islamabad

Basic Concepts Computer Organization and Assembly Language – NUCES Ameen Chilwan 1
 Next …
 Welcome to COAL 2020
 Assembly-, Machine-, and High-Level Languages
 Assembly Language Programming Tools
 Programmer’s View of a Computer System
 Data Representation

Basic Concepts Computer Organization and Assembly Language – NUCES Ameen Chilwan 2
 Data Representation
 Binary Numbers
 Hexadecimal Numbers
 Base Conversions
 Integer Storage Sizes
 Binary and Hexadecimal Addition
 Signed Integers and 2's Complement Notation
 Binary and Hexadecimal Subtraction
 Carry and Overflow
 Character Storage

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 Binary Numbers

 Digits are 1 and 0

1 = true
0 = false
 MSB – most significant bit
 LSB – least significant bit
 Bit numbering:
MSB LSB
1011001010011100
15 0

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 Binary Numbers
1 1 1 1 1 1 1 1
 Each digit (bit) is either 1 or 0 27 26 25 24 23 22 21 20

 Each bit represents a power of 2:

Every binary
number is a
sum of powers
of 2

Basic Concepts Computer Organization and Assembly Language – NUCES Ameen Chilwan 5
 Converting Binary to Decimal

Weighted positional notation shows how to calculate the

decimal value of each binary bit:
Decimal = (dn-1  2n-1) + (dn-2  2n-2) + ... + (d1  21) + (d0  20)
d = binary digit

binary 00001001 = decimal 9:

(1  23) + (1  20) = 9

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 Convert Unsigned Decimal to Binary

 Repeatedly divide the decimal integer by 2. Each

remainder is a binary digit in the translated value:

least significant bit

most significant bit

stop when
37 = 100101 quotient is zero
Basic Concepts Computer Organization and Assembly Language – NUCES Ameen Chilwan 7
 Hexadecimal Integers
Binary values are represented in hexadecimal.

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 Converting Binary to Hexadecimal
• Each hexadecimal digit corresponds to 4 binary bits.
• Example: Translate the binary integer
000101101010011110010100 to hexadecimal:

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 Converting Hexadecimal to Decimal
 Multiply each digit by its corresponding power of 16:
 Decimal = (d3  163) + (d2  162) + (d1  161) +
(d0  160)
 d = hexadecimal digit
 Examples:
Hex 1234 = (1  163) + (2  162) + (3 
161) + (4  160) =
 Decimal 4,660
Hex 3BA4 = (3  163) + (11 * 162) + (10 
161) + (4  160) =
 Decimal 15,268
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Converting Decimal to Hexadecimal
 Repeatedly divide the decimal integer by 16. Each
remainder is a hex digit in the translated value:

least significant digit

most significant digit

stop when
quotient is zero

Decimal 422 = 1A6 hexadecimal

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 Integer Storage Sizes
byte 8

word 16
Standard sizes: doubleword 32

quadword 64

What is the largest unsigned integer that may be stored in 20 bits?

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 Binary Addition
 Start with the least significant bit (rightmost bit)
 Add each pair of bits
 Include the carry in the addition, if present

carry: 1

0 0 0 0 0 1 0 0 (4)

+ 0 0 0 0 0 1 1 1 (7)

0 0 0 0 1 0 1 1 (11)
bit position: 7 6 5 4 3 2 1 0

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 Hexadecimal Addition
 Divide the sum of two digits by the number base (16).
The quotient becomes the carry value, and the
remainder is the sum digit.
1 1
36 28 28 6A
42 45 58 4B
78 6D 80 B5

21 / 16 = 1, remainder 5

Important skill: Programmers frequently add and subtract the

addresses of variables and instructions.

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 Signed Integers
 Several ways to represent a signed number
 Sign-Magnitude
 Biased
 1's complement
 2's complement
 Divide the range of values into 2 equal parts
 First part corresponds to the positive numbers (≥ 0)
 Second part correspond to the negative numbers (<
0)
 Focus will be on the 2's complement representation
 Has many advantages over other representations
 Used widely in processors to represent signed
integers

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Representation of negative numbers
 Is a representation of negative numbers possible?
 Unfortunately:
 you can not just stick a negative sign in front of a binary
number. (it does not work like that)
 There are three methods used to represent
negative numbers.
 Signed magnitude notation
 Excess notation
 Two’s complement notation

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 Signed Magnitude Representation
 Unsigned: - and + are the same.
 In signed magnitude
 the left-most bit represents the sign of the integer.
– 0 for positive numbers.
– 1 for negative numbers.
 The remaining bits represent to magnitude of the
numbers.

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 Example

 Suppose 10011101 is a signed magnitude representation.

 The sign bit is 1, then the number represented is negative

position 7 6 5 4 3 2 1 0
Bit pattern 1 0 0 1 1 1 0 1
contribution 24 23 22 20
-

 The magnitude is 0011101 with a value 24+23+22+20= 29

 Then the number represented by 10011101 is –29.

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 Exercise 1

1. 3710 has 0010 0101 in signed magnitude notation. Find

the signed magnitude of –3710 ?

2. Using the signed magnitude notation find the 8-bit

binary representation of the decimal value 2410 and -
2410.

3. Find the signed magnitude of –63 using 8-bit binary

sequence?

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 Disadvantage of Signed Magnitude
 Addition and subtractions are difficult.
 Signs and magnitude, both have to carry out the
required operation.
 They are two representations of 0
 00000000 = + 010
 10000000 = - 010
 To test if a number is 0 or not, the CPU will need to see
whether it is 00000000 or 10000000.
 0 is always performed in programs.
– Therefore, having two representations of 0 is inconvenient.

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 Signed-Summary
 In signed magnitude notation,
 The most significant bit is used to represent the sign.
 1 represents negative numbers
 0 represents positive numbers.
 The unsigned value of the remaining bits represent The
magnitude.
 Advantages:
 Represents positive and negative numbers
 Disadvantages:
 two representations of zero,
 Arithmetic operations are difficult.

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 Excess Representation
(cont.)
– For a given fixed number of bits Excess
Number to
the range is remapped such that Value Notation
store
roughly half the numbers are 0 0000 -8
negative and half are positive. 1 0001 -7
2 0010 -6
3 0011 -5
Example: (as left) 4 0100 -4
5 0101 -3
Excess 8 -> notation for 4 bit numbers 6 0110 -2
7 0111 -1
 Binary value = 8 + excess-8 value 8 1000 0
9 1001 1
 MSB can be used as a sign bit, but 10 1010 2
 If MSB =1, positive number 11 1011 3
 If MSB =0, negative number 12 1100 4
 Excess Representation is also called 13 1101 5
bias 14 1110 6
15 1111 7

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 Biased-Excess-N
 Uses a pre-specified number N as a biasing value.
 A value is represented by the unsigned number which is
N greater than the intended value.
0 is represented by N,
 −N is represented by the all-zeros bit pattern.

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 Two's Complement Representation
 Positive numbers
 Signed value = Unsigned value 8-bit Binary Unsigned Signed
 Negative numbers value value value
00000000 0 0
 UnSigned value = Signed value + 2n
00000001 1 +1
 n = number of bits
00000010 2 +2
 Negative weight for MSB
... ... ...
 Another way to obtain the signed
value is to assign a negative weight 01111110 126 +126
to most-significant bit 01111111 127 +127
10000000 128 -128
1 0 1 1 0 1 0 0
10000001 129 -127
-128 64 32 16 8 4 2 1
... ... ...
11111110 254 -2
= -128 + 32 + 16 + 4 = -76
11111111 255 -1
Basic Concepts Computer Organization and Assembly Language – NUCES Ameen Chilwan 24
 Forming the Two's Complement
starting value 00100100 = +36

step1: reverse the bits (1's complement) 11011011

step 2: add 1 to the value from step 1 + 1

sum = 2's complement representation 11011100 = -36

Sum of an integer and its 2's complement must be zero:
00100100 + 11011100 = 00000000 (8-bit sum)  Ignore Carry

The easiest way to obtain the 2's complement of a

binary number is by starting at the LSB, leaving all the
0s unchanged, look for the first occurrence of a 1. Leave
this 1 unchanged and complement all the bits after it.

Basic Concepts Computer Organization and Assembly Language – NUCES Ameen Chilwan 25
 Sign Bit
Highest bit indicates the sign. 1 = negative, 0 = positive
sign bit

1 1 1 1 0 1 1 0
Negative

0 0 0 0 1 0 1 0 Positive

If highest digit of a hexadecimal is > 7, the value is negative

Examples: 8A and C5 are negative bytes
A21F and 9D03 are negative words
B1C42A00 is a negative double-word
Basic Concepts Computer Organization and Assembly Language – NUCES Ameen Chilwan 26
 Sign Extension
Step 1: Move the number into the lower-significant bits
Step 2: Fill all the remaining higher bits with the sign bit
 This will ensure that both magnitude and sign are correct
 Examples
Sign-Extend 10110011 to 16 bits
10110011 = -77 11111111 10110011 = -77

Sign-Extend 01100010 to 16 bits

01100010 = +98 00000000 01100010 = +98

 Infinite 0s can be added to the left of a positive number

 Infinite 1s can be added to the left of a negative number
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 Two's Complement of a Hexadecimal
 To form the two's complement of a hexadecimal
Subtract each hexadecimal digit from 15
Add 1
 Examples:
2's complement of 6A3D = 95C2 + 1 = 95C3
2's complement of 92F0 = 6D0F + 1 = 6D10
2's complement of FFFF = 0000 + 1 = 0001
 No need to convert hexadecimal to binary

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 Binary Subtraction
 When subtracting A – B, convert B to its 2's complement
 Add A to (–B)
00001100 00001100
– 00000010 + 1 1 1 1 1 1 1 0 (2's complement)
0 0 0 0 1 0 1 0 (same result)
 Carry is ignored, because
 Negative number is sign-extended with 1's
 You can imagine infinite 1's to the left of a negative number
 Adding the carry to the extended 1's produces extended zeros

Practice: Subtract 00100101 from 01101001.

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 Hexadecimal Subtraction
 When a borrow is required from the digit to the left,
add 16 (decimal) to the current digit's value
16 + 5 = 21

-1 11
C675 C675
- +
A247 5DB9 (2's complement)
242E 242E (same result)

 Last Carry is ignored

Practice: The address of var1 is 00400B20. The address of the next
variable after var1 is 0040A06C. How many bytes are used by var1?

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 Ranges of Signed Integers
The unsigned range is divided into two signed ranges for positive
and negative numbers

Practice: What is the range of signed values that may be stored in 20 bits?

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 Carry and Overflow
 Carry is important when …
 Adding or subtracting unsigned integers
 Indicates that the unsigned sum is out of range
 Either < 0 or >maximum unsigned n-bit value
 Overflow is important when …
 Adding or subtracting signed integers
 Indicates that the signed sum is out of range
 Overflow occurs when
 Adding two positive numbers and the sum is negative
 Adding two negative numbers and the sum is positive
 Can happen because of the fixed number of sum bits

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 Carry and Overflow Examples
 We can have carry without overflow and vice-versa
 Four cases are possible
1 1 1 1 1 1

0 0 0 0 1 1 1 1 15 0 0 0 0 1 1 1 1 15
+ +
0 0 0 0 1 0 0 0 8 1 1 1 1 1 0 0 0 248 (-8)

0 0 0 1 0 1 1 1 23 0 0 0 0 0 1 1 1 263
(+7)
Carry = 0 Overflow = 0 Carry = 1 Overflow = 0

1 1 1 1

0 1 0 0 1 1 1 1 79 1 1 0 1 1 0 1 0 218 (-38)
+ +
0 1 0 0 0 0 0 0 64 1 0 0 1 1 1 0 1 157 (-99)

1 0 0 0 1 1 1 1 143 0 1 1 1 0 1 1 1 375
(-113) (-137)
Carry = 0 Overflow = 1 Carry = 1 Overflow = 1
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 Character Storage
 Character sets
 Standard ASCII: 7-bit character codes (0 – 127)
 Extended ASCII: 8-bit character codes (0 – 255)
 Unicode: 16-bit character codes (0 – 65,535)
 Unicode standard represents a universal character
set
 Defines codes for characters used in all major languages
 Used in Windows-XP: each character is encoded as 16 bits
 UTF-8: variable-length encoding used in HTML
 Encodes all Unicode characters
 Uses 1 byte for ASCII, but multiple bytes for other characters
 Null-terminated String
 Array of characters followed by a NULL character

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 Printable ASCII Codes
0 1 2 3 4 5 6 7 8 9 A B C D E F
2 space ! " # $ % & ' ( ) * + , - . /
3 0 1 2 3 4 5 6 7 8 9 : ; < = > ?
4 @ A B C D E F G H I J K L M N O
5 P Q R S T U V W X Y Z [ \ ] ^ _
6 ` a b c d e f g h i j k l m n o
7 p q r s t u v w x y z { | } ~ DEL

 Examples:
 ASCII code for space character = 20 (hex) = 32 (decimal)
 ASCII code for 'L' = 4C (hex) = 76 (decimal)
 ASCII code for 'a' = 61 (hex) = 97 (decimal)
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 Control Characters
 The first 32 characters of ASCII table are used for control
 Control character codes = 00 to 1F (hex)
 Not shown in previous slide
 Examples of Control Characters
 Character 0 is the NULL character  used to terminate a string
 Character 9 is the Horizontal Tab (HT) character
 Character 0A (hex) = 10 (decimal) is the Line Feed (LF)
 Character 0D (hex) = 13 (decimal) is the Carriage Return (CR)
 The LF and CR characters are used together
 They advance the cursor to the beginning of next line
 One control character appears at end of ASCII table
 Character 7F (hex) is the Delete (DEL) character

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 Terminology for Data Representation
 Binary Integer
Integer stored in memory in its binary format
Ready to be used in binary calculations
 ASCII Digit String
A string of ASCII digits, such as "123"
 ASCII binary
String of binary digits: "01010101"
 ASCII decimal
String of decimal digits: "6517"
 ASCII hexadecimal
String of hexadecimal digits: "9C7B"
Basic Concepts Computer Organization and Assembly Language – NUCES Ameen Chilwan 37
 Summary
 Assembly language helps you learn how software is constructed at
the lowest levels
 Assembly language has a one-to-one relationship with machine
language
 An assembler is a program that converts assembly language
programs into machine language
 A linker combines individual files created by an assembler into a
single executable file
 A debugger provides a way for a programmer to trace the execution of
a program and examine the contents of memory and registers
 A computer system can be viewed as consisting of layers. Programs
at one layer are translated or interpreted by the next lower-level layer
 Binary and Hexadecimal numbers are essential for programmers
working at the machine level.

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