introduction to computer systems and software lecture 2 of 2 simon coupland [email protected]

Introduction to Computer Systems and Software

Lecture 2 of 2

Simon Coupland

[email protected]

Representation of Data Within the Computer Contents:

Decimal and Binary Integer Numbers Binary Addition Signed Binary Numbers Overflow Hexadecimal Numbers Number Conversion Real Numbers Character Encoding

Introduction

A few terms: A bit – a single Binary digIT, 0 or 1 A byte – eight bits A word – one or more bytes Integer – whole number Real number – a number with decimal points Binary – Base 2 numbers Octal – Base 8 numbers Decimal – Base 10 numbers (everyday numbers) Hexadecimal – Base 16 numbers

Decimal and Binary Numbers

We all use decimal numbers Base 10 numbers Example: 124

Digit 7 6 5 4 3 2 1 0

Digit Value 107 106 105 104 103 102 101 100

Digit Value 10M 1M 100k 10k 1k 100 10 1

Example 0 0 0 0 0 1 2 4


Computers use binary numbers Base 2 numbers: Example: 124

Bit 7 6 5 4 3 2 1 0

Bit Value 27 26 25 24 23 22 21 20

Bit Value 128 64 32 16 8 4 2 1

Example 0 1 1 1 1 1 0 0


More binary numbers:

00010011 = 16 + 2 + 1

= 19

01011101 = 64 + 16 + 8 + 4 + 1

= 93

11111111 = 255

00000000 = 0

Binary Addition

When adding binary numbers we use binary logic Binary Addition Truth Table 1:

A B A + B Carry

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

Binary Addition Binary Addition Truth Table 2:

A B Carry (in) A + B Carry (out)

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

Binary Addition

Binary Addition Example:

00010001

+ 00011101

=

Binary Addition


00010001

+ 00011101

= 0

1

Binary Addition


00010001

+ 00011101

= 10

Binary Addition


00010001

+ 00011101

= 110

Binary Addition


00010001

+ 00011101

= 1110

Binary Addition


00010001

+ 00011101

= 01110

1

Binary Addition


00010001

+ 00011101

= 101110

Binary Addition


00010001

+ 00011101

= 0101110

Binary Addition


00010001

+ 00011101

= 00101110

Binary Addition

Binary Addition Q1:

00101101

+ 00011001

=

Binary Addition

Binary Addition Q1:

00101101

+ 00011001

= 01000110

Binary Addition

Binary Addition Q2:

00001110

+ 00001111

=

Binary Addition

Binary Addition Q2:

00001110

+ 00001111

= 00011101

Binary Addition

Binary Addition Q3:

01011001

+ 00111011

=

Binary Addition

Binary Addition Q3:

01011001

+ 00111011

= 10010011

Negative Binary Numbers

Sign-true Magnitude Left most bit holds sign Example: -10

Bit 7 6 5 4 3 2 1 0

Bit Value sign 26 25 24 23 22 21 20

Bit Value +/- 64 32 16 8 4 2 1

Example 1 0 0 0 1 0 1 0


Ones complement All 1’s and 0’s are switched When negative, result = -255 + value Example: -10

Bit 7 6 5 4 3 2 1 0

Bit Value 27 26 25 24 23 22 21 20

Bit Value 128 64 32 16 8 4 2 1

Example 1 1 1 1 0 1 0 1


Twos complement Left most bit holds sign When negative, result = -128 + result Example: -10

Bit 7 6 5 4 3 2 1 0

Bit Value Sign 26 25 24 23 22 21 20

Bit Value +/-128 64 32 16 8 4 2 1

Example 1 1 1 1 0 1 1 0


Conversion to Twos complement Ones complement the byte/word Add 1

Example: 00001010 +10

11110101 Ones complement + 00000001 = 11110110

Why Use Twos Complement?

Because addition rules still work:

00010110 22

+ 10001000 + -120

= 10011110 = -98

Overflow

Overflow is when the number of bits is too small to store the result of an arithmetic operation

Example (twos complement) :

01011001 89

+ 01111011 + 123

= 11010011 = -45

Overflow

Overflow can be easily detected for signed binary numbers

Errors can then be corrected Adding two positive numbers should give a

positive result Adding two negative numbers should give a

negative result Adding a positive and negative number

together can never result in overflow. Why?

Hexadecimal Numbers

Writing code with long binary numbers would be cumbersome and error prone

A hexadecimal digit can take 16 values (0-F) One hex digit can represent a four bit word Examples:

Decimal Hexadecimal Binary

4 4 0100

10 A 1010

74 4A 01001010

Number Conversion

Binary to Hexadecimal: From the least significant (rightmost) bit split the

binary number into groups of four bits. Each 4 bits has a hexadecimal equivalent Example:0001001110011110

0001 0011 1001 1110

1 3 9 E

139E

Number Conversion

Hexadecimal to Binary: Convert each hexadecimal digit into its 4 bit binary

equivalent Join the all 4 bit words together Example:A42F

A 4 2 F

1010 0100 0010 1111

1010010000101111

Number Conversion

Question Hexadecimal to Binary:

Convert D36B into binary

Number Conversion

Question Hexadecimal to Binary:

Convert D36B into binaryD 3 6 B

1101 0011 0110 1011

Number Conversion

Question Binary to Hexadecimal:

Convert 1001101001001010 into hexadecimal

Number Conversion

Question Binary to Hexadecimal:

Convert 1001101001001010 into hexadecimal1001 1010 0100 1010

9 A 8 A

9A8A

Number Conversion

Decimal to Binary Use the following algorithm, begin with LSB

int dec_value = some_number;

int next_bit;

while(dec_value > 0)

{

next_bit = dec_value % 2;

dec_value = dec_value / 2;

}

Number Conversion

Decimal to Binary Convert 42 to binary:

dec_value /2 %2 Result

42 21 0 0

21 10 1 10

10 5 0 010

5 2 1 1010

2 1 0 01010

1 0 1 101010

Number Conversion

Question Convert 39 to binary:

Number Conversion

Question Convert 39 to binary:

dec_value /2 %2 Result

39 19 1 1

19 9 1 11

9 4 1 111

4 2 0 0111

2 1 0 00111

1 0 1 100111

Number Conversion

Binary to Decimal Use the following algorithm, begin with MSB

int dec_value = 0;

int bit_value;

int bit_index = MSB_index;

while(bit_index >= 0)

{

bit_value = word[bit_index];

dec_value = dec_value * 2 + bit_value;

bit_index--;

}

Number Conversion

Binary to Decimal Convert 011010 to decimal:

bit_index bit_value dec_value

5 0 0

4 1 1

3 1 3

2 0 6

1 1 13

0 0 26

Number Conversion

Binary to Decimal Question, convert 101110 to decimal:

Number Conversion

Binary to Decimal Question, convert 101110 to decimal:

bit_index bit_value dec_value

5 1 1

4 0 2

3 1 5

2 1 11

1 1 23

0 0 46

Number Conversion

Decimal to Hexadecimal Use the following algorithm, begin with LSB

int dec_value = some_number;

char hex_digit;

while(dec_value > 0)

{

hex_digit = to_hex(dec_value % 16);

dec_value = dec_value / 16;

}

Number Conversion

Decimal to Hexadecimal Convert 1863 to hexadecimal

dec_value dec_value / 16 dec_value % 16 hex_digit

1863 116 7 7

116 7 4 4

7 0 7 7

Number Conversion

Decimal to Hexadecimal Question, convert 1437 to hexadecimal

Number Conversion

Decimal to Hexadecimal Question, convert 1437 to hexadecimal

dec_value dec_value / 16 dec_value % 16 hex_digit

1437 89 13 D

89 5 9 9

5 0 5 5

Number Conversion

Hexadecimal to Decimal Use the following algorithm, begin with MSB

int dec_value = 0;

char digit_value;

while(!word.off)

{

digit_value = to_dec(hex_digit);

dec_value = dec_value * 16 + digit_value;

}

Number Conversion

Hexadecimal to Decimal Convert F2AC to decimal

Index hex_value dec_value result

3 F 15 15

2 2 2 242

1 A 10 3882

0 C 12 62124

Number Conversion

Hexadecimal to Decimal Convert E59A to decimal

Number Conversion

Hexadecimal to Decimal Convert E59A to decimal

Index hex_value dec_value result

3 E 14 14

2 5 5 229

1 9 9 3673

0 A 10 58778

Real Numbers

A number n is a real number if n % 1 != 0 Take a lot of processing power/time Often math co-processors are used to

perform arithmetic on these numbers

Fixed Point Real Numbers

Similar to integer representation 8 bit example:

Bit 8 7 6 5 4 3 2 1

value -/+ 23 22 21 20 2-1 2-2 2-3

value -/+ 8 4 2 1 0.5 0.25 0.125

Fixed Point Real Numbers

Maximum number is small Accuracy is limited Not used because of these reasons

Floating Point Real Numbers

Floating point numbers are held in two parts Mantissa Exponent

In 0.3876 104

The mantissa is 0.3876 The exponent is 4

Floating Point Real Numbers

Example (not IEEE float):

00000000000000110010100111110011mantissa exponent

Mantissa = 809 Exponent = -13 809 10-13

0.0000000000809

Character Encoding

ASCII EBCDIC UNICODE ISO-8859-1

Character Encoding

ASCII American Standard Code for Information Interchange Characters are represented by integers Table in your notes

A = 65 B = 66 a = 96 z = 123 @ = 64

Character Encoding

ASCII Special functions characters are also encoded Examples:

Carriage return = 13 Escape = 27 Space = 32

Recap

Binary numbers can be signed of unsigned Twos complement gives simple addition Overflow - when the result of a arithmetic

operation is too large to be represented Hexadecimal numbers – base 16 Real numbers use floating point Characters are represented as intergers

introduction to computer systems and software lecture 2 of 2 simon coupland [email protected]

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