Lecture No. 4

Number Systems

32-bit f.p. number (recap)

32-bit Floating point format Sign bit 1 Exponent bits 8 Mantissa bits 23 Exponent represented as Biased 127

Range of f.p. numbers (recap)

Largest positive/negative number 2127

Smallest positive/negative number 2-126

The number Zero Exponent = 00000000 Mantissa = 000 0000

0000 0000 0000 0000 The number infinite

Exponent = 11111111 Mantissa = 000 0000 0000 0000 0000 0000

Arithmetic operations on floating point numbers

Addition Adding mantissas after adjusting exponents

Subtraction Subtracting mantissas after adjusting

exponents Multiplication

Multiplying mantissas and adding exponents Division

Dividing mantissas and subtracting exponents

64-bit f.p. number (recap)

64-bit Double-Precision floating Point format Sign bit 1 Exponent bits 11 Mantissa bits 52 Exponent represented as Biased 1023

f.p. numbers (recap)

How do systems differentiate between number representations?

Defining and Declaring Data Types.

Hexadecimal Numbers (recap)

Hexadecimal Number System Base 16 number system 0 to F Used to represent large binary numbers

Counting in Hexadecimal (recap)

Decimal Binary Hexadecimal Decimal Binary Hexadecimal

0 0000 0 8 1000 8

1 0001 1 9 1001 9

2 0010 2 10 1010 A

3 0011 3 11 1011 B

4 0100 4 12 1100 C

5 0101 5 13 1101 D

6 0110 6 14 1110 E

7 0111 7 15 1111 F

Binary-Hexadecimal conversion (recap)

Binary to Hexadecimal Conversion 11010110101110010110 1101 0110 1011 1001 0110 D 6 B 9 6

Hexadecimal to Binary Conversion FD13 1111 1101 0001 0011

Hexadecimal-decimal conversion (recap)

Hexadecimal to Decimal Conversion Indirect Method

Hexadecimal →Binary → Decimal Sum-of-Weights

Decimal-Hexadecimal Conversion (recap)

Decimal to Hexadecimal Conversion Indirect Method

Decimal →Binary → Hexadecimal Repeated Division by 16

Hexadecimal Arithmetic (recap)

Hexadecimal Addition Carry generated

Hexadecimal Subtraction Borrow weight 16

Octal Number System

Base 8 0, 1, 2, 3, 4, 5, 6, 7 Representing Binary in compact form

11011000001102 = 154068 Not commonly used in the presence of

Hexadecimal Number System

Counting in Octal

Octal digit represented by a 3-bit binary Decimal 8 represented by 2-digit Octal

Counting in Octal

Decimal Binary Octal

0 000 0

1 001 1

2 010 2

3 011 3

4 100 4

5 101 5

6 110 6

7 111 7

Counting in Octal

Decimal Octal Decimal Octal Decimal Octal

8 10 16 20 24 30

9 11 17 21 25 31

10 12 18 22 26 32

11 13 19 23 27 33

12 14 20 24 28 34

13 15 21 25 29 35

14 16 22 26 30 36

15 17 23 27 31 37

Binary-Octal Conversion

Binary to Octal Conversion Octal to Binary Conversion

Octal-Decimal Conversion

Octal to Decimal Conversion Indirect Method

Octal →Binary → Decimal Sum-of-Weights

Decimal-Octal Conversion

Decimal to Octal Conversion Indirect Method

Decimal →Binary → Octal Repeated Division by 8

Octal Addition & Subtraction

Octal Addition Carry generated

Octal Subtraction Borrow weight 8

Binary to Octal Conversion

011010110101110010110 011 010 110 101 110 010 110 3 2 6 5 6 2 6

1011011101001 1 011 011 101 001 001 011 011 101 001 1 3 3 5 1

Octal to Binary Conversion

1726 001 111 010 110

Sum-of-Weights

4037

(4 x 83) + (0 x 82) + (3 x 81) + (7 x 80)

(4 x 512) + (0 x 64) + (3 x 8) + (7 x 1)

2048 + 0 + 24 + 7

2079

Repeated Division by 8

Number Quotient Remainder

2079 259 7 (O0)

259 32 3 (O1)

8 4 0 (O2)

4 0 4 (O3)

Octal Addition

Carry 1

7602

+ 4771

14573

Octal Subtraction

Borrow 11

7602

- 4771

2611

Excess Code BCD Code Gray Code

Alternate Representations

Excess Code

A bias is added to Binary Code Used by floating point numbers

Excess-8 Code

Decimal 2’s Comp.

Excess-8 Decimal 2’s Comp.

Excess-8

0 0000 1000 -8 1000 0000

1 0001 1001 -7 1001 0001

2 0010 1010 -6 1010 0010

3 0011 1011 -5 1011 0011

4 0100 1100 -4 1100 0100

5 0101 1101 -3 1101 0101

6 0110 1110 -2 1110 0110

7 0111 1111 -1 1111 0111

Binary Code to represent decimal digits 0-9 Used by Decimal Number Displays

BCD (Binary Coded Decimal) Code

BCD (Binary Coded Decimal) Code

Decimal BCD Decimal BCD

0 0000 5 0101

1 0001 6 0110

2 0010 7 0111

3 0011 8 1000

4 0100 9 1001

BCD Addition

23 0010 001145 0100 010168 0110 1000

23 0010 001149 0100 100172 0110 1100

1100 is illegal BCD number

BCD Addition

Add a 0110 (6) to an invalid BCD number Carry added to the most significant BCD digit

23 0010 0011

49 0100 1001

72 0110 1100

0110

0111 0010

Gray Code

Binary Code more than 1 bit change Electromechanical applications of digital

systems restrict bit change to 1 Shaft encoders Braking Systems

Un-Weighted Code

Gray Code

Decimal Gray Binary

0 000 000

1 001 001

2 011 010

3 010 011

4 110 100

5 111 101

6 101 110

7 100 111

Gray Code Application

Binary Gray Code

AB

C

A

CB

Alphanumeric Code

Numbers, Characters, Symbols ASCII 7-bit Code American Standard Code for Information

Interchange 10 Numbers (0-9) 26 Lower Case Characters (a-z) 26 Upper Case Characters (A-Z) 32 Control Characters Punctuation and Symbols

Alphanumeric Code

Extended ASCII 8-bit Code Additional 128 Graphic characters Unicode 16-bit Code

ASCII Code

Numbers 0 to 9 ASCII 0110000 (30h) to 0111001 (39h) Alphabets a to z ASCII 1100001 (61h) to 1111010 (7Ah) Alphabets A to Z ASCII 1000001 (41h) to 1011010 (5Ah) Control Characters ASCII 0000000 (0h) to 0011111 (1Fh)

Error Detection

Digital Systems are very Reliable Errors during storage or transmission Parity Bit

Even Parity Odd Parity

Odd Parity Error Detection

Original data 10011010 With Odd Parity 110011010 1-bit error 110111010 Number of 1s even indicates 1-bit error 2-bit error 110110010 Number of 1s odd no error indicated 3-bit error 100110010 Number of 1s even indicates error