CONTENTS
Module-1 (Part-1)
❖ Number systems and conversions
❖ Binary arithmetic operations
❖ r’s and (r-1)’s compliment
❖ Binary signed and unsigned numbers
❖ Weighted and non-weighted binary codes
Number Systems 1
CONTENTS
Lecture-3
❖ Conversion of different number systems (cont…)
❖ Binary Arithmetic
Number Systems 2
CONVERSION BETWEEN NUMBER BASES
✓ The possibilities:
Decimal Binary
Octal Hexadecimal
Number Systems 3
CONVERSION FROM BINARY TO OCTAL
Decimal Binary
Octal Hexadecimal
Number Systems 4
CONVERSION FROM BINARY TO OCTAL
✓ Technique:
1. Group bits in threes, starting on right for integer part and from left for fractional part.
2. Ignore leading zeros and add zeros after decimal if necessary, to form group of three
bits.
3. Convert to octal digits
✓ Example: (1011010111)2 = (N)8
(1011010111)2 = (1327)8
Number Systems 5
CONVERSION FROM BINARY TO OCTAL
✓ Example: (1010111000.0101)2 = (N)8
1 010 111 000 . 010 100
1 2 7 0 . 2 4
(1010111000.0101)2 = (1270.24)8
Number Systems 6
CONVERSION FROM BINARY TO HEXADECIMAL
Decimal Binary
Octal Hexadecimal
Number Systems 7
CONVERSION FROM BINARY TO HEXADECIMAL
✓ Technique:
1. Group bits in fours, starting on right for integer part and from left for fractional part.
2. Ignore leading zeros and add zeros after decimal if necessary, to form group of four bits.
3. Convert to hexadecimal digits
✓ Example: (1010111011)2 = (N)16
(1010111011)2 = (2BB)16
Number Systems 8
CONVERSION FROM BINARY TO HEXADECIMAL
✓ Example: (1010111000.0101)2 = (N)16
0010 1011 1000 . 0101
2 B 8 . 5
(1010111000.0101)2 = (2B8.5)16
Number Systems 9
CONVERSION FROM OCTAL TO BINARY
Decimal Binary
Octal Hexadecimal
Number Systems 10
CONVERSION FROM OCTAL TO BINARY
✓ Technique:
1. Convert each octal digit to a 3-bit equivalent binary representation
✓ Example: (705)8 = (N)2
(705)8 = (111000101)2
Number Systems 11
CONVERSION FROM OCTAL TO BINARY
✓ Example: (1472.16)8 = (N)2
1 4 7 2 . 1 6
1 100 111 010 . 001 110
(1472.16)8 = (1100111010.00111)2
Number Systems 12
CONVERSION FROM OCTAL TO HEXADECIMAL
Decimal Binary
Octal Hexadecimal
Number Systems 13
CONVERSION FROM OCTAL TO HEXADECIMAL
✓ Technique:
1. Use binary as an intermediary
✓ Example: (1472.16)8 = (N)16
1 4 7 2 . 1 6
1 100 111 010 . 001 110 (1472.16)8 = (33A.38)16
001 100 111 010 . 001 11000
3 3 A . 3 8
Number Systems 14
CONVERSION FROM HEXADECIMAL TO BINARY
Decimal Binary
Octal Hexadecimal
Number Systems 15
CONVERSION FROM HEXADECIMAL TO BINARY
✓ Technique:
1. Convert each hexadecimal digit to a 4-bit equivalent binary representation
✓ Example: (10AF)16 = (N)2
(10AF)16 = (1000010101111)2
Number Systems 16
CONVERSION FROM HEXADECIMAL TO BINARY
✓ Example: (A5BF.0C)16 = (N)2
A 5 B F . 0 C
1010 0101 1011 1111 . 0000 1100
(A5BF.0C)16 = (1010010110111111.000011)2
Number Systems 17
CONVERSION FROM HEXADECIMAL TO OCTAL
Decimal Binary
Octal Hexadecimal
Number Systems 18
CONVERSION FROM HEXADECIMAL TO OCTAL
✓ Technique:
1. Use binary as an intermediary
✓ Example: (A5BF.0C)16 = (N)8
A 5 B F . 0 C
1010 0101 1011 1111 . 0000 1100 (A5BF.0C)16 = (122677.03)8
001010 0101 1011 1111 . 0000 1100
1 2 2 6 7 7 . 0 3
Number Systems 19
ASSIGNMENT QUESTION
✓ Fill the table:
Number Systems 20
Binary Arithmetic
Number Systems 21
BINARY ADDITION
✓ Two 1-bit values:
Number Systems 22
BINARY ADDITION
✓ Two n-bit values:
▪ Add individual bits
▪ Propagate carries
✓ Example:
1 1 1 1 1 1 carries
1
1 1 1 0 1 6 1
+ 1 0 1 1 1 + 2 3
--------------------- ----------
1 0 1 0 1 0 0 8 4
Number Systems 23
BINARY ADDITION
✓ Two n-bit values:
▪ Add individual bits
▪ Propagate carries
✓ Example:
Number Systems 24
BINARY SUBTRACTION
✓ Two 1-bit values:
With Borrow 1
Number Systems 25
BINARY SUBTRACTION
✓ Two n-bit values:
▪ Subtract individual bits with borrow if necessary
✓ Example:
10- decimal value 2
10
0 0 10 Borrows
1 1 1 1 0 1 6 1
- 1 0 1 1 1 - 2 3
--------------------- ----------
1 0 0 1 1 0 3 8
Number Systems 26
BINARY MULTIPLICATION
✓ Two 1-bit values:
Number Systems 27
BINARY MULTIPLICATION
✓ Example:
1 0 1 1 1 2 3
X 1 0 1 0 X 1 0
----------------------- ----------
0 0 0 0 0 2 3 0
1 0 1 1 1
0 0 0 0 0
1 0 1 1 1
-----------------------
1 1 1 0 0 1 1 0
Number Systems 28
ASSIGNMENT QUESTIONS
1. Add: (1000111)2 + (1110)2
2. Subtract: (1110000)2 – (1111)2
3. Multiply: (101)2 x (10101)2
Number Systems 29
