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NUMBER SYSTEMS, OPERATION AND CODES
Lecture 2
Digital Design
Dr. PO Kimtho
Department of Computer Sciences
Norton University (NU)
Numbers & Codes
Numbering Systems
Decimal numbering system (Base 10)
Binary numbering system (Base 2)
Hexadecimal numbering system (Base 16)
Octal numbering system (Base 8)
Number Conversion
Binary Arithmetic
1s and 2s Complements of Binary Numbers
Numbers & Codes (cont..)
Signed Numbers
Arithmetic Operations with Signed Numbers
Other Number Codes
Binary-Coded-Decimal (BCD)
ASCII codes
Gray codes
Digital Codes & Parity
Numbering Systems
0 ~ 9
0 ~ 1
0 ~ 7
0 ~ F
Decimal (base 10)
Binary (base 2)
Octal (base 8)
Hexadecimal (base 16)
Num. Systems (Characteristics)
The digits are consecutive.
The number of digits is equal to the size of the base.
Zero is always the first digit.
When 1 is added to the largest digit, a sum of zero and a carry of one results.
Numeric values determined by the implicit positional values of the digits.
00000000
00000001
00000010
00000011
00000100
00000101
00000110
00000111
00001000
00001001
00001010
00001011
00001100
00001101
00001110
00001111
000
001
002
003
004
005
006
007
010
011
012
013
014
015
016
017
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Binary Octal Hex Dec
Numbering Systems (Cont.)
Numbering System (Decimal)
Also called the Base 10 system
Have 10 digits : 0 9
The position for each digit in the decimal number indicates the magnitude of the quantity represented and can be assigned a weight
Numbering System (Decimal)
The weight for whole numbers are positive powers of ten that increase from right to left
105 104 103 102 101 100
For fractional numbers, the weights are negative powers of ten that decrease from left to right
102 101 100 . 10-1 10-2 10-3.
Decimal point
Significant Digits
Binary: 11101101
Hexadecimal: 1D63A72A
Most significant digit Least significant digit
Question: How many bits does the numbers represent?
Decimal: 19635724
Bsince binary digits are properly known as bits, therefore
the term Most Significant Bit (MSB) and Least Significant
Bit (LSB) are widely used.
Numbering System (Binary)
Also called the Base 2 system
The binary number system is used to model the series
of electrical signals computers use to represent
information
0 represents the no voltage or an off state
1 represents the presence of voltage or an on state
Number Conversion
Any Radix (base) to Decimal Conversion
Binary to Decimal Conversion
Decimal value of any binary number can be found by adding weights of all bits that are 1 and discarding the weights of all bits that are 0
Solve this..
(a) 10102
(b) 101112
Answer : ?
Answer : ?
(c) 10101102 Answer : ?
Convert the following binary numbers to decimal
Decimal to Binary Conversion
For whole number conversion, use the repeated division-by-2 process and record the remainder
For fractional number conversion, use repeated multiplication by 2 until the fractional product is 0
or until the desired number of decimal places is
reached
2 5 = 12 + 1 2
1 2 = 6 + 0 2
6 = 3 + 0 2
3 = 1 + 1 2
1 = 0 + 1 2 MSB LSB 2510 = 1 1 0 0 1 2
Remainder
Decimal to Binary Conversion
Whole number
Decimal to Binary Conversion
Carry . 0 1 0 1 0.3125 x 2 = 0.625 0 0.625 x 2 = 1.25 1 0.25 x 2 = 0.50 0 0.5 x 2 = 1.00 1
The Answer: 1 1 0 0 1.0 1 0 1
MSB LSB
Fractional number
Solve this..
(a) 3910
(b) 5810
Answer : ?
Answer : ?
(c) 0.37510 Answer : ?
Convert the following decimal numbers to binary
Binary Arithmetics
Binary Addition
Binary Subtraction
Binary Multiplication
Binary Division
Binary Addition
Four basic rules for adding binary digits (bits) are:
0 + 0 = 0 (Sum of 0 with a carry of 0)
0 + 1 = 1 (Sum of 1 with a carry of 0)
1 + 0 = 1 (Sum of 1 with a carry of 0)
1 + 1 = 1 0 (Sum of 0 with a carry of 1)
Examples Perform the following binary additions:
(a) 100 + 10
1 0 0
1 0 +
1 1 0 (Answer)
(b) 111 + 11
1 1 1
1 1 +
1 0 1 0 (Answer)
Solve this..
(a) 11 + 01
(b) 111 + 110
Answer : ?
Answer : ?
(c) 1001 + 101:
Answer : ?
Perform the following binary additions:
Binary Arithmetics
Binary Addition
Binary Subtraction
Binary Multiplication
Binary Division
Binary Subtraction
Four basic rules for subtracting binary digits (bits) are:
0 - 0 = 0
1 - 1 = 0
1 - 0 = 1
1 0 - 1 = 1 (0 1 with a borrow of 1)
Examples Perform the following binary subtractions:
(a) 101 011
1 0 1
0 1 1 -
0 1 0 (Answer)
(b) 110 101
1 1 0
1 0 1 -
0 0 1 (Answer)
Solve this..
(a) 101 100
(b) 1110 - 11
Answer : ?
Answer : ?
(c) 1100 - 1001:
Answer : ?
Perform the following binary subtractions
Binary Arithmetics
Binary Addition
Binary Subtraction
Binary Multiplication
Binary Division
Binary Multiplication
Four basic rules for muliplying binary digits (bits) are:
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
Examples
Multiply 111 and 101:
1 1 1
1 0 1 x
1 1 1
0 0 0
1 1 1
1 0 0 0 1 1 (Answer)
Solve this..
(b) 110 x 111:
(c) 1101 x 1010:
Answer : ?
Answer : ?
(a) 11 x 11:
Answer : ?
Binary Arithmetics
Binary Addition
Binary Subtraction
Binary Multiplication
Binary Division
Binary Division
Division in binary follows the same procedure as division in
decimal
Example:
Perform the binary divisions of 110 11
1 1 0 1 1
1 0
1 1
0 0 0
(Answer)
Solve this..
(a) 100 10
Answer : ?
(b) 1100 100:
Answer : ?
Divide the binary numbers as indicated:
1s Complement
Changing all the 1s to 0s and all the 0s to 1s
Example:
1 1 0 1 0 0 1 0 1 Binary number
0 0 1 0 1 1 0 1 0 1s complement
2s Complements
Find the 1s complements of the numbers
1 1 1 0 1 0 1 0 1 Binary number
0 0 0 1 0 1 0 1 0 1s complement
Step 1:
Step 2: Add 1 to the 1s complements
0 0 0 1 0 1 0 1 0 1s complement
+ 1 Add 1
0 0 0 1 0 1 0 1 1 2s complement
Solve this..
(a) 00010110
Answer : ?
(b) 10010001
Answer : ?
Determine the 2s complement of each binary number:
Signed Numbers
Left most is the sign bit 0 is for positive, 1 is for negative
Sign & magnitude
00011001 = +25
sign bit magnitude bits
Sign-Magnitude Numbers
Sign bit
0 = positive
1 = negative
31 bits for magnitude
0110010.. 00101110010101
The left-most is the sign bit and the remaining bits are the magnitude bits
Signed Numbers (Cont.) 1s complement
The negative number is the 1s complement of the corresponding positive number
Example
+25 is 00011001 So, -25 is 11100110
2s complement
The positive number same as sign magnitude and 1s complement
The negative number is the 2s complement of the corresponding positive number.
Example
+25 is 00011001 So, -25 is 11100111
Solve this..
Express +19 and -19 (as an 8-bit number) in
i. sign magnitude
ii. 1s complement
iii. 2s complement
Answer : ?
Answer : ?
Answer : ?