Lecture 02 - Logic Design(Number Systems).pptx

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Computer Logic Design
Lecture 2
Number Systems
LECTURERS:DAROON ALI &SAZAN KAMAL
BAYAN UNIVERSITY

Outlines
Base of Number Systems
Decimal Numbers
Binary Numbers
 MSB and LSB
 Binary to Decimal Conversion
 Decimal To Binary Conversion
Hexadecimal Numbers
 Conversions
Octal Numbers
 Conversions

Number Systems
The numeric system we use daily is the decimal system,
 but this system is not convenient for machines
 since the information is handled codified in the shape of On or Off bits;
A base of a number system defines the range of values that a digit may have. For example,
 base 2 Binary number has only Two different values (0 and 1).
 base 10 Decimal number has Ten different values (0,1,2,3,4,5,6,7,8 and 9).
 And etc.…

Decimal Numbers
The position of each digit in a weighted number system is assigned a weight based on the base
of the system.
 The base of decimal numbers is ten, because only ten symbols (0 through 9) are used to represent any
number.
The column weights of decimal numbers are powers of ten that increase from right to left
beginning with 100 =1:
 105 104 103 102 101 100
For fractional decimal numbers, the column weights are negative powers of ten that decrease
from left to right:
 102 101 100. 10-1 10-2 10-3 10-4

Decimal Numbers
Decimal numbers can be expressed as the sum of the products of each digit times the column
value for that digit.
 Thus, the number 9240 can be expressed as
Example: Express the number 480.52 as the sum of values of each digit.
(9 x 103) + (2 x 102) + (4 x 101) + (0 x 100)
or
9 x 1,000 + 2 x 100 + 4 x 10 + 0 x 1
480.52 = (4 x 102) + (8 x 101) + (0 x 100) + (5 x 10-1) +(2 x 10-2)

Binary Numbers
For digital systems, the binary number system is used. Binary has a radix of two and uses the
digits 0 and 1 to represent quantities.
The column weights of binary numbers are powers of two that increase from right to left
beginning with 20 =1:
 …25 24 23 22 21 20.
For fractional binary numbers, the column weights are negative powers of two that decrease
from left to right:
 22 21 20. 2-1 2-2 2-3 2-4 …

LSB and MSB
Binary number can be a stream of 0 and 1
 For Example :
 10010010
 010101
 111001
 1101100
The first bit from the left is called Most Significant Bit (MSB)
 Because of its significance on the number.
The first bit from the right is called Least Significant Bit (LSB)
 Because of its low significance on the number.
1001101
MSB LSB
Second MSB

Binary-to-Decimal Conversion
The decimal equivalent of a binary number can be determined by adding the column values of
all of the bits that are 1 and discarding all of the bits that are 0.
Example: Convert the binary number 100101.01 to decimal.
Solution:

Decimal-to-Binary Conversion
 You can convert a decimal whole number to binary by reversing the procedure.
 Write the decimal weight of each column and place 1’s in the columns that sum to the decimal
number This Method is called Sum - Of - Weights Method.
 Example : Convert the decimal number 49 to binary.
 Solution :
 The column weights double in each position to the right.
 Write down column weights until the last number is larger than the one you want to convert.

Repeated Division-by-2 Method
A systematic method of converting whole numbers from decimal to binary is the repeated
division-by-2 process.
 For example, to convert the decimal number 12 to binary,
 begin by dividing 12 by 2.
 Then divide each resulting quotient by 2 until there is a 0 whole-number quotient.
 The remainders generated by each division form the binary number.
 The first remainder to be produced is the LSB (least significant bit) in the binary number,
 and the last remainder to be produced is the MSB (most significant bit).
 This procedure is shown in the following steps for converting the decimal number 12 to binary.

Example:
 Convert the following decimal
numbers to binary:
 (a) 19 (b) 45

Converting Decimal Fractions to Binary
Sum-oF-Weights
 The sum-of-weights method can be applied to fractional decimal numbers, as shown in the
following example:
 0.625 = 0.5 + 0.125 = 2-1 + 2-3 = 0.101
 There is a 1 in the 2-1 position, a 0 in the 2-2 position, and a 1 in the 2-3 position.

Repeated Multiplication by 2
As you have seen, decimal whole numbers can be converted to binary by
repeated division by 2.
Decimal fractions can be converted to binary by repeated multiplication by 2.
For example, to convert the decimal fraction 0.3125 to binary,
 begin by multiplying 0.3125 by 2
 and then multiplying each resulting fractional part of the product by 2 until the fractional product is
zero or until the desired number of decimal places is reached.
 The carry digits, or carries, generated by the multiplications produce the binary number.
 The first carry produced is the MSB, and the last carry is the LSB. This procedure is illustrated as
follows:

Example : Convert the decimal fraction 0.188 to binary by repeatedly multiplying the fractional
results by 2.
Solution:

Hexadecimal Numbers
The hexadecimal number system has sixteen characters.
They are:
 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Hexadecimal is a weighted number system. The column weights are
powers of 16, which increase from right to left.
4096 256 16 1
163 162 161 160

Binary-to-Hexadecimal Conversion
Converting a binary number to hexadecimal is a straightforward procedure.
Simply break the binary number into 4-bit groups, starting at the right-most bit and replace each
4-bit group with the equivalent hexadecimal symbol.

Hexadecimal-to-Binary Conversion
To convert from a hexadecimal number to a binary number, reverse the process and replace
each hexadecimal symbol with the appropriate four bits.

Hexadecimal-to-Decimal Conversion
One way to find the decimal equivalent of a hexadecimal number is
 first convert the hexadecimal number to binary and then convert from binary to decimal.

Another way to convert a hexadecimal number to its decimal equivalent is
 multiply the decimal value of each hexadecimal digit by its weight and then take the sum of
these products.
 The weights of a hexadecimal number are increasing powers of 16 (from right to left).
 For a 4-digit hexadecimal number, the weights are
4096 256 16 1
163 162 161 160

Example : Express 1A2F16 in decimal.
Solution :
 Start by writing the column weights:
 4096 256 16 1
 ( 1 A 2 F)16
= 1(4096) + 10(256) +2(16) +15(1) = 6703

Decimal-to-Hexadecimal Conversion
Repeated division of a decimal number by 16 will produce the equivalent hexadecimal number,
formed by the remainders of the divisions.
The first remainder produced is the least significant Digit (LSD).
Each successive division by 16 yields a remainder that becomes a digit in the equivalent
hexadecimal number.

Decimal-to-Hexadecimal Conversion

Octal Numbers
Like the hexadecimal number system, the octal number system provides a convenient way to
express binary numbers and codes.
The octal number system is composed of eight digits, which are
 0, I, 2, 3, 4, 5, 6, 7

Octal-to-Decimal Conversion
Since the octal number system has a base of eight,
each successive digit position is an increasing power of eight, beginning in the right-most
column with 8°.
The evaluation of an octal number in terms of its decimal equivalent is accomplished by
multiplying each digit by its weight and summing the products,
 as illustrated here for (2374),

Decimal-to-Octal Conversion
A method of converting a decimal number to an octal number is the repeated division-by- 8
method,

Octal-to-Binary Conversion
Because each octal digit can be represented by a 3-bit binary number, it is very easy to convert
from octal to binary. Each octal digit is represented by three bits as shown below.
To convert an octal number to a binary number, simply replace each octal digit with the
appropriate three bits.

Binary-to-Octal Conversion
Conversion of a binary number to an octal number is the reverse of the
octal-to-binary conversion.
The procedure is as follows:
 Start with the right-most group of three bits and, moving from right to left,
 convert each 3-bit group to the equivalent octal digit.
 If there are not three bits available for the left-most group, add either one or two
zeroes to make a complete group.
 These leading zeroes do not affect the value of the binary number

Lecture 02 - Logic Design(Number Systems).pptx

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