Number system in Digital Electronics

CONTENTS
Number System
Representation of Numbers of Different
Radix
Conversion of Numbers from one Radix to
Another Radix
Complement of Number
Binary Arithmetic

What is Number System ?
•A system for representing number of certain type.
• Example:
–There are several systems for representing the
–counting numbers.
– These include the usual base “10” or decimal system : 1,2,3
,…..10,11,12,..99,100,…

System Base Symbols
Used by
humans?
Used in
computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa-
decimal
16 0, 1, … 9,
A, B, … F
No No
Common Number System

Decimal Binary Octal
Hexa-
decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
Counting

Counting
Decimal Binary Octal
Hexa-
decimal
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

Conversion Among Bases
Hexadecimal
Decimal Octal
Binary

•Group into 3's starting at least significant symbol (if the
number of bits is not evenly divisible by 3, then add 0's at
the most significant end)
• write 1 octal digit for each group
e.g.: (1010101)2 to ( )8
001 010 101
1 2 5
Answer = 1258
Binary to Octal

•For each of the Octal digit write its binary equivalent
e.g.: (257)8 to ( )2
Answer = (010101111)2
Octal to Binary
2 5 7
010
101
111

Binary to Hexadecimal
• Group into 4's starting at least significant symbol (if the
number of bits is not evenly divisible by 4, then add
0's at the most significant end)
• write 1 hex digit for each group.
e.g.: (1010111011)2 to ( )16
10 1011 1011
2 BB
Answer = (2BB)16

Hexadecimal to Binary
• For each of the Hex digit write its binary equivalent (use 4 bits to
represent).
e.g.: (25A0)16 to ( )2
2 5 A 0
0010
0101 1010
0000
Answer = (0010010110100000)2

• Steps:
1.Convert octal number to its binary equivalent
2.Convert binary number to its hexadecimal equivalent
e.g.: (635.27)8 to ( )16
1
6 3 5 . 2 7
110 011 101 . 010 111000 00
1 9 D . 5 C
Octal to Hexadecimal

• Steps:
1.Convert hexadecimal number to its binary equivalent
2.Convert binary number to its octal equivalent
e.g.:
1
Hexadecimal to Octal
A 3 B . 7
1010 0011 1011 . 0111 00
5 0 7 3 . 3 4

Any Base to Decimal
Converting from any base to decimal is done by multiplying
each digit by its weight and summing.
e.g.:
Binary to Decimal
1011.112 = (1x23 ) + (0x22 ) + (1x21 ) + (1x20) + (1x2-1) + (1x2-2)
= 8 + 0 + 2 + 1 + 0.5 + 0.25
= 11.7510

Decimal to Any Base
Steps:
1. Convert integer part
( Successive Division Method )
2. Convert fractional part
( Successive Multiplication Method )

Steps in Successive Division Method
1. Divide the integer part of decimal number by desired
base number, store quotient (Q) and remainder (R)
2. Consider quotient as a new decimal number and
repeat step1 until quotient becomes 0
3. List the remainders in the reverse order
Steps in Successive Multiplication Method
1. Multiply the fractional part of decimal number by
desired base number
2. Record the integer part of product as carry and
fractional part as new fractional part
3. Repeat steps 1 and 2 until fractional part of product
becomes 0 or until you have many digits as necessary
for your application
4. Read carries downwards to get desired base number

e.g.: (125)10 to ( )2
Answer : (1111101)2

1’s Complement
The 1’s complement of a binary number is the number
that results when we change all 1’s to zeros and the zeros
to ones.
1 1 0 1 0 0 1 0
NOT OPEARATION
0 0 1 0 1 1 0 1

2’s Complement
The 2’s complement the binary number that results when
add 1 to the 1’s complement. It’s given as,
2’s complement = 1’s complement + 1
•The 2’s complement form is used to represent negative
numbers.
Example: Express 35 in 8-bit 2’s complement form.
Solution:
35 in 8-bit form is 00100011
0 0 1 0 0 0 1 1
1 1 0 1 1 1 0 0
+ 1
--------------------
1 1 0 1 1 1 0 1

9’s Complement
The nines' complement of a decimal digit is the number
that must be added to it to produce 9. The complement of
3 is 6, the complement of 7 is 2.
Example: Obtain 9’s complement of 7493
Solution:
9 9 9 9
- 7 4 9 3
--------
2 5 0 6 9’s complement

10’s Complement
The 10’s complement of the given number is obtained by
adding 1 to the 9’s complement. It is given as,
10’s complement = 9’s complement + 1
Example: Obtain 10’s complement of 7493
Solution:
9 9 9 9 2 5 0 6
- 7 4 9 3 + 1
-------- ----------
2 5 0 6 2 5 0 7 10’s complement

Binary Addition
The addition consists of four possible elementary
operations:
Sr no. Operations
0. 0+0=0
1. 0+1=1
2. 1+0=1
3. 1+1=10 (0 with carry of 1)
In the last case, sum is of two digits: Higher Significant bit
is called Carry and lower significant bit is called Sum.

Binary Addition
e.g.:
1 1 0 0
+ 0 1 1 0
1 0 0 1 0
Carry

The subtraction consists of four possible elementary
operations:
Binary Subtraction
In case of second operation the minuend bit is smaller
than the subtrahend bit, hence 1 is borrowed.
Sr no. Operations
0. 0-0=0
1. 0-1=1(borrow 1)
2. 1-0=1
3. 1-1=0

Binary Subtraction
e.g.:
0 1 0 1
- 0 1 1 0
1 1 1 1

Binary Multiplication
Rules for Binary Multiplication are:
Sr no. Operations
0. 0*0=0
1. 0*1=0
2. 1*0=0
3. 1*1=1
e.g.: Multiply 110 by 10
1 1 0
* 1 0
0 0 0
+ 1 1 0 0
1 1 0 0

Binary Division
Rules for Binary Division are:
Sr no. Operations
0. 0/0=0
1. 1/0=0
2. 0/1=0
3. 1/1=1
e.g.: Divide 110 by 10
1 1
1 0 1 1 0
1 0
0 1 0
1 0
0 0

REFERENCES
1. “Digital Electronics” By A.P.Godse and Dr.D.A.Godse
2. “Digital Electronics” By A.Anandkumar

Number system in Digital Electronics

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