types of number system conversion among them

1. Number System
DIGITAL ELECTRONICS
Ms.P.Sivalakshmi .M.E.,
Assistant Professor/ECE
R.M.K College of Engineering & Technology
DEPARTMENT OF ELECTRONICS AND COMMUNICATION
ENGINEERING

2. CONTENTS
• NumberSystem
• RepresentationofNumbersofdifferentRadix
• ConversionofNumbersfromoneRadixtoanotherRadix
• ComplementofNumber
• BinaryArithmetic

3. WHAT IS NUMBER SYSTEM ?
Asystemfor representingnumberof certaintype.
Example:
• Thereareseveralsystemsfor representingthecounting
numbers.
• Theseincludetheusualbase“10”or decimalsystem
1,2,3…..10,11,12,..99,100,…

4. HOW THE COMPUTER GETS THE
ANSWER
A computer understands information
composed of only zeros and ones.
The decimal number system is
convenient for the programmer.
The computer uses binary digits for its
operation.
8
+1
=9

5. NUMBER SYSTEM CHART

6. 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

7. DECIMAL NUMBER SYSTEM
0,1,2,3,4,5,6,7,8,9.DIGITS
10BASE
7
9*101
8*100
5*102
4*103
4598

8. BINARY NUMBER SYSTEM
DIGITS
2BASE
0,1
0*22
1*21
1*20
1*23
1011

9. HEXADECIMAL NUMBER SYSTEM
DIGITS
16BASE
• 0,1,2,3,4,5,6,7,
• 8,9,A,B,C,D,E,F
5*161
D*160
A*1621*163
1A5D

10. OCTAL NUMBER SYSTEM
DIGITS
8BASE
0,1,2,3,4,5,6,7,8
7*81
3*80
2*82
5*83
5273

12. CONVERSION AMONG
BASES
Hexadecimal
Decimal Octal
Binary

13. • Groupinto3'sstartingatleastsignificantsymbol.
• write1octaldigit for eachgroup .
Example:
Convert (1010101)2 to()8
001 010 10
1
1 2 5
Answer= 1258
BINARY TO OCTAL
ifthe numberofbits
isnotevenlydivisible
by3,thenadd0'sat
the mostsignificant
end.

14. Answer= (010101111)2
OCTAL TO BINARY
Example: Convert(257)8
2 57
010
101
111

15. BINARY TO HEXADECIMAL
• Group into 4's starting at leastsignificant symbol write 1 hexdigit for
eachgroup.
E.g Convert(1010111011)2to ()16
10 1011 1011
B2 B
Answer= (2BB)16
if the number of
bits is not evenly
divisible by 4,
then add 0's at
the most
significant end.

16. HEXADECIMAL TO BINARY
• For each of the Hexa digit write its binary equivalent.
E.g: Convert(25A0)16to ()2
25A0
0010
0101 1010
0000
Answer= (0010010110100000)2
use 4 bits
to
represent

17. Steps
 Convertoctalnumbertoitsbinary equivalent.
 Convertbinarynumbertoitshexadecimalequivalent.
E.g.: Convert (635.27)8 to()16
6 3 5 . 2 7
110 011 101 . 010 111000 00
1 9 D . 5 C
OCTAL TO HEXADECIMAL

18. Steps
1.Convert hexadecimal numberto itsbinaryequivalent.
2.Convertbinarynumberto itsoctalequivalent
E.g.Convert(A3B.7)16
HEXADECIMAL TO OCTAL
A 3 B . 7
1010 0011 1011 . 0111 00
5 0 7 3 . 3 4

19. ANY BASE TO DECIMAL
 Convertingfromanybasetodecimalisdonebymultiplying each
digit byits weightand summing.
Binaryto Decimal
1011.112 = (1x23 ) + (0x22 )+ (1x21) + (1x20). (1x2-1)+ (1x2-2)
= 8 + 0 + 2 + 1 . 0.5 + 0.25
= 11.7510

20. DECIMAL TO ANY BASE
Decimal To any
Base
Integer Part
Successive
Division Method
Fractional Part
Successive
Multiplication
Method

21. Steps inSuccessiveDivisionMethod
1. Dividetheintegerpartofdecimalnumberbydesired basenumber,storequotient(Q)and
remainder(R).
2. Consider quotient asanewdecimal number and repeat Step1untilquotientbecomes 0.
3. Listthe remainders in thereverse order.
Steps in SuccessiveMultiplication Method
1. Multiplythefractionalpartofdecimal numberby desiredbasenumber.
2. Recordtheintegerpartof productascarryand fractionalpartasnewfractional part.
3. RepeatSteps1and2untilfractionalpartofproduct becomes0oruntilyouhavemanydigits
asnecessaryforyourapplication.
4. Readcarriesdownwards togetdesiredbasenumber.

22. E.G.:(125)10 TO( )2
Answer: (1111101)2
Decimal To Binary

23. 1’S COMPLEMENT
The 1’s complement of a binary number is the number that
resultswhenwechangeall1’sto zerosandthezeros to ones.
1 1 0 1 0 0 1 0
NOT OPEARATION
0 0 1 0 1 1 0 1

24. 2’S COMPLEMENT
The2’scomplementthebinarynumberthatresultswhen add1to the
1’scomplement.It’sgivenas,
2’s complement = 1’s complement + 1
Example: Express35in 8-bit 2’scomplementform.
35in 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

25. 9’S COMPLEMENT
The nines'complementof adecimaldigit isthenumber thatmustbeaddedto
ittoproduce9.Thecomplementof 3is6,thecomplementof 7is 2.
Example:Obtain 9’scomplementof 7493
Solution:
9 9 99
- 749 3
2506 9’s complement

26. 10’S COMPLEMENT
The10’scomplementof thegivennumberisobtainedby adding 1to
the9’scomplement.Itisgivenas,
10’s complement = 9’s complement + 1
Example:Obtain10’scomplementof 7493
Solution:
9999 2506
- 749 3 + 1
2506 2507 10’s complement

27. BINARY ADDITION
EXAMPLE
1
+ 0
1
1
0
1
0
0
1 0 0 1 0
Carry

28. • Thesubtractionconsistsof four
possibleelementaryoperations.
BINARY SUBTRACTION
• Incaseof secondoperationtheminuend
bit issmaller thanthesubtrahendbit,
hence1isborrowed.
0-0=0
0-1=1(borrow 1)
1-0=1
1-1=0
0
- 0
1
1
0
1
1
0
1 1 1 1
11 Borrow
EXAMPLE

29. BINARY MULTIPLICATION &
DIVISION
*
e.g.: Multiply 110 by 10
1 1
1
0
0
0 0 0
+ 1 1 0
1 1 0 0
e.g.: Divide110 by 10 1 1
10 1 1 0
1 0
0 1 0
1 0
0 0

30. REFERENCES
“Digital Electronics” By A.Anandkumar