The document discusses different number systems including binary, decimal, octal, and hexadecimal. It provides details on: - How each system uses different digits or "radix" (e.g. binary uses 0 and 1) - Converting between the different number systems by grouping digits and representing them in other bases - Methods for converting integers and fractions between decimal and other number systems using successive division or multiplication.

1. Number System
Unit II: Chapter 5

2. Binary Number System
•A system uses only the digits 0 and 1 as codes.
• The word ‘bit’ is the abbreviation for binary digit.
• The binary number has 4 bits i.e. 0000 or 1111 called as nibble.
• The binary number with 8 bits i.e. 00000000 or 11111111 called as
byte.

3. Number
System
Decimal Base
10(0-9)
Binary
Base
2(0,1)
Octal Base 8
(0-7)
Hexa
decimal
Base 16
(0-9, A – F)

4. Decimal Binary Octal
Hexa-
decim
al
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
Numbers Equivalent

5. Counting
Decimal Binary Octal
Hexa-
decim
al
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

6. Conversion Among Bases
Hexadecimal
Decimal Octal
Binary

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

9. For Binary to Decimal conversion

15. • 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

16. Answer= (010101111)2
Octal to Binary
•Foreachof theOctaldigit writeitsbinaryequivalent
e.g.: (257)8to ()2
2 57
010
101
111

17. 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
themostsignificantend)
•write 1 hex digit for each group.
e.g.:(1010111011)2to ()16
10 1011 1011
B
2 B
Answer= (2BB)16

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

19. • Steps:
1.Convertoctalnumberto itsbinary equivalent
2.Convertbinarynumberto itshexadecimalequivalent
e.g.: (635.27)8 to ()16
6 3 5 . 2 7
110 011 101 . 010 111
000 00
1 9 D . 5 C
Octal to Hexadecimal

20. • Steps:
1.Convert hexadecimal numberto itsbinaryequivalent
2.Convertbinarynumberto itsoctalequivalent
e.g.:
Hexadecimal to Octal
A 3 B . 7
1010 0011 1011 . 0111 00
5 0 7 3 . 3 4

21. Any Base to Decimal
Convertingfromanybaseto decimalisdone bymultiplying
eachdigit byitsweightand summing.
e.g.:
Binaryto Decimal
1011.112 = (1x23 ) + (0x22 )+ (1x21 ) + (1x20)+ (1x2-1) + (1x2-2)
= 8 + 0 + 2 + 1 + 0.5 + 0.25
= 11.7510

22. Decimal to Any Base
Steps:
1. Convertintegerpart
( SuccessiveDivisionMethod )
2. Convertfractionalpart
( SuccessiveMultiplicationMethod )

23. Stepsin SuccessiveDivisionMethod
1. Dividetheintegerpartof decimalnumberbydesired
basenumber,storequotient(Q) andremainder(R)
2. Consider quotient as a new decimal number
and repeatstep1untilquotientbecomes 0
3. Listtheremaindersin thereverse order
Stepsin SuccessiveMultiplicationMethod
1. Multiply the fractional part of decimal number by
desiredbasenumber
2. Record the integer part of product as carry and
fractionalpartasnewfractional part
3. Repeat steps 1 and 2 until fractional part of product
becomes0or untilyouhavemanydigitsasnecessary for
yourapplication
4. Readcarriesdownwardsto getdesiredbasenumber