NUMBER SYSTEM

Number System

• Number Base B => B symbols –Base 16(Hexa):0, 1,……9, A ,…, E, F–Base 10 (Decimal): 0, 1, 2,……, 7, 8, 9–Base 8(Octal): 0, 1, 2, 3, 4, 5, 6, 7–Base 2 (Binary): 0, 1

Number System (cont)

• Number representation: • d31d30 ... d2d1d0 is a 32 digit number• value = d31x B31 + d30 x B30 + ... + d2 x B2 + d1 x

B1 + d0 x B0

Decimal Numbers: Base 10

• Base or Radix is 10• Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

• Example:41210 = 4x102 + 1x101 + 2x100

= 400 + 10 +2 = 412

Binary Numbers: Base 2

• Base or Radix is 2

• Digits: 0, 1

• Example:1012 = 1x22 + 0x21 + 1x20

= 4 + 0 +1 = 5

Octal Numbers: Base 8

• Base or Radix is 8

• Digits: 0, 1, 2, 3, 4, 5, 6, 7

• Example:1238 = 1x82 + 2x81 + 3x80

= 64 + 16 +3 = 83

Hexadecimal Numbers: Base 16

• Digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F:–A 10–B 11–C 12–D 13–E 14–F 15

• Example:1216 = 1x161 + 2x160

= 16 +2 = 18

Conversions

Decimal to Base N Base N to Decimal

Successive Division by N(Remainder Method)

Multiplication with power of N

Decimal to Binary

Conversion of 2510 to Binary

2 252 12 -12 6 - 02 3 - 0 1 - 1

2510=110012

• Binary to decimal conversion

11001 1X20 = 1 0X21 = 0 0X22 = 0 1X23 = 8 1X24 = 16

2510

Decimal to Octal

Conversion of 12510 to Octal

8 1258 15 - 5 1 - 7

12510=1758

• Octal to decimal conversion

175 5X80 = 5 7X81 = 56 1X82 = 64

125

1758 = 12510

Conversion of 8410 to Octal

8 848 10 - 4 1 - 2

8410=1248

• Octal to decimal conversion

124 4X80 = 4 2X81 = 16 1X82 = 64

84

1248 = 8410

Decimal to Hexa

Conversion of 45010 to Hexadecimal

16 45016 28 - 2 1 - 12

45010=1C216

• Hexa to decimal conversion

1C2 2X160 = 2 12X161 = 192 1X162 = 256

450

1C216 = 45010

Conversion of 38510 to Hexadecimal

16 38516 24 - 1 1 - 8

38510=18116

• Hexa to decimal conversion

181 1X160 = 1 8X161 = 128 1X162 = 256

385

18116 = 38510

Other Conversions

• BINARYOCTAL• BINARYHEXA• OCTALHEXA etc,.

• BINARYOCTAL• OCTAL – BASE 8 = 23

• 3 DIGIT BINARY NUMBERExample:110012 OCTAL

11 001

011 001 3 1 110012=318

Example:11100112 OCTAL

1 110 011

001 110 011 1 6 3

11100112=1638

• BINARYHEXA• HEXA – BASE 16 = 24

• 4 DIGIT BINARY NUMBERExample:110012 HEXA

1 1001

0001 1001 1 9 110012=1916

Example:11100112 OCTAL

1 110 011

001 110 011 1 6 3

11100112=1638

• OCTALHEXA• OCTALBINARYHEXAExample:318 HEXA

3 1 011 001

0001 1001 1 9318=1916

• HEXA OCTAL• Example: 1BD OCTAL

1 B D0001 1011 1101

000 110 111 101 0 6 7 5

1BD = 6758

Conversion of 25.12510 to Binary

2 252 12 -12 6 - 02 3 - 0 1 - 1

2510=110012

0.125 X 2 = 0.2500.250 X 2 = 0.5 0.5 X 2 = 1.0

25.12510 = 11001.0012

• Binary to decimal conversion11001.001

1X2-3 = 0.125 0X2-2 = 0 0X2-1 = 0 1X20 = 1 0X21 = 0 0X22 = 0 1X23 = 8 1X24 = 16

25.12510

Conversion of 84.25010 to Octal

8 848 10 - 4 1 - 2

8410=1248

0.250 X 8 = 2.000

25.12310 = 124.28

• Octal to decimal conversion

124.2

2X8-1 = 0.25 4X80 = 1 2X81 = 0 1X82 = 0 25.25010

ASSIGNMENT

DECIMAL TO BINARY

• 43• 79• 107• 241• 476

DECIMAL TO OCTAL

• 39• 152• 284• 758• 2534

DECIMAL TO HEXA

• 75• 826• 3128• 7359• 4756

DECIMAL TO OTHER

• 63 BASE4• 279 BASE6• 161 BASE7

BINARY OCTAL,HEXA

• 1010101• 11010101011• 111010110111• 1011110111• 101000101

OCTAL HEXA

• 473• 1235• 724• 5774• 4625

• 75.12510 BASE2

• 356.52310 OCTAL

• 527.4210 HEXA