Number Systems

Demo CSE Introduction to Number System By : RK Khatri

Introduction

to

Number Systems


Outlines

• Introduction• Common number systems.• Counting• Conversions• Binary to Other number systems• Others to Binary number system• Decimal to Other number systems• Others to Decimal number system• Octal to Hexadecimal number system• Hexadecimal to Octal number system


INTRODUCION• We are familiar with the decimal number system in which digits are 0, 1,

2, 3, 4, 5, 6, 7, 8 and 9.We use these numbers in and as phone number, age, score, weight, etc.

• The computer uses binary digits for its operation. Binary means 2. • In the binary system, there are only two digits 0 and 1. • How Computer works? The programmer or user feeds instruction and

data in alphabets and decimal digits through the keyboard. These are converted to binary bits. Final result is again converted back to monitor in human readable form.

• Other number systems are Octal n Hexadecimals.Octal means 8 and hexadecimals means 16.Octal n hexadecimals are used in Computers internal processes.

jayganesh

IntroductionWe are familiar with the decimal number system in which digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.The computer uses binary digits for its operation. In the binary system there are only two digits 0 and 1. The programmer feeds instruction and data in alphabets and decimal digits. But for the operation of the computer these are converted to binary bits.This tutorial deals with the conversion of binary numbers to decimal numbers and vice versa. It also deals with hexadecimal and octal system. Computer circuitry are usually designed to process hexadecimal or octal number.


Common Number Systems

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

(In place of Hex)

Hexa-decimal

16 0, 1, … 9,

A, B, … F

No No

(Coding, color

coding, registry)


Quantities/Counting (1 of 3)

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 7p. 33




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




Hexa-decimal

16 10000 20 10

17 10001 21 11

18 10010 22 12

19 10011 23 13

20 10100 24 14

21 10101 25 15

22 10110 26 16

23 10111 27 17 Etc.




Hexa-decimal

24 10000 30 18

25 10001 31 19

26 10010 32 1A

27 10011 33 1B

28 10100 34 1C

29 10101 35 1D

30 10110 36 1E

31 10111 37 1F

32 11000 40 20 Etc.


Conversion Among Bases

• The possibilities:

Hexadecimal

Decimal Octal

Binary


Quick Example

2510 = 110012 = 318 = 1916

Base


Binary to Others

• 110112 = ?10 = Sum of (bit x 2 weight )… weight = position of the bit from right.

1x24 + 1x23 + 0x22 + 1x21 + 1x20 = 27

• 110112 = ?8 = Make pairs of 3 bits.011 011 = 33

• 110112 = ?16 = Make pairs of 4 bits.0001 1011 = 1 B


Others to Binary• 2710 = ?2 = Divide by 2… read remainders in reverse !! 2 | 27

2 |13 | 12 |6 | 12 |3 | 02 |1 | 1 | 1

= 11011

• 338 = ?2 = Convert in pairs of 3 bits.011 011

• 1B16 = ?2 = Convert in pairs of 4 bits.0001 1011


Decimal to Others• 2710 = ?2 = Divide by 2… read remainders in reverse !! 2 | 27

2 |13 | 12 |6 | 12 |3 | 02 |1 | 1 | 1

= 11011

• 27710 = ?8 = Divide by 8… read remainders in reverse !! 8 | 277

8 |34 | 58 |4 | 2 | | 4

= 425


Decimal to Others• 27710 = ?16 = Divide by 16… read remainders in reverse !! 16 |277

16 |17 | 516 |1 | 1 | | 1

= 115


Others to Decimal

• 110112 = ?10 = Sum of (bit x 2 weight )… weight = position of the bit from right.

1x24 + 1x23 + 0x22 + 1x21 + 1x20 = 27• 1238 = ?10 = Sum of (bit x 8 weight )…

weight = position of the bit from right.1x82 + 2x81 + 3x80 = 83• 3B16 = ?10 = Sum of (bit x 16 weight )…

weight = position of the bit from right.3x161 + 11x160 = 59


Octal to Hexadecimal

• Use binary as an intermediaryOctal > Binary > Hexadecimal

10768 = ?16

1 0 7 6

001 000 111 110

2 3 E

10768 = 23E16


Hexadecimal to Octal

• Use binary as an intermediaryHexadecimal > Binary > Octal

23E16 = ?8

2 3 E

0010 0011 1110

1 0 7 6

23E16 = 10768


Exercise – Convert ...

Don’t use a calculator!


Hexa-decimal

33

1110101

703

1AF

Skip answer Answer


Exercise – Convert …


Hexa-decimal

33 100001 41 21

117 1110101 165 75

451 111000011 703 1C3

431 110101111 657 1AF

Answer


Fractions

• Binary to decimal

pp. 46-50

10.1011 => 1 x 2-4 = 0.06251 x 2-3 = 0.1250 x 2-2 = 0.01 x 2-1 = 0.50 x 20 = 0.01 x 21 = 2.0 2.6875


Fractions

• Decimal to binary

p. 50

3.14579

.14579x 20.29158x 20.58316x 21.16632x 20.33264x 20.66528x 21.33056

etc.11.001001...


Exercise – Convert ...

Don’t use a calculator!


Hexa-decimal

29.8

101.1101

3.07

C.82

Skip answer Answer


Exercise – Convert …


Hexa-decimal

29.8 11101.110011… 35.63… 1D.CC…

5.8125 101.1101 5.64 5.D

3.109375 11.000111 3.07 3.1C

12.5078125 1100.10000010 14.404 C.82

Answer


Thank you

Number Systems

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