Number system

What is Number?


We cannot imagine our life without number.

It is symbol to represent quantity for mathematical manipulations.

It is used to express quantities as basis for counting, comparing, performing calculations and representing value.



Number system is basic element to represent the actual amount or value.

In early days when there were no means of counting, people use to count with the help of fingers, stones, sticks, etc. These methods were not adequate and had many limitations.

There are different numbers system that use different digit or symbol to represent numbers.

Number of digits used to represent that number system is known as base or radix of number system.

Different number system have their own base value. Depending upon the base the number system are classified into following categories

Decimal number system(0-9)

  • It consist of ten digit i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 with the base 10. 
  • Each number can be used individually or they can be grouped to form a numeric value as 85,48,35,456 etc.
  • All other number system are derived from this number system
  • Developed by Hindu Arabic mathematicians during 8-11 centuries

  

0 1 2 3 4 5 6 7 8 9
10 11 12

                            99

100

0 
1
10   
11
100
101
110
110 





binary number system(0,1)

  • The Binary Number System consist of only two digits– 0 and 1. 
  • Since this system use two digits, it has the base 2. 
  • All digital computer use this number system and convert the data input from the decimal format into its binary equivalent.



Decimal

Binary

0

0

1

1

2

10

3

11

4

100

5

101

6

110

7

111

8

1000

9

1001

10

1010

D===>B

conversion into decimal into binary number

rules :

1) divide by base of binary 2

2)  write the remainder in the right side and write quotient to the bottom

3) Repeat until the quotient is 0

4) Answer will be obtain from reverse order of remainder

Convert: (27)10=(?)2


therefore, (27)10=(11011)2

conversion from binary to decimal


rule:

1) multiply each digit of binary number by 2 and write + between all the number

2)raise power on 2 by 0 on rightmost 2 and increase power on 2 as you go left

3) sum with all the calculations will be equivalent decimal


(10101)2=(?)10

 

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

=16+0+4+0+1

=21

 

Therefore, (10101)2 = (21)10

octal number system(0,1,2,3,4,5,6,7)

  • In the Octal Number System it consist of 8 digits i.e. 0, 1, 2, 3, 4, 5, 6, 7 with a base 8.]
  • The sequence of octal number goes as 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, …..as go on.
  • See each successive number after 7 is a combination of two or more unique symbols of octal system.

Decimal

0

1

2

3

4

5

6

7

8

9

10

Octal

0

1

2

3

4

5

6

7

10

11

12

Binary

0

1

10

11

100

101

110

111

1000

1001

1010

Decimal to octal number system

rules:

  1. Divide given number by 8
  2. Then write the quotient under the number as new number
  3. Write the remainder in right side
  4. Repeat the steps 1,2,and 3 until quotient becomes 0
  5. Write the remainder from bottom to top order to get required octal number


octal to Decimal number system
  1. Multiply each digit of octal number by 8
  2. raise power of 8 increasing by 1 from righty to left starting from 0
  3. add product to get the required decimal number

Octal to binary

Rules

  1. Separate the given number individually, assuming it to be an octal number
  2. Convert it into binary and place it in the combination of 3 bit each
binary to octal
Rules
  1. Separate the given binary number into group of three bit from right to left and add zero in left most side if required to make a compete set
  2. Replace each group into its equivalent from binary table



hexa decimal system(0-9,A,B,C,D,E,F)

0,1,2,3,4,5,6,7,8,9,10,11,12,....19,20,21

0,1,10,11,100



D======>B

DIVIDE BY 2 

REMAINDER FROM BOTTOM

D=======>O

DIVIDE BY 8

REMAINDER FROM BOTTOM

D========>H

DIVIDE BY 16

REMAINDER FORM BOTTOM


B======>D

EACH MULTIPLY BY 2 RAISE POWER STRAING FROM 0 FROM FIRST PLACE VALUE

O======>D

EACH MULTIPLY 8 RAISE POWER STRAING FROM 0 FROM FIRST PLACE VALUE

H======>D

EACH MULTIPLY 16 RAISE POWER STRAING FROM 0 FROM FIRST PLACE VALUE


B=======>O

TAKE LAST 3 DIGIT FROM LAST

B=======>H

TAKE LAST 4 DIGIT FROM LAST

0========>B

EACH 3 BINARY DIGIT SERIALY OF EQUIVALENT OCTAL NUMBER 

H=========>B

EACH 4 BINARY DIGIT SERIALY OF EQUIVALENT OCTAL NUMBER 


O======H

CHANGE OCTAL INTO BINARY FIRST AND THEN CHANGE BINARY INTO EQUIVALENT HEXADECIMAL



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