Number System

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Number System
The number system of a number is determined by the Radix of a number.Radix means the base of the number which is obtained by the number of symbols present in that number System.They are as follow:-

Sr.No.	Number System	Radix	Symbols
1.	Binary number System	2	0,1
2.	Octal Number System	8	0,1,2,3,4,5,6,7
3.	Decimal Number System	10	0,1,2,3,4,5,6,7,8,9
4.	Hexadecimal Number System	16	0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

1. Binary Number System :- The number System with Radix 2 are said to be binary numbers.It contains only 2 symbols 0 and 1.For Examples (1110)₂,(1001)₂,(1101)₂,(1010)₂,(1010.111)₂etc.

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2. Octal Number System   :- The number System with Radix 8 are said to be octal numbers.It contains 8 symbols 0,1,2,3,4,5,6 and 7.For Examples (257)₂,(346)₈,(412)₈,(728)₈,(72.39)₈,etc.

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3. Decimal Number System :- The number System with Radix 10 are said to be decimal numbers.It contains 10 symbols 0,1,2,3,4,5,6,7,8 and 9.For Examples (123)₁₀,(786)₁₀,(157)₁₀,(928)₁₀,(28.625)₁₀,etc.

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4. Hexadecimal Number System :- The number System with Radix 16 are said to be hexadecimal numbers.It contains only 16 symbols 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F.Here A represent for number 10,B represents for number 11,C represents for number 12,D represents for number 13,E represents for number 14,F represents for number 15,For Examples (1ABC)₁₆,(25DE)₁₆,(85C.3D)₁₆,(0AB)₁₆,etc.
Note that for hexadecimal number should start with symbols of numbers only that is 0,1,2,3,4,5,6,7,8,9 and not with A,B,C,D,E,F .If there is a possibility of number to star with alphabets then write 0(zero before that number example (0AB)₁₆,(0ADE)_16,etc.

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Conversion of numbers Systems:

# Convert Binary to Decimal  :- 
To convert the binary number into decimal we have to multiply each binary number with 2 and then take the raise to power from Least Significant Bit (LSB) side starting with 0 and so on.
1.(1010)₂ = 1*2³ + 0*2²+ 1*2¹ + 0*2⁰
                  = 1*8 + 0*4 +1*2 + 0*1
                  = 8 + 0 + 2 + 0
                  =(10)₁₀
2.(101)₂ = 1*2² + 0*2¹ + 1*2⁰
                = 1*4 + 0*2 + 1*1
                =  4 + 0 +1
                = (5)₁₀
3.(1001)₂ = 1*2³ + 0*2²+ 0*2¹ + 1*2⁰
                  = 1*8 + 0*4 +0*2 + 1*1
                  = 8 + 0 + 0 +1
                  =(9)_10
4..(1111)₂ = 1*2³ + 1*2²+ 1*2¹ + 1*2⁰
                  = 1*8 + 1*4 +1*2 + 1*1
                  = 8 + 4 + 2 +1
                  =(15)₁₀

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# Convert Decimal to Binary :-
 To convert the decimal number to binary divide the number by 2 to get integers note down the remainders in front of them simultaneously and then write remainders in reverse order from bottom to top.
  1.(157)₁₀
Solution : -

2	157	Remainders
2	78	1
2	39	0
2	19	1
2	9	1
2	4	1
2	2	0
2	1	1
2	0	1

Answer=(11011101)₂

2.(9)₁₀
Solution : -

2	9	Remainders
2	4	1
2	2	0
2	1	0
2	0	1

Answer=(1001)₂

Note:- Binary counting can be done by taking only those values of numbers from similarly like starting from decimal numbers only those numbers which contains symbols 0 and 1.That is 0,1,2,3,4,5,6,7,8,9,10,11,12,13,............,99,100,101,102,103,....,110,111,...and so on. Cancel the numbers that contains symbols other than 0 and 1.So the counting will be:

Decimal Numbers	Binary numbers	Hexadecimal Numbers
0	0	0
1	1	1
2	10	2
3	11	3
4	100	4
5	101	5
6	110	6
7	111	7
8	1000	8
9	1001	9
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F

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# Convert octal to decimal :- 
To convert the octal number into decimal we have to multiply each octal  number with 8 and then take the raise to power from Least Significant Bit (LSB) side starting with 0 for integers and so on.

1. (157)₈ = 1*8² + 5*8¹ + 7*8⁰
              = 1*64 + 5*40 + 7*1
              = 64 + 200 +7
              = (271)₁₀
2.(459)₈ = 4*8² + 5* 8¹ + 9*8⁰
              = 4*64+5*8+9*1
              =256 + 40 +9
              =(305)₁₀
 But for fractional numbers that is for the numbers with decimal point we have to multiply fractional numbers with 8 raise to power of -1 from decimal point and so on -2,-3,.. .
 1.(351.123)₈   =  3*8² + 5*8¹ + 1*8⁰ + 1*8^-1 + 2*8^-2 + 3*8^-3
                        =  3*64 + 5*8 + 1*1 + 1*1/8¹ + 2*1/8² + 3*1/8³
                        =3*64 + 5*8 + 1*1 + 1*1/8 + 2* 1/64 + 3* 1/512
                        =192 + 40 + 1 + 1*0.125 + 2*0.015625 + 3* 0.001953125
                        = 192 + 40 + 1 + 0.125 + 0.03125 +  0.005859375
                        = (233.162109375)₁₀

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