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:-
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)2,etc.
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.
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.
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.
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
But for fractional numbers that is for the numbers with decimal point we have to multiply fractional numbers with 2 raise to power of -1 from decimal point.
1. (1111.101) =1*23 + 1*22+ 1*21 + 1*20 + 1*2-1+ 0*2-2 + 1*2-3
= 1*8 + 1*4 + 1*2 + 1*1+ 1*1/21+ 0*1/22 + 1*1/22
= 1*8 + 1*4 + 1*2 + 1*1+ 1*1/2 + 0*1/4 + 1*1/8
= 1*8 + 1*4 +1*2 + 1*1+1*0.5 + 0*0.25 + 1*0.125
= 8 + 4 + 2 + 1 + 0.5 + 0 + 0.125
=(15.625)10
# 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 : -
Answer=(11011101)2
2.(9)10
Solution : -
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:
# 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
# Convert decimal to octal:- To convert the decimal number to octal divide the number by 8 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 : -
Answer=(532)8
1.(304)10
Solution : -
Answer=(064)8
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)2,etc.
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.
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.
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.
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
But for fractional numbers that is for the numbers with decimal point we have to multiply fractional numbers with 2 raise to power of -1 from decimal point.
1. (1111.101) =1*23 + 1*22+ 1*21 + 1*20 + 1*2-1+ 0*2-2 + 1*2-3
= 1*8 + 1*4 + 1*2 + 1*1+ 1*1/21+ 0*1/22 + 1*1/22
= 1*8 + 1*4 + 1*2 + 1*1+ 1*1/2 + 0*1/4 + 1*1/8
= 1*8 + 1*4 +1*2 + 1*1+1*0.5 + 0*0.25 + 1*0.125
= 8 + 4 + 2 + 1 + 0.5 + 0 + 0.125
=(15.625)10
# 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 |
| 0 | 1 |
2.(9)10
Solution : -
| 2 | 9 | Remainders |
| 2 | 4 | 1 |
| 2 | 2 | 0 |
| 2 | 1 | 0 |
| 0 | 1 |
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,
| 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 |
# 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
# Convert decimal to octal:- To convert the decimal number to octal divide the number by 8 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 : -
| 8 | 157 | Remainders |
| 8 | 19 | 5 |
| 8 | 2 | 3 |
| 0 | 2 |
1.(304)10
Solution : -
| 8 | 304 | Remainders |
| 8 | 38 | 0 |
| 8 | 4 | 6 |
| 0 | 4 |
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