Number system in computer binary octal decimal hexadecimal
Description:
Number system in computer binary octal decimal hexadecimal- this is a very detailed article about the number system in a computer.
| Binary()2 | octal()8 | Decimal ()10 | Hexadecimal ()16 |
| 0000 | 0 | 0 | 0 |
| 0001 | 1 | 1 | 1 |
| 0010 | 2 | 2 | 2 |
| 0011 | 3 | 3 | 3 |
| 0100 | 4 | 4 | 4 |
| 0101 | 5 | 5 | 5 |
| 0110 | 6 | 6 | 6 |
| 0111 | 7 | 7 | 7 |
| 1000 | 10 | 8 | 8 |
| 1001 | 11 | 9 | 9 |
| 1010 | 12 | 10 | A |
| 1011 | 13 | 11 | B |
| 1100 | 14 | 12 | C |
| 1101 | 15 | 13 | D |
| 1110 | 16 | 14 | E |
| 1111 | 17 | 15 | F |
Number system:
The number system is used in a different areas of the day to day life. Each of the numbers of systems has its own importance in its area. Each number system starts its unique value from 0 and ends with its limits maximum value. The different number system is discussed as under.
Binary Number system:
Binary number systems normally implemented in electronic devices. Due to its internal structure, the internal structure of the electronic devices is based on ICs (integrated circuit) and ICs are made of logical gates. Logical gates operate on only two values 0 and 1. It is also called computer language due to its operation. Because the computer system is based on an electronic circuit. Each unique value in the binary system is called BIT, or Digit. The binary number system has base two-2. We can extract its base from its name “Bi” normally uses to represent two.
To find the unique value of the binary number system given below formula is used.
{0, base-1}
By putting the value in the above formula:
{0, 2-1}
The unique value of the binary number system are:
{0, 1}
Octal number system:
Octal number system normally used in a seven-segment display system. This technique is implemented in digital clocks calculators digital scoreboards and traffic counters etc. each represented value has its own seven-pin structure and number can be extracted by activation of the pins needed for any number. Oct word used for 8, the base of the octal number system is 8 so total unique values in octal number system are 8.
To find the unique value of the octal number system given below formula is used.
{0, Base-1}
By putting the value in the above formula:
{0, 8-1}
The unique value of the octal number system are:
{0, 7}
Now by putting the remaining value after 1 and before 7, we can get a complete set of the octal number system unique values:
{0,1,2,3,4,5,6,7}
Decimal number system:
The decimal number system is used in daily life for calculation purposes. The decimal number system has a base of 10 so total unique numbers in a decimal number system are 10. Each value of the decimal number system will be unique from other values. Each value of the decimal number system is called a number.
To find the unique value of the decimal number system given below formula is used:
{0, Base-1}
By putting the value in the above formula:
{0, 10-1}
The unique value of the decimal number system are
{0, 9}
Now by putting the remaining values after 1 and before 9, we can get a complete set of decimal number system unique values.
{0,1,2,3,4,5,6,7,8,9}
Hexadecimal number system:
Hexadecimal number system has a total of 16 unique values we can find the unique values by its name. Hexa is used for 6 and decimal for 10. So by combining both values with each other, we can find the total unique values of the Hexadecimal number system which is 16. Hexadecimal number system normally used in the computer system for addressing purpose. different data we store in the computer each of them has a unique address to reside in the computer memory. When we click on that file or folder computer system extract the requested file or folder from memory via its address. The base of the hexadecimal number system is 16.
To find unique values of the hexadecimal number system given below formula is used
{0, Base-1}
By putting the values in the above formula:
{0, 16-1}
Unique values of Hexadecimal number system are:
{0, 15}
Now by putting the remaining values after 1 and before 15(F), we can get a complete set of the Hexadecimal number system unique values. The main point with hexadecimal is that after 9 each new value represented by upper case letters of the English from A to F.
{0, F}
Now by putting the remaining values after 1 and before 9 and from A to F, we can get a complete set of the Hexadecimal number system unique values.
{0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}
Conversion from one number system to others:
Conversion from Binary to octal:
To convert any number from binary to octal number system follow the given below steps:
- Make three, three digits from right to left.
- If the last tri(three digits group) remains incomplete means only one digit or two digits add required zeroes to its leftmost side of the MSB.
- Check the tri digit equal value written against it in octal tables.
- Replace each tri group of digits with that octal value.
Example: (10101011001)2= (?)8
Solution:
10 101 011 001
010 101 011 001
2 5 3 1
Answer (2531)8
Example: (10011001110110.100101)2 = (?)8
Solution:
10 011 001 110 110 . 100 101
010 011 001 110 110 . 100 101
2 3 1 6 6 . 4 5
Answer=(23166.45)8
Conversion from binary to decimal:
Example: (10010101)2 = (?)
Solution:
=127 + 026 + 025 + 124 + 02 3 + 022 + 021 + 120
=1128 + 064 + 032 + 116 + 08 + 04 + 02 + 11
=128 + 0 + 0 + 16 + 0 + 4 + 0 + 1
=128 + 16 + 4 + 1
=149
Answer: 149
Example: (100101.101)2 = (?)
Solution:
=125 + 024 + 023 + 122 + 021 + 120 + 12-1 + 02-2 + 1*2-3
=132 + 016 + 08 + 14 + 02 + 11 + 1(1/2) + 0(1/4) + 1*(1/8)
=32 + 0 + 0 + 4 + 0 + 1 + 0.5 + 0 + 0.125
=32+4+1+0.5+0.125
=37.625
Answer: 37.625
Conversion from Binary to Hexadecimal:
To convert any number from binary to hexadecimal number system follow the given below steps:
- Make three, four digits group from right to left.
- If the last tetra(four digits group) remains incomplete means only one-digit two or three digits add required zeroes to its leftmost side of the MSB.
- Check the tetra digit equal value written against it in the hexadecimal table.
- Replace each tri group of digits with that hexadecimal value.
Example: (11101001001110101)2 = (?)16
Solution:
0001 1101 0010 0111 0101
1 D 2 7 5
Answer: (1D275)16
Example: (1010101101010101101110.10011100) = (?)16
Solution:
10 1010 1101 0101 0110 1110. 1001 1100
0010 1010 1101 0101 0110 1110. 1001 1100
2 A D 5 6 E. 9 C
Answer: (2AD56E.9C)16
Conversion from octal to binary:
Example: (764324)8 = (?)2
Solution:
7 6 4 3 2 4
111 110 100 011 010 100
Answer: (111110100011010100)2
Example: (265473.723)8 = (?)2
Solution:
2 6 5 4 7 3. 7 2 3
010 110 101 100 111 011. 111 010 011
Answer: (010 110101100111011.111010 011)2
Conversion from Octal to Decimal:
Example: (234721)8 = (?)10
Solution:
=285 + 384 + 483 + 782 + 281 +180
=232768 + 34096 + 764 + 2 *8 + 11
=65536 + 12288 + 2048 + 448 +16 + 1
=80337
Answer: (80337)10
Example: (6571.342)8 = (?)10
Solution:
=683 + 582 + 781 + 180 + 38-1 + 48-2 + 2*8-3
=6512 + 564 + 78 + 11 + 3(1/8) + 4(1/64) + 2*(1/512)
=3072 + 320 + 56 + 1 + 3/8 +4/64 + 0.00390625
= 3449+320+56+1+0.375+0.0625+ 0.00390625
=3449+0.44140625
=3449.044140625
Answer: (3449.044140625)10
Conversion from octal to hexadecimal:
To convert any given value from octal to hexadecimal number systems. First of all convert value into a binary number system than convert the resultant value into the hexadecimal number systems.
Conversion from decimal to binary:
To select the values after calculation we take it in two phases
- From left to right in the last statement after calculation.
- From bottom to top reminder binary BIT against each statement.
Example: (9867)10 = (?)2
Solution:
| 2 | 9867 |
| 2 | 4933—–1 |
| 2 | 2466—–1 |
| 2 | 1233—–0 |
| 2 | 616——1 |
| 2 | 308——0 |
| 2 | 154——0 |
| 2 | 77——–0 |
| 2 | 38——–1 |
| 2 | 19——–0 |
| 2 | 9———1 |
| 2 | 4———1 |
| 2 | 2———0 |
| 1———0 |
answer: (9867)10 =(10011010001011)2
when fraction exists in any given value conversion will take place in two phases
- convert the left-hand side of the value dividing by 2.
- Multiply right-hand side of the value with 2
- If the result has 0 at the start place it after the fraction
- If the result has non zero value also place 0 follows the previous value
- If the result has 1 at the start before the fractions place it after previous value
- Apply the multiplication process till the last value came 1 or 0.
Example: (839.625)10 = (?)2
L.H.S
| 2 | 839 |
| 2 | 419—–1 |
| 2 | 209—–1 |
| 2 | 104—–1 |
| 2 | 52——0 |
| 2 | 26——0 |
| 2 | 13——0 |
| 2 | 6——–1 |
| 2 | 3——–0 |
| 1——–1 |
R.H.S
0.625*2 = 1.25——-1
0.25*2 = 0.50——–0
0.50*2 = 1.00——-1
Answer: (893.625)10 = (1101000111.101)2
Conversion from decimal to octal:
Example: (98563)10 = (?)8
Solution:
| 8 | 98563 |
| 8 | 12320—–3 |
| 8 | 1540——-0 |
| 8 | 192———4 |
| 8 | 24———-0 |
| 3————0 |
Answer: (98562)10 = (300403)8
Example: (986547.3125)10 = (?)8
L.H.S
| 8 | 986547 |
| 8 | 123318—–3 |
| 8 | 15414——-6 |
| 8 | 1926——–6 |
| 8 | 240———6 |
| 8 | 30———–0 |
| 3————6 |
R.H.S
0.3125*8 = 2.5——-2
0.5*8 = 4.00———-4
Answer: (986547.3125)10 = (3606663.24)8
Conversion from Decimal to HexaDecimal:
Example: (9468462410)10 = (?)16
Solution:
| 16 | 9468462410 |
| 16 | 598028900—-A |
| 16 | 3737806——-4 |
| 16 | 2336050——-6 |
| 16 | 146003——–2 |
| 16 | 9125———–3 |
| 16 | 570————5 |
| 16 | 35————-A |
| 16 | 2—————3 |
Answer: (9468462410)10 = (23A53264A)16
Example: (9201476.9765625)10 = (?)16
Solution:
L.H.S
| 16 | 9201476 |
| 16 | 575092—–4 |
| 16 | 35943——-4 |
| 16 | 2246——–7 |
| 16 | 140———6 |
| 16 | 8————C |
R.H.S
9765625*16 = 15.625———F
0.625*16 = 10———A
Answer: (9201476.9765625)10 = (8C6744.FA)16
Conversion from Hexadecimal to Binary:
Example: (F67B2A)16 = (?)2
Solution:
F 6 7 B 2 A
111 0110 0111 1011 0010 1010
Answer: (F67B2A)16 = (11101100111101100101010)2
Example: (79AE3.CA1)16 = (?)2
Solution:
7 9 A E 3. C A 1
0111 1001 1010 1110 0011 1100 1010 0001
Answer: (79AE3.CA1)16 = (01111001101011100011110010100001)2
Conversion from Hexadecimal to octal:
To convert any given value from the hexadecimal number systems to the octal number systems. First of all, convert the value into a binary number systems than convert the resultant value into the octal number systems.
Conversion from hexadecimal to decimal:
Example: (9C6B)16 = (?)10
Solution:
=9163 + C162 + 6161 + B160
=9163 + 12162 + 6161 + 11160
=94096 + 12256 + 616 + 111
=36864 + 3072 + 96 + 11
=40043
Answer: (9C6B)16 = (40043)10
Example: (A5E7.1F3)16 = (?)10
Solution:
=A16 3+ 5162 + E161 + 7160 + 116-1 + F16-2 + 3*16-3
= 1016 3 + 5162 + 14161 + 7160 + 116-1 + 1516-2 + 3*16-3
= 104096 + 5256 + 1416+ 7160 + 116-1 + 1516-2 + 3*16-3
= 40960 + 5256 + 1416+ 71 + 1(1/16) + 15(1/16) + 3(1/16)
= 40960 + 1280 + 224 + 7 + 1(1/16) + 15(1/16) + 3*(1/16)
= 40960 + 1280 + 224 + 7 + 0.0625 + 150.0625 + 30.0625
= 40960 + 1280 + 224 + 7 + 0.0625 + 0.9375 + 0.1875
= 42471 + 1.1875
= 42472 .1875
Answer: (A5E7.1F3)16 = (42472.1875)10
