Numbering System: In our real life we normally use 10 based numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) by which we can show all the calculations. If you are given only 2 numbers (0 &1,) what can you do? The numbers sequence0, 1, 10, 11, 100, 101, 110, 111, 1000,… … and so on.
If you are given 3 numbers (0 1 2) what it will be?
0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121, 122, 200………….
Now we are going to discuss some special numbering systems
1. Binary numbering system
2. Hexadecimal numbering system
3. Decimal numbering system
Binary system: The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1.
Binary systems can only have 2 states, on and off, unlike analogue systems which can have varying states. These 1s and 0s are commonly referred to as a bit (Binary Digit). You can show binary numbers like this 1012 or 101b .
Bits: A single binary digit (like "0" or "1") is called a "bit". For example 11010 is five bits long. The word bit is made up from the words "binary digit"
Hexadecimal System: A Hexadecimal Number is based on the number 16. The Hexadecimal numbers look the same as the decimal numbers up to 9, but then there are the letters (‘A’,’B’,’C’,’D’,’E’,’F’) in place of the decimal numbers 10 to 15. You can show hexadecimal numbers like this 10E16 or 10EH .
Decimal System: 10 based numbering system which we are using in general mathematical calculations. You can show binary numbers like this 19110 or 191d
Table of the Numbering system
Binary
|
0
|
1
|
10
|
11
|
100
|
101
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110
|
111
|
1000
|
1001
|
1010
|
1011
|
1100
|
1101
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1110
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1111
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Decimal
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
Hexadecimal
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
A
|
B
|
C
|
D
|
E
|
F
|
Octal
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0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
10
|
11
|
12
|
13
|
14
|
15
|
16
|
17
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Converting numbers:
i)Binary to decimal:
1011100101011b to decimal number
Here you have to think like this way a) how many digits. Start from the lower case. You can also use table for your calculation like the following way
Digit
|
12
|
11
|
10
|
9
|
8
|
7
|
6
|
5
|
4
|
3
|
2
|
1
|
0
|
No.
|
1
|
0
|
1
|
1
|
1
|
0
|
0
|
1
|
0
|
1
|
0
|
1
|
1
|
1x212+0x211+1x210+1x29+1x28+0x27+0x26+1x25+0x24+1x23+0x22+1x21+1x20
= 4096+0+1024+512+256+0+0+32+0+8+0+2+1=5931
actually you can remove 0s at the time of calculation
1x211+1x210+1x29+1x28+1x25+1x23+1x21+1x20
= 4096+1024+512+256+32+8+2+1 =5931
To understand more clearly
1 0 1 0 1
| | | | |__ 1 x 20 =1
| | | |____ 0 x 21 =0
| | |______ 1 x 22 =4
| | ________ 0 x 23 =0
|___________ 1 x 24 =16
21
|
So the result is 101012 =2110
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For fractional numbers
0.101 1x1/21+0x1/22+1x1/23 =0.5+0+0.125 = 0.625
ii)Hexadecimal to decimal: same as binary to decimal
1EF16 =(…)10
Digit
|
2nd
|
1st
|
0th
|
No.
|
1
|
E
|
F
|
1x162+Ex161+Fx160
=1x16x16 + 14x16 + 15x1 = 256+224+15=495
For fractional numbers
0.E01 = Ex1/161+0x1/162+1x1/163 =0.875+0+0.000244 = 0.875244
Now you may have question, why we are taking power of 2 for binary & power of 16 for hexadecimal?
Let me explain with a decimal no. which we use in daily life..
324.12 = 3x100 + 2x10+ 4 =3x102+2x101+4x100 + 1x1/101+2x1/102 , so base power is important for all numbers.
By this system you can show conversion from any other types to decimal easily
(xyz.uvw)base = x.base2+y.base1+z.base0 +u.1/base1+v.1/base2+w.1/base3
iii) Decimal to binary: As binary is a 2 based number which means there should be a circulation or repetition of your number after 2 numbers whereas in decimal it changes after 10 numbers. It will be calculate such a way there should be a repetition of digits in every two. We can get it by dividing by 2.
(123)10 = (……)2
 |
You are dividing the number by two until it dissolved. After each division you will get reminder 1 or 0 and the division will continue until the result being 1 or 0 at last.
Place your new binary no. from the last result to the first reminder
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By this process you will get your number as
111101112
The result is
(123)10 = (11110111)2
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iv) Hexadecimal to Binary: Inside a hexadecimal character there contain 4 binary digits.eg F=1111 or E=1110 and so on. It is very easy to change hexadecimal to binary
(EF0)h = (…….)b
E = 1110, F=1111, 0=0000; hence (EF0)h= (1110 1111 0000)b
v) Binary to hexadecimal: It is the reverse order of the previous operation(iv). Just separate them by 4 bits for getting one hexadecimal no. start from lower bit
(100001110100011)2 = (….)16
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So the result will be (100001110100011)2 = (43A3)16
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vi) Decimal to Hexadecimal: It’s almost same with converting from decimal to binary. Remember, we divide it by 2 until the result come 0 or 1. This time we will divide by 16 as we are going to get 16 based hexadecimal numbers. Eg (5902)10=(…)16
 590210 = 170E16 |
ß You can see at the left which is the same process with converting decimal to binary. We got result (5902)10=(170E)16
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