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Number Conversion

Number conversion is the fundamental operation of any digital systems.

There are different bases like base2, base8, base10 and base 16

Base – 10 Base 2
(Binary)
Base 8
(Octal)
Base 16
(Hexadecimal)
0 0000 00 0
1 0001 01 1
2 0010 02 2
3 0011 03 3
4 0100 04 4
5 0101 05 5
6 0110 06 6
7 0111 07 7
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

The above table shows the various base systems.

Converting between one base to another base is of importance.  Usually all the numbering systems of day to day life is done using Base 10 or Decimal system. So it is necessary to convert

  • Other Base to Decimal
  • Decimal to other base

Any number represented as

anan-1an-2an-3…..a1a0 . a-1a-2……..a-n

We will see one by one

Decimal to Binary (Base 10 to Base 2)

(19.456)10 – (?.?)2

In the above, the number 19.456 has to be converted to binary or base2 which will have only 0s and 1s.

First we will take 19

Division Quotient Remainder Remarks
19/2 9 1 a0=1
9/2 4 1 a1=1
4/2 2 0 a2=0
2/2 1 0 a3=0
1/2 0 1 a4=1

so the conversion is 10011

Secondly we will take (0.456)10

Multiplication Whole number decimal Remarks
0.456 * 2 0 0.912 a-1=0
0.912 * 2 1 0.824 a-2=1
0.824 * 2 1 0.648 a-3=1
0.648 * 2 1 0.296 a-4=1

Therefore (0.456)10 is (0.0111)2

Finally (19.456)10 = (10011.0111)2

Binary (Base 2) to Decimal (Base 10) Conversion

(11110.0111)2 = (?.?)10

Let us take the whole portion 11110

1 1 1 1 0 . 0 1 1 1
a4 a3 a2 a1 a0 . a-1 a-2 a-3 a-4
1 * 24 1 * 23 1 * 22 1 * 21 0 * 20 . 0 * 2-1 1 * 2-2 1 * 2-3 1 * 2-4
16 8 4 2 0 . 0.5 0.25 0.125 0.0625

Total is 30.9375

Decimal to Octal (Base 10 to Base 8)

(19.456)10 – (?.?)8

In the above, the number 19.456 has to be converted to binary or base2 which will have only 0s and 1s.

First we will take 19

Division Quotient Remainder Remarks
19/8 2 3 a0=3
2/8 0 2 a1=2

so the conversion is 23

Secondly we will take (0.456)10

Multiplication Whole number decimal Remarks
0.456 * 8 3 0.648 a-1=3
0.648 * 8 5 0.184 a-2=5
0.824 * 8 1 0.472 a-3=1

Therefore (0.456)10 is (0.351)8

Finally (19.456)10 = (23.351)8

Octal (Base 8) to Decimal (Base 10) Conversion

(337.64)8 = (?.?)10

Let us take the whole portion 11110

3 3 7 . 6 4
a2 a1 a0 . a-1 a-2
3 * 82 3 * 81 7 * 80 . 6 * 8-1 4 * 8-2
192 24 7 . 0.75 0.0625

Total is 267.8125

Decimal to Hexadecimal (Base 10 to Base 16)

(19.456)10 – (?.?)16

In the above, the number 19.456 has to be converted to binary or base2 which will have only 0s and 1s.

First we will take 19

Division Quotient Remainder Remarks
19/16 1 3 a0=3
1/16 0 1 a1=1

so the conversion is (13)16

Secondly we will take (0.456)10

Multiplication Whole number decimal Remarks
0.456 * 16 7 0.296 a-1=7
0.296 * 16 4 0.736 a-2=4
0.736 * 16 B 0.776 a-3=B

Therefore (0.456)10 is (0.74B)16

Finally (19.456)10 = (13.74B)16

Octal (Base 8) to Decimal (Base 10) Conversion

(1AB.62)16 = (?.?)10

Let us take the whole portion 11110

1 A B . 6 2
a2 a1 a0 . a-1 a-2
1 * 162 10 * 161 11 * 160 . 6 * 16-1 2 * 16-2
256 160 11 . 0.375 0.007

Total is (427.382)10

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