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