number systems consider a decimal

Question

 

Number systems

 

Consider a decimal number:               7654.32

 

Short hand for:            7 * 103 + 6*102 + 5* 101 + 4*100 + 3*10^-1 + 2*10^-2

 

Likewise binary number:                     1011.011

 

In full:                         1*23 + 0*22 + 1* 21 + 1*20 + 0* 2^-1 + 1* 2^-2 + 1*2^-3

 

In general the number:                        x­2 x1 x0 . x-1 x-2 x-3

 

In base B is equal to:   x­₂  B² + x₁  B¹ + x₀  B⁰ + x_-1  B^-1 + x_-2  B^-2 + x_-3 B^-3

 

 

Conversion table.

 

Decimal (Base 10)	Binary (Base 2)	Octal (Base 8)	Hexadecimal (Base 16)
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

 

 

 

Convert decimal to binary by repeated division by two.

 

 e.g. 2510 in binary?

 

25	2  =	12	remainder 1
12	2  =	6	remainder 0
6	2  =	3	remainder 0
3	2  =	1	remainder 1
1	2  =	0	remainder 1

 

Read remainders from bottom up.

Convert decimals less than one to binary by repeated multiplication by two.

 

 e.g. 0.62510 in binary?

 

0.625	2  =	1.25	whole number 1
0.25	2  =	0.5	whole number 0
0.5	2  =	1.0	whole number 1

 

Read whole numbers from top down.

 

Therefore 25.62510 = 11001.1012