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