Number system, Computer Engineering

 

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:   x­ B2 + x1  B1 + x 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

 

Posted Date: 8/20/2012 10:59:23 PM | Location : United States







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