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Number Systems

The base 10 system

103

102

101

100

1000

100

10

1

0

2

3

5

235 =  2 x 102 + 3 x 101 + 5 x 100

The base 2 system

      235 in the base 2 system

29

28 

27

26

25

24

23

22 

21 

20

512

256

128 

64

32

16

8

4

2

1

0

0

1

1

1

0

1

0

1

1

235 =  1 x 27  + 1 x 26  + 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 1 x 21 +1 x 20

Therefore, in an 8-bit system, the representation of the base 10 number 235 would be

1 1 1 0 1 0 1 1 

In a 16 bit system the representation would be 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1      

Why Not use Other Bases?    

The reason computers use only the base 2 system for basic operations (as opposed to some other more convenient to humans) is not theoretical. There is no theoretical reason why some other base is not used. For example, computers could use the standard base 10 system. One approach would be to assign voltages to each of the numbers.

For example, 0 volts could represent 0, 5 volts could represent 1, etc. and 45 volts could represent the number 9.

The problem is one of stability and the resultant reliability. Over time, voltages would begin to wander and result in errors in calculation. One the other hand, the base 2 system - on or off - is the most stable and hence the most reliable.

The Hexadecimal Reporting System

Humans have difficulty in thinking in the binary system, particularly if the number is large. It is not natural to keep track of huge amounts of 0s and 1s. To simplify reporting, the computer frequently states information in the hexadecimal system.

 This system represents the decimal numbers 0 to 9 as 0 to 9. The numbers from 10 to 15 are represented by the characters A to F.

 Example

Bit Pattern

Hexadecimal Representation

0000

0

0001

1

0010

2

0011

3

0100

4

0101

5

0110

6

0111

7

1000

8

1001

9

1010

A

1011

B

1100

C

1101

D

1110

E

1111

F

This notation takes advantage of the fact that bit patterns within a machine tend to have lengths in multiples of four. In particular, hexadecimal notation uses a single tymbol to represent four bits, meaning that a string of twelve bits can be represented by only three symbols.


For example, the 16 bit pattern


    1010010011001000


can be represented as


    A4C8