Overview
of a Computer - Data Representation
·
The first computers used vacuum
tubes to hold data
·
Vacuum tubes have two states - ON
and OFF
·
An ON state represents a 1
·
An OFF state represents a 0
·
Eight vacuum tubes strung
together can represent an 8 digit string of 0s and 1s
·
We use decimal (base 10) numbers
in our daily life
·
A decimal number is a string of
digits whose values are drawn from the set {0,1,2,3,4,5,6,7,8,9}
·
In general, a number system is
simply a way of representing numbers
·
A number system has a base (the
number of digits used in the number system)
· Consider a number in a base b number system:
· The value of this number is:
·
A binary number has a base of 2 where the valid digits are 0 or 1
·
E.g., 1001 binary == 9 decimal
(1*8 + 0*4 + 0*2 + 1)
·
An octal number has a base of 8 where the valid digits are 0 through 7
·
E.g., 031 octal == 25 decimal
(3*8 + 1)
·
A decimal number has a base of 10 where the valid digits are 0
through 9
·
E.g., 2000 decimal == 2000
·
A hexadecimal number has a base of 16 where the valid digits are 0
through F, i.e. {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}
·
E.g., xABBA hex == 43962 decimal
(10*4096 + 11*256 + 11*16 + 10)
Powers of 2
 2^0 |
1
|
2^11
|
2048 2K
|
|
2^1
|
2
|
2^12
|
4096 4K
|
|
2^2
|
4
|
2^13
|
8192 8K
|
|
2^3
|
8
|
2^14
|
16,384 16K
|
|
2^4
|
16
|
2^15
|
32,768 32K
|
|
2^5
|
32
|
2^16
|
65,536 64K
|
|
2^6
|
64
|
2^17
|
131,072 128K
|
|
2^7
|
128
|
2^18
|
263,144 256K
|
|
2^8
|
256
|
2^19
|
524,288 512K
|
|
2^9
|
512
|
2^20
|
1,048,576 1M
|
|
2^10
|
1024 1K
|
2^21
|
2,097,152 2M
|
1 KILO = 2^10
= 1024
1 MEG = 2^20 = 1024*1024 = 1,048,576
1 GIGA = 2^30
= 1024*1024*1024 = 1,073,741,824
To
evaluate a binary number, say 101101, simply add up the corresponding powers of 2:
101101 = 1× 25 + 0× 24
+ 1× 23 + 1× 22 + 0× 21 +1× 20
= 32 + 0 + 8 + 4 + 0 + 1
= 45