Now, Lets get down to the level of Study of Std. 4
and 5.
In small Classes we used to expand numbers, right?
Lets do that again for the number 123:
100
|
10
|
1
|
(102)
|
(101)
|
(100)
|
1
|
2
|
3
|
1x100
= 100
|
2x10
= 20
|
3x1
= 3
|
And 100+20+3 = 123
Similarly for, 256, We have,
100
|
10
|
1
|
(102)
|
(101)
|
(100)
|
2
|
5
|
6
|
2x100
= 200
|
5x10
= 50
|
6x1
= 6
|
And 200+50+6 =
256
Now We notice, that we are simply multiplying the
numbers on various places of the numbers with the corresponding powers of 10
and then adding it all up. right?
Now, Why do we multiply it with powers of 10 and not
any other number?(Why not 3, 5 or 200?). This is known as Decimal Notation
The Answer is because we use 10 digits to represent the numbers i.e. (0,1,2.3,4,5,6,7,8,9).
Now, just imagine what if there were just two numbers and whatever we had to represent we had to do so with just these two numbers. What would we do?
A: We would simply multiplying the corresponding digits with the powers of 2 instead of 10.
Makes sense, right?
Then let’s do that for 1111011 :
26
|
25
|
24
|
23
|
22
|
21
|
20
|
64
|
32
|
16
|
8
|
4
|
2
|
1
|
1
|
1
|
1
|
1
|
0
|
1
|
1
|
64x1
|
32x1
|
16x1
|
8x1
|
4x0
|
2x1
|
1x1
|
And 64+32+16+8+0+2+1 = 123
You can see for yourself that there is no other representation for the number 123.
Similarly if we try it for 100000000,
28
|
27
|
26
|
25
|
24
|
23
|
22
|
21
|
20
|
256
|
128
|
64
|
32
|
16
|
8
|
4
|
2
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
256x1
|
128x0
|
64x0
|
32x0
|
16x0
|
8x0
|
4x0
|
2x0
|
1x0
|
And 256 + 0 +0 +0 +0 +0+0+0+0 = 256.
Thus, We see that every number in Decimal has a
unique representation in Binary and like that we may store that number in a
Computer in Transistors.
Exercise
Convert the following to
Binary:
1) 129 2)25 3) 250 4)259
5) 999
Convert the Following to
Decimal Notation:
1) 1101110 2)1001000 3) 111110000 4)1
5) 000110