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Binary Number System
A Binary Number is made up of only 0s and 1s.
Everywhere, except for computer-related operations, the main
system of mathematical notation today is the decimal system, which is a base-10 system. As in other number systems, the position of a
symbol in a base-10 number denotes the value of that symbol in terms of
exponential values of the base. That is, in the decimal system, the quantity
represented by any of the ten symbols used - 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 - depends
on its position in the number.
Unlike the decimal system, only two digits - 0, 1 - suffice to represent a number in the binary
system. The binary system plays a crucial role in computer science and technology.
The first 20 numbers in the binary notation are 1, 10, 11, 100, 101, 110, 111,
1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011,
10100, the origin of which may be better understood if they are re-written in the
following way:
1:
00001
11: 01011
2:
00010
12: 01100
3:
00011
13: 01101
4:
00100
14: 01110
5:
00101
15: 01111
6:
00110
16: 10000
7:
00111
17: 10001
8:
01000
18: 10010
9:
01001
19: 10011
10:
01010
20: 10100
Any decimal number
can be converted into the binary system by summing the appropriate multiples of
the different powers of two.
For example, starting from the right, 10101101 represents (1 x 20)
+ (0 x 21) + (1 x 22)
+ (1 x 23) + (0 x 24)
+ (1 x 25) + (0 x 26)
+ (1 x 27) = 173. This example can be used for
the conversion of binary numbers into decimal numbers.
For the conversion of decimal numbers to binary numbers, the same principle can be used, but the
other way around. Thus, to convert, first find the highest power of two that does not exceed the given number, and place a 1 in the
corresponding position in the binary number. For example, the highest power of two in the decimal number 519 is 29 = 512. Thus,
a 1 can be inserted as the 10th digit, counted from the right: 1000000000.
In the remainder, 519 - 512 = 7, the highest power of 2 is 22 = 4, so the third zero from the right can be replaced by a 1: 1000000100. The
next remainder, 3, consists of the sum of two powers of 2: 21 + 20, so the first and second zeros from the
right are replaced by 1: 519 = 10000001112.
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