Binary numbers are numbers built on the number base 2. (See Lecture 3 for more detail on number systems and number bases.)
The number system in today's daily life uses decimal numbers; in other words, its numbers are built on the number base 10: The number 1976 is understood to mean "One thousand nine hundred seventy-six" or in mathematical notation
1x1000 + 9x100 + 7x10 + 6x1
The factors 1, 10, 100 and 1000 are the successive powers of the number base 10:
1 = 100
10 = 101
100 = 102
1000 = 103
The choice of 10 as number base was made for convenience. In principle, any number can be used as a number base. Leibnitz introduced the system of binary calculation, which uses the number 2 as its base. Its factors are
2 = 20
4 = 21
8 = 22
16 = 23
32 = 24
64 = 25
128 = 26
256 = 27
512 = 28
1024 = 29
When written in binary notation the number 1976 therefore becomes 1111011100, which stands for
1x1024 + 1x512 + 1x256 + 1x128 + 0x64 + 1x32 + 1x16 + 1x8 + 0x4 + 0x2
The binary system may appear impractical because its number notation gets rather long. But it is extremely easy to use, since it only uses the numerals 0 and 1, which makes all calculations very easy.
More importantly, the numerals 0 and 1 can be represented as two physical states of a medium, for example positive and negative voltage or south and north magnetization, which is the basis of electronic computing. All electronic computers therefore operate in the binary system and "translate" the results of their calculations into the decimal system for our convenience only when the numbers are displayed on the screen or printed on paper.