Binary System of Numeration

a system of numeration constructed on the positional principle of writing numbers with the base 2. Only two symbols are used—the digits 0 and 1. Here, as in every positional system, the value of a number also depends on the position it occupies. The number 2 is considered as a unit of the second order and written thus: 10 (read as “one, zero”). Each unit of the next order is twice as large as the preceding unit, that is, these units make up the number sequence 2, 4, 8, 16, . . . , 2ⁿ, . . . . To convert a number from the decimal number system into the binary system, the number is divided successively by 2 and the remainders are written in inverse order, from the last to the first— for example, 43 = 21 . 2 + 1,21 = 10.2 + 1,10 = 5 . 2 + 0, 5 = 2. 2 + 1,2 = 1.2 + 0, 1 = 0.2 + 1. Thus the binary representation of the number 43 is 101011. So 101011 in the binary number system stands for 1 . 2⁰ + 1 . 2¹ + 0 . 2² + 1 . 2³ + 0 . 2⁴ + 1 . 2⁵.

All arithmetic operations are accomplished very easily in the binary system. For instance, the multiplication table is reduced to the single equality 1–1 = 1. However, notation in the binary number system is laborious. For instance, the number 9000 has 14 digits. But owing to the fact that only two digits are employed in the system, it is often useful in theoretical problems and for calculations on digital computers.