Number system

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Introduction and notation

In a Number system of base N, there are N basic symbols (or digits) that correspond to the first N natural numbers including zero. To generate the rest of the numbers, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by N.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base is added in subscript to the right of the number, like this: numberbase. Numbers without subscript are considered to be decimal.

The binary system

The most basic system is the binary system, N=2, so there are two symbols, 0 and 1. The binary number 101102 means 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20 = 2210 (in the familiar decimal notation).

Non-integers can be represented by using negative powers, which are set off from the other digits by means of a radix point (called a decimal point in base 10). The binary number 11.012 thus means 1 x 21 + 1 × 20 + 0 × 2-1 + 1 × 2-2 which equals 3.2510.

Additional information

Note that no matter in which base, numbers have terminating or repeating expansions if and only if they are rational. A number which is terminating in one base may get a repeating expansion in another (thus 0.310 = 0.0100110011001...2), and vice versa. An irrational number stays unperiodic (infinite amount of unrepeating digits) in all bases. Thus, for example in base 2, π = 3.1415926...10 can be written down as the unperiodic 11.001001000011111...2.

The system that humans use has base 10 because we have ten fingers on our hands. Base 60 has also appeared from time to time (hence the division of an hour to 60 minutes and an minute to 60 seconds).

Electronic components (first vacuum tubes then transistors) may have only 2 possible states: open (1) and closed (0). Because this is exactly the set of binary digits, and because arithmetics in a binary system are the easiest to describe electronically (using Boolean algebra), the binary system became natural for electronic computers. It is used to perform integer arithmetic in almost all electronic computers (the only exception being the exotic base-3 and base-10 designs that were discarded very early in the history of computing). Note however that a computer does not treat all of its data as integers. Thus, some of it may be treated as texts and program data. Real numbers (variables that can be not whole) are usually written down in the floating point notation, that has different rules of arithmetic.

If N is a prime number, one can define base-N numbers whose expansion to the left never stops; these are called the p-adic numbers.

Specific number systems