Real number

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Real numbers include within them all of these other kinds of numbers:

  • The positive integers, 1, 2, 3, ...
  • Zero and the negative integers
  • Fractions, like 355/113
  • Any decimal representation which terminates (comes to an end), like 6.023, because this is just a way of writing a fraction (in this case, 6023/1000)
  • Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractions
  • Irrational numbers, like π = 3.1415926525..., whose decimal representations never repeat or terminate.

Formal definition

Formally, real numbers are defined as the unique field which is ordered, complete, and Archimedean. The reals can be constructed from the rationals by means of Dedekind cuts or Cauchy sequences, i.e. it is the completion of the metric space of rational numbers.

Infinity

The real numbers do not include or (infinity and minus infinity). However, there are non-standard models of real numbers which include or include both and .

There is no largest real number, because you can always make a real number larger by adding 1 (or 137.035 or 6.023·1023) to it, and no smallest real number, because you can always make a real number smaller by subtracting from it.

Every real number is finite. One way to see this is to observe that you cannot subtract infinity from itself—the result is indeterminate—but, for any real number x, then x - x = 0, exactly.

It is sometimes convenient to have a set of numbers that does include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 754-1985; this system is built in to modern processor chips. It provides for values which print out as INF and -INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or -INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."

Real line

The real numbers can be thought of as a line, called the real line. Each real number represents a point on the real line. However, it is a mistake to think of the real line as a row of individual points, like beads. There is no real number “just to the right” of a given real number. This is because the real numbers, like the rational numbers, are a dense set, so points accumulate around each other.[1]

The real line is useful as a coordinate system for graphing functions. Thus, the x-axis and y-axis are both instances of the real line. The real line is the basis for geometric measurements, and more generally for ideas in metric topology.

Notes and references