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Line 1: − '''Real numbers''' can be thought of as numbers which can be represented by some infinite or finite decimal representation, such as 0.707106781187...+ − − They are called "real" because they actually exist in the real world, i.e. measures, weights, temperatures and so on are all real numbers, as opposed to the [[imaginary number]]s, which are just an abstract concept, but do not really exist. − − In classical physics, measurements of things that can vary smoothly and continuously, like [[speed]] or [[temperature]], are treated as real numbers. − − In a number line representation, the real numbers correspond to all the points on a geometric line. The distance between any two points on a line is a real number. − − In computer programming, some computer languages such as [[FORTRAN]] include a ''real'' data type that is intended to represent real numbers.<ref>In reality, the actual values the computer uses are very-high-precision fractions which can equal or approximate real numbers</ref> − − The real numbers include within them all of these other kinds of numbers: − − *The "natural numbers" or 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{{prove}} − *[[Irrational numbers]], like <math>\sqrt{10} = 3.162277660168...</math>π = 3.1415926525..., whose decimal representations never repeat or terminate. − − ==Formal definition== − − Formally, real numbers are defined as the unique [[Field (mathematics)|field]] which is [[ordered]], [[Complete (mathematics)|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 <math>\infty</math> or <math>-\infty</math> (infinity and minus infinity). However, there are non-standard models of real numbers which include <math>\infty</math> or include both <math>\infty</math> and <math>-\infty</math>. − − There is no largest real number, because you can always make a real number larger by adding 1 (or 137.035 or 6.023·10<sup>23</sup>) 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." − − ==Notes and references== − <references/> − − [[Category:Mathematics]]
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#REDIRECT [[Real number]]