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Line 14: − + − − ==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 compeletion 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, 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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*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 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}}
*Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractions{{prove}}
*[[Irrational numbers]], like &sqrt;10 = 3.162277660168... π = 3.1415926525..., whose decimal representations never repeat or terminate.{{prove}}
*[[Irrational numbers]], like