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preface boilerplate, add stuff about connectedness from "rational number". Maybe it's too advanced, but if it belongs anywhere, it belongs here.
{{Math-m}}
{{Math-m}}
The '''real numbers''' are a set of numbers with extremely important theoretical and practical properties. They can be considered to be the numbers used for ordinary measurement of physical things like length, area, weight, charge, etc. They are the 4<sup>th</sup> item in this hierarchy of types of [[number]]s:
The '''real numbers''' are a set of numbers with extremely important theoretical and practical properties. They can be considered to be the numbers used for ordinary measurement of physical things like length, area, weight, charge, etc. Mathematicians denote the set of real numbers with an ornate capital letter: <math>\mathbb{R}</math>. They are the 4<sup>th</sup> item in this hierarchy of types of [[number]]s:
*The "[[natural number]]s", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)
*The "[[natural number]]s", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)
Real numbers are typically represented by a decimal (or any other base) representation, as in 3.1416. It can be shown that any decimal representation that either terminates or gets into an endless repeating pattern is rational. The other numbers are real numbers that are ''irrational''. Examples are <math>\sqrt{10} = 3.162277660168...\,</math> and <math>\pi = 3.1415926535...\,</math>. These decimal representations neither repeat nor terminate.
Real numbers are typically represented by a decimal (or any other base) representation, as in 3.1416. It can be shown that any decimal representation that either terminates or gets into an endless repeating pattern is rational. The other numbers are real numbers that are ''irrational''. Examples are <math>\sqrt{10} = 3.162277660168...\,</math> and <math>\pi = 3.1415926535...\,</math>. These decimal representations neither repeat nor terminate.
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==Formal definition==
==Formal definition==
<!-- [This paragraph is all wrong, and doesn't belong here in any case. The topic should be covered in the compter arithmetic page, perhaps pointing here.] 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."-->
<!-- [This paragraph is all wrong, and doesn't belong here in any case. The topic should be covered in the compter arithmetic page, perhaps pointing here.] 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."-->
==Topological properties==
{{Math-a}}
In the field of [[topology]], an important difference between the rational and real numbers is that the rational numbers are "totally disconnected"<!--No wikilink; I doubt that it would ever change color from red.-->, whereas the real numbers are [[connected (topology)|connected]]. To see this, observe that the set of numbers strictly less than <math>\sqrt{2}</math> is open in both sets, but it is also closed in the rationals but not in the reals. In each case, it has a [[limit point]], namely, <math>\sqrt{2}</math>, which it clearly does not contain. In the reals, the fact that <math>\sqrt{2}</math> is not contained makes the set not closed, whereas in the rationals, it doesn't matter because that point doesn't exist. In the rationals, the fact that the set is both open and closed makes it disconnected from its complement. Any open set in the rationals contains situations like this, which makes the rationals totally disconnected.
==History==
==History==
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Calculus]]
[[Category:Topology]]