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least upper bound property!
The real line is useful as a [[coordinate system]] for [[graph]]ing [[functions]]. Thus, the [[x-axis]] and [[y-axis]] are both instances of the real line. The real line is the basis for geometric [[measurement]]s, and more generally for ideas in [[metric topology]].
The real line is useful as a [[coordinate system]] for [[graph]]ing [[functions]]. Thus, the [[x-axis]] and [[y-axis]] are both instances of the real line. The real line is the basis for geometric [[measurement]]s, and more generally for ideas in [[metric topology]].
==What is the problem? Aren't rational numbers good enough?==
Any real-world measurement that anyone could possibly make, one can make as accurately as one wants with rational numbers. For example, one can calculate the ratio of the circumference of a circle to its diameter to within one part is a trillion using the number 3.1415926535898 (<math>\pi\,</math> itself is irrational.) Put another way, you never have to worry about the difference between the rationals and the reals in a lumber yard or a laboratory. The technical term that topologists use for this state of affairs is that the rationals are [[dense subset|dense]].
The shortcoming of the rationals, that is overcome by defining the reals, is a somewhat subtle theoretical point. The most direct example is that, if one lived in a world with only rational numbers, 2 has no square root, even though it obviously should.
::One can easily prove that there is no rational number m/n such that (m/n)<sup>2</sup> = 2. The factors of m<sup>2</sup> all come in pairs, as do the factors of n<sup>2</sup>. But the factors of m<sup>2</sup> must be the same as the factors of n<sup>2</sup> except for a single extra factor of 2.
The theoretical property that the rational numbers lack is called the ''least upper bound'' property.
:Definition: A number B is an ''upper bound'' for a set of numbers if no element of the set is greater than B. (There is also the notion of a lower bound.)
For example, 10 is an upper bound for the open interval <math>(3, 6)\,</math>. 7 is also an upper bound, as is 6. 5 is not. 2 is a lower bound.
Some sets do not have upper bounds. For example, all rational or real numbers, or all odd integers.
:Definition: A number L is a ''least upper bound'' (often abbreviated "lub") if it is an upper bound and no other upper bound is smaller. (There is also the notion of a greatest lower bound, abbreviated "glb".) 6 is the lub of the open interval <math>(3, 6)\,</math>. 3 is its glb. 6 and 3 are also the lub and glb of the closed interval <math>[3, 6]\,</math>—the inclusion of the endpoints makes no difference.
A set has the ''least upper bound property'' if every set that has an upper bound has a least upper bound. There is also a ''greatest lower bound property'', and any reasonable set having one property has the other.
The least upper bound property is extremely important in caclulus and analysis. It is essential for many theorems, notably the ''[[mean value theorem]]'' and the ''intermediate value theorem''.
==Infinity==
==Infinity==
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Calculus]]