Real Number

The set of all Rational and Irrational numbers is called the real numbers, or simply the ``reals,'' and denoted $\Bbb{R}$ . The set of real numbers is also called the Continuum, denoted

The real numbers can be extended with the addition of the Imaginary Number i, equal to $\sqrt{-1}$ . Numbers of the form

, where

and

are both real, are then called Complex Numbers. Another extension which includes both the real numbers and the infinite Ordinal Numbers of Georg Cantor

is the Surreal Numbers.

Pick two real numbers

and

at random in

with a Uniform Distribution. What is the Probability $P_{\rm even}$ that

, where

denotes Nint, the nearest Integer to

, is Even? The answer may be found as follows (Putnam Exam).

$\displaystyle P\left({a<{x\over y}<b}\right)$	$\textstyle =$	$\displaystyle \left\{\begin{array}{ll} P(ay<x<by) & \mbox{}\\ P\left({{x\over b}<y<{x\over a}}\right)& \mbox{}\end{array}\right.$
	$\textstyle =$	$\displaystyle \left\{\begin{array}{ll} \int_0^1 \int_{ay}^{by} dx\,dy={\textsty... ...}^{x/a} dy\,dx = {1\over 2a}-{1\over 2b} & \mbox{for $1<a<b$}\end{array}\right.$	(1)

$\displaystyle P_{\rm even}$	$\textstyle =$	$\displaystyle P\left({0<{x\over y}<{\textstyle{1\over 2}}}\right)+\sum_{n=1}^\i... ... P\left({2n-{\textstyle{1\over 2}}<{x\over y}<2n+{\textstyle{1\over 2}}}\right)$
	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}({\textstyle{1\over 2}}-0)+\sum_{n=1}^\inft... ...ver 2(2n-{\textstyle{1\over 2}})}-{1\over 2(2n+{\textstyle{1\over 2}})}}\right]$
	$\textstyle =$	$\displaystyle {1\over 4}+\sum_{n=1}^\infty \left({{1\over 4n-1}+{1\over 4n-1}}\right)$
	$\textstyle =$	$\displaystyle {\textstyle{1\over 4}}+({\textstyle{1\over 3}}-{\textstyle{1\over... ...\over 7}}-{\textstyle{1\over 9}}+\ldots)={\textstyle{1\over 4}}+(1-\tan^{-1} 1)$
	$\textstyle =$	$\displaystyle {5\over 4}-{\pi\over 4}={\textstyle{1\over 4}}(5-\pi)\approx 46.460\%.$	(2)

Plouffe's ``Inverse Symbolic Calculator'' includes a huge database of 54 million real numbers which are algebraically related to fundamental mathematical constants and functions.