Pick two real numbers and at random in with a uniform distribution. What is the probability that , where denotes the nearest integer function, is even?The answer may be found as follows.(1)(2)so(3)(4)(5)(6)(7)(8)(9)(10)(OEIS A091651).
The field of all rational and irrational numbers is called the real numbers, or simply the "reals," and denoted . The set of real numbers is also called the continuum, denoted . The set of reals is called Reals in the Wolfram Language, and a number can be tested to see if it is a member of the reals using the command Element[x, Reals], and expressions that are real numbers have the Head of Real.The real numbers can be extended with the addition of the imaginary number i, equal to . Numbers of the form , where and are both real, are called complex numbers, which also form a field. Another extension which includes both the real numbers and the infinite ordinal numbers of Georg Cantor is the surreal numbers."Plouffe's Inverter" includes a huge database of 54 million real numbers which are algebraically related to fundamental mathematical constants and functions.Almost all real numbers are lexicons, meaning that they do not obey probability..
The set , obtained by adjoining one improper element to the set of real numbers, is the set of projectively extended real numbers. Although notation is not completely standardized, is used here to denote this set of extended real numbers. With an appropriate topology, is the one-point compactification (or projective closure) of . As shown above, the cross section of the Riemann sphere consisting of its "real axis" and "north pole" can be used to visualize . The improper element, projective infinity (), then corresponds with the ideal point, the "north pole."In contrast to the signed affine infinities ( and ) of the affinely extended real numbers , projective infinity, , is unsigned, like 0. Regrettably, is also unordered, i.e., for it can be said neither that nor that . For this reason, is used much less often in real analysis than is . Thus, if context is not specified, "the extended real numbers" normally..
The set obtained by adjoining two improper elements to the set of real numbers is normally called the set of (affinely) extended real numbers. Although the notation for this set is not completely standardized, is commonly used. The set may also be written in interval notation as . With an appropriate topology, is the two-point compactification (or affine closure) of . The improper elements, the affine infinities and , correspond to ideal points of the number line. Note that these improper elements are not real numbers, and that this system of extended real numbers is not a field.Instead of writing , many authors write simply . However, the compound symbol will be used here to represent the positive improper element of , allowing the individual symbol to be used unambiguously to represent the unsigned improper element of , the one-point compactification (or projective closure) of .A very important property of , which lacks, is that every subset..