Editing Naive set theory (section)

== Some important sets ==
There are some ubiquitous sets for which the notation is almost universal. Some of these are listed below. In the list, ''a'', ''b'', and ''c'' refer to [[natural number]]s, and ''r'' and ''s'' are [[real number]]s.
# [[Natural number]]s are used for counting. A [[blackboard bold]] capital '''N''' (<math>\mathbb{N}</math>) often represents this set.
# [[Integer]]s appear as solutions for ''x'' in equations like ''x'' + ''a'' = ''b''. A blackboard bold capital '''Z''' (<math>\mathbb{Z}</math>) often represents this set (from the German ''Zahlen'', meaning ''numbers'').
# [[Rational number]]s appear as solutions to equations like ''a'' + ''bx'' = ''c''. A blackboard bold capital '''Q''' (<math>\mathbb{Q}</math>) often represents this set (for ''[[quotient]]'', because R is used for the set of real numbers).
# [[Algebraic number]]s appear as solutions to [[polynomial]] equations (with integer coefficients) and may involve [[Nth root|radicals]] (including <math>i=\sqrt{-1\,}</math>) and certain other [[irrational number]]s. A '''Q''' with an overline (<math>\overline{\mathbb{Q}}</math>) often represents this set. The overline denotes the operation of [[algebraic closure]].
# [[Real number]]s represent the "real line" and include all numbers that can be approximated by rationals. These numbers may be rational or algebraic but may also be [[transcendental number]]s, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital '''R''' (<math>\mathbb{R}</math>) often represents this set.
# [[Complex number]]s are sums of a real and an imaginary number: <math>r+s\,i</math>. Here either <math>r</math> or <math>s</math> (or both) can be zero; thus, the set of real numbers and the set of strictly imaginary numbers are subsets of the set of complex numbers, which form an [[algebraic closure]] for the set of real numbers, meaning that every polynomial with coefficients in <math>\mathbb{R}</math> has at least one [[Root of a function|root]] in this set. A blackboard bold capital '''C''' (<math>\mathbb{C}</math>) often represents this set. Note that since a number <math>r+s\,i</math> can be identified with a point <math>(r,s)</math> in the plane, <math>\mathbb{C}</math> is basically "the same" as the [[Cartesian product]] <math>\R\times\R</math> ("the same" meaning that any point in one determines a unique point in the other and for the result of calculations, it doesn't matter which one is used for the calculation, as long as multiplication rule is appropriate for <math>\mathbb{C}</math>).