Naive Set Theory - Some Important Sets

Some Important Sets

1. Natural numbers are used for counting. A blackboard bold capital N often represents this set.
2. Integers appear as solutions for x in equations like x + a = b. A blackboard bold capital Z often represents this set (from the German Zahlen, meaning numbers).
3. Rational numbers appear as solutions to equations like a + bx = c. A blackboard bold capital Q often represents this set (for quotient, because R is used for the set of real numbers).
4. Algebraic numbers appear as solutions to polynomial equations (with integer coefficients) and may involve radicals and certain other irrational numbers. A blackboard bold capital A or a Q with an overline often represents this set. The overline denotes the operation of algebraic closure.
5. Real numbers 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 numbers, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital R often represents this set.
6. Complex numbers are sums of a real and an imaginary number: r + si. Here both r and s can equal zero; thus, the set of real numbers and the set of 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 has at least one root in this set. A blackboard bold capital C often represents this set. Note that since a number r + si can be identified with a point (r, s) in the plane, C is basically "the same" as the Cartesian product R×R ("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).