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Irrationals, Transcendentals, & Infinity

Some questions to explore:

  • It has been suggested earlier that the set of prime numbers is infinite.  Primes are a subset of integers.
  • Integers are a subset of Rationals, which are a subset of Reals, and then on to Complex and beyond.
  • If the integers are infinite, and if the real numbers are infinite, are there the same number of both?
  • What about rational numbers?  What about irrational numbers? Are those two sets equal in size?
  • What is an algebraic number?  What is a transcendental number?  How large are those sets?
  • Are all of these infinities equal?  How many infinities are there?  Who thought all of this up?  Are there any other approaches?

A few hints to start with, from Wikipedia:

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x2 − 2 = 0. Another irrational number that is not transcendental is the golden ratio, φ or ϕ, since it is a solution of the polynomial equation x2x − 1 = 0.

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