Numbers ========== *References: most of the definitions and notation in the section are based on [^rudin] or [^meserve]* incommensurable : objects are incommensurable when their ratio isn't rational Real Numbers ------------- The *real numbers* are defined via Dedakind cuts in [^rudin], or [^meserve] (p1-12). Complex Numbers ----------------- The *complex numbers* are constructed as an ordered pair of real numbers. Algebraic and Transcendental Numbers -------------------------------------- *Algebraic numbers* are solutions of polynomials, such as x in $a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$, where all a are real numbers. *Transcendental numbers* are not solutions to any such polynomials. All real numbers are either algebraic or transcendental. Some algebraic numbers aren't real (such as $i = \sqrt{-1}$). They can be rational or irrational. All transcendental numbers are irrational; some are not real. Exercise: is the square root of 5 algebraic or transcendental? $e$ ---------- $e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$ Infinities ---------- *aleph-zero* ($\aleph_0$) is the countably infinite set. Positive integers, integers, and rational numbers are all countably infinite. It is unproven that the real numbers are *aleph-one* ($\aleph_1$). [^rudin]: **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976 [^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve.