======================== Numbers ======================== .. note:: incomplete .. note:: Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ .. contents:: *incommensurable* objects are incommensurable when their ratio isn't rational Real Numbers ================== The *real numbers* are defined via Dedakind cuts in [rudin]_, or [meserve]_ (1-12). Complex Numbers ================== The *complex numbers* are constructed as an ordered pair of real numbers. Algebraic and Transendental Numbers =============================================== *Algebraic numbers* are solutions of polynomials, such as x in :latex:$a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$, 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 :latex:$i = \sqrt{-1}$). They can be rational or irrational. All transcendental numbers are irrational; some are not real. Exersize: is the square root of 5 algebraic or transcendental? e ======== :latex:$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$ Infinities ================== *aleph-zero* (:latex:$\aleph_0$) 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* (:latex:$\aleph_1$). .. [rudin] Principles of Mathematical Analysis (3rd ed):title:, by Walter Rudin. McGraw-Hill, 1976 .. [meserve] Fundamental Concepts of Algebra:title:, by Bruce Meserve.