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diff --git a/math/numbers b/math/numbers deleted file mode 100644 index 541d174..0000000 --- a/math/numbers +++ /dev/null @@ -1,54 +0,0 @@ -======================== -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. |