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