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 diff --git a/math/numbers.page b/math/numbers.pageindex 541d174..6481c75 100644--- a/math/numbers.page+++ b/math/numbers.page@@ -1,54 +1,52 @@-========================-Numbers-========================+---+format: markdown+categories: math+toc: no+... -.. note::- incomplete+# Numbers -.. note:: - Most of the definitions and notation in the section are based on [rudin]_ or [meserve]_ -.. contents::+*References: most of the definitions and notation in the section are based on [rudin] or [meserve]* -*incommensurable*+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).+## Real Numbers++The *real numbers* are defined via Dedakind cuts in [^rudin], or [^meserve] +(p1-12).++## Complex Numbers -Complex Numbers-================== The *complex numbers* are constructed as an ordered pair of real numbers. -Algebraic and Transendental 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+$a_0 x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... a_n = 0$, 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+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. Exersize: is the square root of 5 algebraic or transcendental? -e-========-:latex:$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$+## e+$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$++## Infinities -Infinities-==================-*aleph-zero* (:latex:$\aleph_0$) is the countably infinite set.+*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* (:latex:$\aleph_1$).+It is unproven that the real numbers are *aleph-one* ($\aleph_1$). -.. [rudin] Principles of Mathematical Analysis (3rd ed):title:, by Walter Rudin. McGraw-Hill, 1976+[^rudin] **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976 -.. [meserve] Fundamental Concepts of Algebra:title:, by Bruce Meserve.+[^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve.