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 diff --git a/math/numbers.page b/math/numbers.pageindex 6481c75..3e9473e 100644--- a/math/numbers.page+++ b/math/numbers.page@@ -1,27 +1,25 @@-----format: markdown-categories: math-toc: no-...+Numbers+========== -# Numbers --*References: most of the definitions and notation in the section are based on [rudin] or [meserve]*+*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+: objects are incommensurable when their ratio isn't rational -## Real Numbers+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 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@@ -34,12 +32,15 @@ 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?+Exercise: is the square root of 5 algebraic or transcendental?++$e$+---------- -## e $e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$ -## Infinities+Infinities+---------- *aleph-zero* ($\aleph_0$) is the countably infinite set. @@ -47,6 +48,6 @@ 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+[^rudin]: **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976 [^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve.