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-========================
-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.