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========================
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.
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