Numbers
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*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
Real Numbers
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The *real numbers* are defined via Dedakind cuts in [^rudin], or [^meserve]
(p1-12).
Complex Numbers
-----------------
The *complex numbers* are constructed as an ordered pair of real 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
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 $i = \sqrt{-1}$). They
can be rational or irrational. All transcendental numbers are irrational;
some are not real.
Exercise: is the square root of 5 algebraic or transcendental?
$e$
----------
$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$
Infinities
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*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* ($\aleph_1$).
[^rudin]: **Principles of Mathematical Analysis (3rd ed)**, by Walter Rudin. McGraw-Hill, 1976
[^meserve]: **Fundamental Concepts of Algebra**, by Bruce Meserve.