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# Numbers


*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

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 Transendental 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.

Exersize: is the square root of 5 algebraic or transcendental?

## e
$e = \lim_{x \rightarrow 0} (1+x)^{\frac{1}{x}}$

## Infinities

*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.