# math/numbers

# Numbers

*References: most of the definitions and notation in the section
are based on ^{1} or ^{2}*

- incommensurable
- objects are incommensurable when their ratio isn’t rational

## Real Numbers

The *real numbers* are defined via Dedakind cuts in ^{3}, or ^{4}
(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

*aleph-zero* (ℵ_{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* (ℵ_{1}).