# 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 a0xn + a1xn − 1 + a2xn − 2 + ...an = 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 (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 (1).

1. Principles of Mathematical Analysis (3rd ed), by Walter Rudin. McGraw-Hill, 1976↩︎

2. Fundamental Concepts of Algebra, by Bruce Meserve.↩︎

3. Principles of Mathematical Analysis (3rd ed), by Walter Rudin. McGraw-Hill, 1976↩︎

4. Fundamental Concepts of Algebra, by Bruce Meserve.↩︎