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 + a₁x^n − 1 + a₂x^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 (ℵ₀) 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 (ℵ₁).