Real numbers

http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/Lectures/L5.html

The Axioms

I The algebraic axioms R is a field under + and . This means that (R, +) and (R, .) are both abelian groups and the distributive law (a + b)c = ab + ac holds.

II The order axioms There is a relation > on R. (That is, given any pair a, b then a > b is either true or false). It satisfies:

a) Trichotomy: For any a ∈ R exactly one of a > 0, a = 0, 0 < a is true.

b) If a, b > 0 then a + b > 0 and a.b > 0

c) If a > b then a + c > b + c for any c

Something satisfying axioms I and II is called an ordered field.

A real number is a value of a continuous quantity that can represent a distance along a line.

The set of real numbers is denoted by $\mathbb{R}$

The natural numbers are the basis from which many other number sets may be built by extension: reals by including with the rationals the limits of (converging) Cauchy sequences of rationals

The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials.

The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number).

Included within the irrationals are the transcendental numbers, such as π (3.14159265...).

In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

reals = rational + irrational

The irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

irrationals = $\mathbb{R} \mathbb{Q}$

The set of real numbers has several standard structures:

• order: each number is either less or more than any other number.

• algebraic structure: there are operations of multiplication and addition that promote the set of real numbers into a field.

• measure: intervals along the real line have a specific length, which can be extended to the Lebesgue measure on many of its subsets.

• metric: there is a notion of distance between points.

• geometry: it is equipped with a metric and it is flat.

• topology: there is a notion of open sets.

There are interfaces among these:

• Its order and, independently, its metric structure induce its topology.

• Its order and algebraic structure make it into an ordered field.

• Its algebraic structure and topology make it into a Lie group, a type of topological group.