Math 2.0

Number theory Definition

Irrational number

Builds on Real numbersRational numbers

Definition

A real number xx is irrational when xx is not a rational number. That is, there are no integers aa and bb with b≠0b \neq 0 and x=a/bx = a / b. The set of irrational numbers is ℝ∖ℚ\mathbb{R} \setminus \mathbb{Q}.

Remarks

The number 2\sqrt{2} is irrational. The numbers π\pi and ee are also irrational. The decimal expansion of a real number is eventually periodic exactly when the number is rational. Thus the decimal expansion of an irrational number never repeats. The set ℚ\mathbb{Q} is countable and ℝ\mathbb{R} is not, thus almost all real numbers are irrational.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph
Rational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectIrrational number, definition, no evidence levelIrrational numberdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Rational numbers object
  2. Real numbers object
  3. Irrational number definition

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Irrational_number, lead section
About this page
Tag080T2TXVQV
URIm2://080T2TXVQV
Kinddefinition
WritingW1 stated
MSC11J72
DefinesIrrational number
UsesReal numbers, Rational numbers

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by