Math 2.0

Number theory Theorem

A root of an integer is an integer or irrational

Builds on Irrational numberIntegers

Statement

Let xx be a real number, let nn be a positive integer and let mm be an integer with xn=mx^n = m. If xx is not an integer, then xx is irrational.

Remarks

Thus, a rational number xx with xn∈ℤx^n \in \mathbb{Z} is an integer. For example, 23\sqrt[3]{2} is irrational, because 13<2<231^3 < 2 < 2^3 shows that it is not an integer. The statement follows from the rational root theorem for the polynomial Xn−mX^n - m.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Integers, object, no evidence levelIntegersobjectRational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectIrrational number, definition, no evidence levelIrrational numberdefinitionA root of an integer is an integer or irrational, statement, no evidence levelA root of an integer …statement
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. Integers object
  2. Rational numbers object
  3. Real numbers object
  4. Irrational number definition
  5. A root of an integer is an integer or irrational statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Irrational_number, General roots
About this page
Tag0XWCKJZRYH
URIm2://0XWCKJZRYH
Kindstatement, theorem
WritingW0 stub
MSC11J72
StatusThis copy of the site has no build data. Run m2 status 0XWCKJZRYH.
UsesIrrational number, Integers

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