Math 2.0

Number theory Diophantine approximation and transcendence Theorem

Irrationality of the square root of 2

Builds on Irrational numberSquare root

Statement

The square root 2\sqrt{2} of 22 is an irrational number. That is, there are no integers aa and bb with b≠0b \neq 0 and (a/b)2=2(a / b)^2 = 2.

Remarks

A classical proof: suppose that a/b=2a / b = \sqrt{2} in lowest terms. Then a2=2b2a^2 = 2 b^2, thus aa is even. Then bb is also even, which is a contradiction. The theorem is a special case of the irrationality of the square root of a prime.

History

The ancient Greeks knew this result. Tradition attributes its discovery to the Pythagoreans.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Rational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionIrrational number, definition, no evidence levelIrrational numberdefinitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionIrrationality of the square root of 2, statement, no evidence levelIrrationality of the …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. Rational numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Irrational number definition
  5. Supremum definition
  6. Completeness axiom statement
  7. Existence of square roots of nonnegative reals statement
  8. Square root definition
  9. Irrationality of the square root of 2 statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Square_root_of_2, Proofs of irrationality
  2. Web https://en.wikipedia.org/wiki/Square_root_of_2, History
About this page
Tag0DZGSDHD92
URIm2://0DZGSDHD92
Kindstatement, theorem
WritingW0 stub
MSC11J72
StatusThis copy of the site has no build data. Run m2 status 0DZGSDHD92.
UsesIrrational number, Square root

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