Irrationality of square roots of non-squares

Builds on Fundamental theorem of arithmetic

Statement

Let $n$ be a natural number. Then $\sqrt{n}$ is irrational if and only if $n$ is not a perfect square.

Remarks

If $n = m^2$ , then $\sqrt{n} = m$ is rational. The content is the other direction. A proof: if $\sqrt{n} = a/b$ , then $a^2 = n b^2$ . By unique factorization, each prime has an even exponent in $a^2$ and in $b^2$ , so also in $n$ . Then $n$ is a square. For example, $\sqrt{2}$ , $\sqrt{3}$ and $\sqrt{12}$ are irrational. The same argument shows that the $k$ -th root of $n$ is irrational unless $n$ is a $k$ -th power.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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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.

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Divisibility definition
Prime number definition
Euclid's lemma statement
Existence of a prime divisor statement
Prime factorization definition
Existence of prime factorization statement
Fundamental theorem of arithmetic statement
Irrationality of square roots of non-squares statement

Tag	`0N10YRNNWV`
URI	`m2://0N10YRNNWV`
Kind	statement, theorem
Writing	W0 stub
MSC	11J72
Status	This copy of the site has no build data. Run `m2 status 0N10YRNNWV`.
Uses	Fundamental theorem of arithmetic

Irrationality of square roots of non-squares

Statement

Remarks

Reading path

Further reading

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