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Math 2.0

Number theory Proof

Proof of the irrationality of the square root of a prime

Builds on Irrational number Square root Fraction in lowest terms Euclid's lemma Prime number Greatest common divisor Divisibility

Proof

We argue by contradiction, as in the proof for $\sqrt{2}$ . Let $p$ be a prime number. Since $p > 0$ , the square root $\sqrt{p}$ is a real number with $(\sqrt{p})^2 = p$ .

Suppose that $\sqrt{p}$ is not irrational, that is, $\sqrt{p}$ is rational.
By the existence of a fraction in lowest terms, $\sqrt{p} = a / b$ with integers $a$ and $b$ , $b > 0$ and $\gcd(a, b) = 1$ .
Squaring gives $a^2 / b^2 = p$ . Multiplying by $b^2$ gives $a^2 = p b^2$ .
So $p$ divides $a \cdot a$ . By Euclid's lemma, $p$ divides $a$ . Write $a = p c$ with $c$ an integer.
Then $p^2 c^2 = p b^2$ . Since $p \neq 0$ , dividing by $p$ gives $b^2 = p c^2$ .
So $p$ divides $b \cdot b$ , and Euclid's lemma gives $p \mid b$ .
Thus $p$ is a common divisor of $a$ and $b$ . By the definition of the greatest common divisor, $\gcd(a, b) \ge p \ge 2$ .
This contradicts $\gcd(a, b) = 1$ . So $\sqrt{p}$ is irrational.

Idea

If $\sqrt{p} = a / b$ in lowest terms, then $a^2 = p b^2$ . Because $p$ is prime, $p$ must divide $a$ , and then $p$ must divide $b$ as well. This contradicts the choice of lowest terms.

Reading path

To understand this page from the ground up, read these 65 pages in order. The foundations come first. The path takes the proofs Proof of Bézout's identity, Proof of the generalized Euclid's lemma, Proof of the commutativity of addition by induction, Proof of Euclid's lemma, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of strong induction from ordinary induction, Proof of the existence of square roots of nonnegative reals, Proof of the existence and uniqueness of a fraction in lowest terms, Proof of the well-ordering principle from strong induction, Proof of the division algorithm.

Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Divisibility definition
Element definition
Integers object
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Set bounded above definition
Greatest common divisor definition
Prime number definition
Set definition
Ring of integers statement
Successor function definition
Implication definition
Predicate definition
Irrational number definition
The reals form a field statement
Supremum definition
Coprime integers definition
Ordered pair definition
Addition of natural numbers definition
The successor function is injective statement
Universal quantifier definition
Zero is not a successor statement
The rationals form a field statement
Completeness axiom statement
Cartesian product definition
Subset definition
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
The rationals form an ordered field statement
Binary relation definition
Order on the reals definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Strong induction statement
Preorder definition
Proof of the well-ordering principle from strong induction proof
Partial order definition
Well-ordering principle statement
Total order definition
Proof of the division algorithm proof
Division algorithm statement
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Proof of Bézout's identity proof
The reals form an ordered field statement
Bézout's identity statement
Proof of the existence of square roots of nonnegative reals proof
Proof of the generalized Euclid's lemma proof
Existence of square roots of nonnegative reals statement
Generalized Euclid's lemma statement
Square root definition
Proof of Euclid's lemma proof
Proof of the existence and uniqueness of a fraction in lowest terms proof
Euclid's lemma statement
Fraction in lowest terms statement
Proof of the irrationality of the square root of a prime proof

Where this leads

1 page builds on this one.

Irrationality of the square root of a prime

Further reading

Web https://en.wikipedia.org/wiki/Square_root_of_2, Proof by infinite descent

About this page

Tag	`3T86NQ3NQY`
URI	`m2://3T86NQ3NQY`
Kind	proof
Writing	W0 stub
MSC	11J72
Proves	Irrationality of the square root of a prime
Uses	Irrational number, Square root, Fraction in lowest terms, Euclid's lemma, Prime number, Greatest common divisor, Divisibility
Methods	Proof by contradiction
Changed	4 November 2026

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