Irrationality of Non-Perfect-Square Roots

Problem Statement

Let $a$ be a positive integer. If $a$ is not a perfect square, prove that $\sqrt{a}$ is irrational.

Hints

Attempt the proof independently first. Focus on the key theorem, algebraic transformation, or contradiction setup that links the hypothesis to the target conclusion.

Solutions

Proof by Contradiction

Step 1: Assume the negation

Assume, for contradiction, that $\sqrt{a}$ is rational. Then we can write

\sqrt{a} = \frac{p}{q}

where $p, q \in \mathbb{Z}$ , $q \neq 0$ , and $\gcd(p,q) = 1$ .

Step 2: Square both sides

\begin{aligned} a &= \frac{p^2}{q^2} \\ p^2 &= aq^2 \end{aligned}

Step 3: Compare prime factorizations

By the Fundamental Theorem of Arithmetic, every positive integer has a unique prime factorization.

When an integer is squared, every prime exponent in its factorization becomes even. So:

in $p^2$ , every prime appears with an even exponent;
in $q^2$ , every prime appears with an even exponent.

Now the equation

p^2 = aq^2

shows that $aq^2$ must also have only even prime exponents.

Since $q^2$ already contributes only even exponents, this is possible only if every prime appearing in $a$ also has an even exponent.

Step 4: Derive the contradiction

But if every prime in the factorization of $a$ has an even exponent, then $a$ is a perfect square.

This contradicts the hypothesis that $a$ is not a perfect square.

Conclusion

Therefore, our assumption was false, and $\sqrt{a}$ must be irrational.

\hfill $\square$

Takeaways

Parity of Prime Exponents: A perfect square has only even exponents in its prime factorization
Why the Method Works: Squaring forces all prime exponents in $p^2$ and $q^2$ to be even, so the same must be true for $a$
Generalization: This extends proofs like $\sqrt{2}$ is irrational'' or $\sqrt{23}$ is irrational'' to any positive integer that is not a square
Key Tool: The argument relies on the Fundamental Theorem of Arithmetic and uniqueness of prime factorization

Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills:

HSC Proofs: https://vumaths.com/booklets/hsc-proofs/
HSC Functions: https://vumaths.com/booklets/hsc-functions/
HSC Probability: https://vumaths.com/booklets/hsc-probability/

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