# Maths - Proof Roots of Primes are Irrational

> Source: https://ollybritton.com/notes/a-level/maths/topics/proof-roots-are-irrational/ · Updated: 2025-01-17 · Tags: maths, school, proof

## See Also
- [Maths - Proof by Contradiction](https://ollybritton.com/notes/a-level/maths/topics/proof-by-contradiction/)

## Flashcards
##### When proving irrationality of a surd $x$, what is the setup??
Assume $x = \frac{a}{b}$ where $a$ and $b$ are integers that do not share any factors.

##### Why can't $a$ and $b$ share any factors when proving the irrationality of a surd??
Because it would mean you could simplify $\frac{a}{b}$.

##### How do you get rid the square root over $x$ when proving it's irrationality??
Square both sides.

##### What's the next step after assuming $\sqrt{3} = \frac{a}{b}$??
$$
3 = \frac{a^2}{b^2}
$$

##### How can you rearrange $3 = \frac{a^2}{b^2}$??
$$
3a^2 = b^2
$$

##### Given something like $xa^2 = b^2$, what does this tell us about $b$??
$x$ is a factor of $b$.

##### Why is $3$ a factor of $b$ in $3a^2 = b^2$??
Since $b^2$ is equal to some integer ($a^2$) multiplied by $3$.

##### If $x$ is a prime factor of $b^2$, then why must it also be a prime factor of $b$??
Consider that $b^2$'s prime factorisation must come in pairs, since it's the same as the product of $b$'s prime factors squared.

##### If $b$ is a multiple of $3$, then how could you rewrite $b$??
$$
(3c)
$$

##### What's another way of expressing $3a^2 = b^2$??
* $3a^2 = (3c)^2$
* $3a^2 = 9c^2$

##### What does the statement $3a^2 = 9c^2$ tell you about $a$??
* Simplifying: $a^2 = 3c^2$
* $3$ must be a factor of $a$.

##### What is the core contradiction at the heart of the proof the roots of primes are irrational??
If $a$ and $b$ share factors, then the root can't be written as $\frac{a}{b}$.

##### In one sentence, what is the core reason that the square root of a prime is irrational??
If $\sqrt{x} = \frac{a}{b}$, you can show $x$ is a factor of $a$ and $b$, which means the fraction could be simplified.

##### The proof the square roots of primes are irrational is a proof by...??
Contradiction.

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