# B Perfect square

#### marino

According to you this theorem is correct?

Exercise 1.2 * Proof that $\sqrt{x}$ isn't a rational number if $x$ isn't a perfect square (i.e. if $x=n^2$ for some $n∈ℕ$).

In effect, if $x=\frac{25}{9}$, so $x$ isn't a perfect square, then $\sqrt{x}=\sqrt{\frac{25}{9}}=\frac{5}{3}$ is rational.

#### jedishrfu

Mentor
But for this proof to work:

x must be a member of N too

and so sqrt of x is not a rational number.

@fresh_42 can answer this better.

#### fresh_42

Mentor
2018 Award
There is nothing to add. Either restrict $x\in \mathbb{N}$ or generalize $n \in \mathbb{Q}$.

• jedishrfu

#### PeroK

Science Advisor
Homework Helper
Gold Member
2018 Award
According to you this theorem is correct?

Exercise 1.2 * Proof that $\sqrt{x}$ isn't a rational number if $x$ isn't a perfect square (i.e. if $x=n^2$ for some $n∈ℕ$).

In effect, if $x=\frac{25}{9}$, so $x$ isn't a perfect square, then $\sqrt{x}=\sqrt{\frac{25}{9}}=\frac{5}{3}$ is rational.
To put it another way.

If $x$ is a positive integer, then either $\sqrt{x}$ is a positive integer; or, $\sqrt{x}$ is irrational.

Last edited:
• Math_QED and jedishrfu

#### WWGD

Science Advisor
Gold Member
One way of doing the proof for Rationals is to generalize the argument that $\sqrt 2$ is not Rational to $\sqrt n$.

#### Mark44

Mentor
In effect, if $x=\frac{25}{9}$, so $x$ isn't a perfect square
But here x is a perfect square ($(\frac 5 3)^2 = \frac {25} 9$, but as has already been pointed out, x is not a positive integer.

#### WWGD

Science Advisor
Gold Member
But here x is a perfect square ($(\frac 5 3)^2 = \frac {25} 9$, but as has already been pointed out, x is not a positive integer.
But you need to be careful on how you define a perfect square otherwise every no negative real number is a perfect square.

#### Mark44

Mentor
But you need to be careful on how you define a perfect square otherwise every no negative real number is a perfect square.
Understood, but when we talk about perfect squares, the context is seldom real numbers. In this thread the context was $x \in \mathbb N$ (more specifically, from post #1, $x = n^2$ with $n \in \mathbb N$) and the OP gave an example with $n^2 = \frac {25} 9$.

• WWGD

"Perfect square"

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