Question

Dentify square root of 2 as either rational or irrational, and approximate to the tenths place.

Answer 1

The square root of two is irrational. It means that you can't write it as a fraction:

$\sqrt{2} \neq \cfrac{p}{q}\quad\forall p \in \mathbb{Z},\ q \in \mathbb{Z}\setminus\{0\}$

Here's a proof. Assume that you could write

$\sqrt{2} = \cfrac{p}{q}$

for some integers p and q, and that p and q have no common divisors. If you square both sides you have

$2 = \cfrac{p^2}{q^2} \implies p^2 = 2q^2$

So, $p^2$ is even, which means that also p is even, and thus there exists some number k such that $p = 2k$. The expression becomes

$p^2 = 2q^2 \implies 4k^2 = 2q^2 \iff q^2 = 2k^2$

So, also $q^2$ is even, and thus q is also even.

But then, p and q are both even, whereas we assumed that they had no common divisors. Contraddiction.

As for the value, any calculator will give you an approximation that starts with

$\sqrt{2} = 1.4142\ldots$

so, it is best approximated by 1.4.

Answer 2

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