Question

Prove tha 15+17root3 is an irrational number

Answer 1

First of all, let's make the following observations:

15 is an integer, so in particular it is rational. The sum of two rational is rational. This means that $15+17\sqrt{3}$ is irrational if and only if $17\sqrt{3}$ is irrational.

Similarly, since 17 is rational and the multiplication of two rationals is rational, we have that $17\sqrt{3}$ is irrational if and only if $\sqrt{3}$ is irrational.

The proof that $\sqrt{3}$ is irrational is the following: suppose by contradiction that we could write

$\sqrt{3}=\dfrac{a}{b}$

where a and b are two integers with no common factors.

Squaring both sides, we have

$\dfrac{a^2}{b^2}=3 \iff a^2 = 3b^2$

So, $a^2$ is a multiple of 3. This happens if and only if a itself is a multiple of 3. So, we can write $a=3k$ and the expression becomes

$(3k)^2 = 3b^2 \iff 9k^2 = 3b^2 \iff 3k^2 = b^2$

So, $b^2$ is a multiple of 3 as well, and this happens if and only if b is itself a multiple of 3.

So, we started with the assumption that we could write $\sqrt{3}$ as a reduced fraction, but this assumption led to a contradiction.

We deduce that $\sqrt{3}$ is irrational, and so are $17\sqrt{3}$ and $15+17\sqrt{3}$