Why $\sqrt{2} \ne \frac{7}{5}$ (or similar sort of thing)

Why does $\sqrt{2} \ne \frac{7}{5}$?  Well, if it did, we could draw a square of side 5 and diagonal 7

Not in proportion!

Then, by removing 5 from the diagonal we could create a second square of side 2 and diagonal 3, which would mean $\sqrt{2} = \frac{3}{2}$, which is a  different fraction to $\frac{7}{5}$.

Actually, we can easily generalize this argument to show that if

$$
\sqrt{2} = \frac{a}{b}
$$
for some whole numbers $a$ and $b$, then
$$
\sqrt{2} = \frac{c}{d}
$$

for some smaller whole numbers $c \lt a$ and $d \lt b$.  Repeating this argument over and over leads us to the conclusion that $\sqrt{2}$ is a whole number itself!  This is clearly incorrect and leads us to the conclusion that we can't in fact write $\sqrt{2}$ as a fraction.

This is not a particularly modern way to prove the existence of irrational (non-fraction) numbers, but it is - supposedly - how the ancient Greeks originally did it!  Unfortunately the person who discovered this fact, Hippasus of Metapontum, found that the philosophers of his time preferred simplicity to truth, and - the story goes - was drowned at sea.

2000 years later Galileo found the same thing.