I have in my presence a mathematics teacher, who asserts that
$$ \frac{a}{b} = \frac{c}{d} $$
Implies:
$$ a = c, \space b=d $$
She has been shown in multiple ways why this is not true:
$$ \frac{1}{2} = \frac{4}{8} $$
$$ \frac{0}{5} = \frac{0}{657} $$
For me, these seem like valid (dis)proofs by contradiction, but she isn't satisfied. She wants a 'more mathematical' proof, and I can't think of any.
I'm worried that if she isn't convinced, it may be detrimental to some students. Is there another way to systematically demonstrate the untruth of her conjecture?
EDIT: Since the answer which worked was from a comment, but each answer is also very good, I'm upvoting all of them instead of accepting a specific one. Feel free to close this question for being too open if so a moderator desires.
Best Answer
You can prove that all the numbers are equal ;-)
Let's assume that for all $a,b,c,d \in \mathbb{R}$, $b \neq 0$, $d \neq 0$ we have
$$ \frac{a}{b} = \frac{c}{d}\quad \text{ implies }\quad a = c\ \text{ and }\ b = d. \tag{$\spadesuit$}$$
Now take any two numbers, say $p$ and $q$, and write
$$\frac{p}{p} = \frac{q}{q}.$$
Using claim $(\spadesuit)$ we have $p = q$. For the special case, where one of them equals zero (e.g. $q$), use $$\frac{2p}{2p} = \frac{p+q}{p+q}.$$
I hope this helps ;-)