I have in my presence a mathematics teacher, who asserts that
She has been shown in multiple ways why this is not true:
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.
You can prove that all the numbers are equal 😉
Let’s assume that for all a,b,c,d∈R, b≠0, d≠0 we have
ab=cd implies a=c and b=d.
Now take any two numbers, say p and q, and write
Using claim (♠) we have p=q. For the special case, where one of them equals zero (e.g. q), use 2p2p=p+qp+q.
I hope this helps 😉