this post was submitted on 11 Jan 2024
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Before i ask my question, know that my math is all the way in the back of my head and i didnt get too far in math at school.
Wdym irrational numbers dont work? -3 * -pi would be the same as 3*pi, no?
I always assumed if all factors of the multiplication are negative, it results in the same as the positive variant, no matter the numbers ( real, fractal, irrational, .. )
3 pi = pi + pi +pi
Sure thats okay, but what about e * pi?
Multiplying two negative irrational numbers together will still give you a positive number, it's just that you can't prove this by treating multiplication as repeated addition like you can multiplication involving integers (note that 3 is an integer, 3 is not irrational, the issue is when you have two irrationals).
So, for example with e * pi, pi isn't an integer. No matter how many times we add e to itself we'll never get e * pi.
Try it yourself: Assume that we can add e to itself k (a nonnegative integer) times to get the value e * pi. Then e * pi = ke follows by basic properties of algebra. If we divide both sides of this equation by e we find that pi=k. But we know k is an integer, and pi is not an integer. So, we have reached a contradiction and this means our original assumption must be false. e * pi can't be equal to e added to itself k times (no matter which nonnegative integer k that we pick).
Thanks for that! Helped understand it a bit more, i think. So its a case of it not working on irrational numbers, its just that we cant prove it because we cant calculate the multiplication of 2, right? Somehow, my mind has issues with the e*pi = ke. Id say that ke = e * pi is impossible because k is an integer and pi isnt, no? It could never be equals, i think.
The issue is the proving part. We can't use repeated addition trickery (at least not in an obvious way) to show a product of two irrational negative numbers is positive. It's definitely still true that a product of two negative numbers is positive, just that proving it in general requires a different approach.
Yes this is correct. The ke example is for a proof by contradiction. We are assuming something is true in order to show it forces us to be able to conclude something ridiculous/false. Since the rest of our reasoning was correct, then it must have been our starting assumption that was wrong. So, we have to conclude our starting assumption was wrong/false.
I think all they mean is you can't write it out since irrational numbers have no end.
You're correct in that the principle still applies in exactly the same way.