Less by less is more
At this point, I can't refrain from sharing a little story. Fortunately for them, my two daughters-in-law are much better at maths than I am. The first one even studied sciences. So we would talk about maths from time to time, and one day, I explained that I never understood why multiplying two negative numbers gives a positive result. Enthusiastic, she, and her mother, decided to give it a go:
- Well, it's simple, obvious, as soon as you do the maths, you...
- Okay, but I'm not really following you. Give a real-life example...
— …
Despite their patience and kindness, I never got my explanation. And that gave me an idea: at the time, I was quite active in a community of Spip users (a French CMS) which included a maths teacher who was also quite active. I had visited his website, where he gave a very honest account of his struggles as a new maths teacher in the Paris suburbs. It's always a logical thing to send newbies to the toughest spots.
It was possible to comment on the posts he published, so I asked: “Can anyone explain to me why two negative numbers multiplied together give a positive result?” I got at least thirty answers from maths enthusiasts, each more incomprehensible than the last, involving elevators going up -3 times -5 floors, guys paying off their debts, and those were the least abstruse answers. To sum up, among these eminent figures, including quite a few math teachers, not one was able to explain simply why the product of two negative numbers was positive. One even said to me, “I can't explain it to you, but I could demonstrate it. But... you wouldn't understand.” Oh dear... Just for fun, I just tried asking Google (my friend?) the question again. The result was pages of unintelligible stuff, I quote at random :
“I place myself in the field of complex numbers. Multiplying two complex numbers is the same as multiplying the modulus of each of these numbers and adding their arguments. The modulus of a negative real number is its absolute value and its argument is pi radians. The product of two negative real numbers has as its modulus the product of the two absolute values and as its argument pi + pi = 2.pi radians, or 0 radians in the geometric circle. And it is positive reals that have 0 radians as their argument. The result is therefore a positive real. It is beautiful and complex. And at the same time so simple!”
QED... Why do we bother, eh? I don't know what a complex number, a modulus, an argument, pi radians, a positive real number is, nor why I'm being told about a geometric circle (are there any non-geometric ones?). I found pages and pages of the same stuff. What is this subject that the teachers are unable to give a clear answer to a simple and legitimate question?
