Who would have believed that of all the subjects that I have talked about in my blog that MATHS would be the most inflammatory topic? Who? You? Yeah right.
I am, as Mike points out below, a pure mathematician. I am not against applied maths as that is what you should do with maths eventually. Apply it. I mean mucking around with primes is fun (“fun” is being used relatively here), but you also do cool cryptographical stuff with them.
But let’s see what my visitors have written (as per my policy of “you write a good comment, I’ll post it up”).
ben.run said...Good on you Ben! I did enjoy the odd piece of integral calculus cake now and then. But nothing beats those pesky little letters.
The best kind of math has no practical application at all. Oh I loved the pure algebra's. None of this calculus crap for me :-)
mike said...Ahhh, I had forgotten the old i versus j deal for engineers. And I would have to agree with our next commentator that the electricity example is liable to confuse kids more than help them (don’t worry though I will disagree with him too).
Sorry to bust in on you pure mathematicians, but I have to have my 2c as an engineer! I was going to come out swinging and lay into you pureheads, but I have to agree with all you're saying. Often you just have to accept things for what they are - e.g. it's better to bend your mind to accept imaginary numbers than try to bend
imaginary numbers to your mind.
Having said that, as an engineer, we need real-world examples, as that's what engineers do. I guess, then, the use of the mathematics defines how it should be presented. If the use is for a real-world application, then a real-world example should be used. If it's not for a real-world application, or is a fundamental concept, then real-world applications or analogies should probably be left out, as they may confuse the issue later. As for imaginary numbers, unfortunately, engineers have to use them quite a lot (and we use j instead of i, because, as we all know, i is the symbol
for instantaneous current). It's definitely more a tool than a reflection of
reality though - I mean, what the hell is imaginary power??
Hmmmm...an application of two negatives making a positive is tough. Well, I guess there is the old AC current x AC voltage = Power. If current and voltage are in phase, even though they change in time from positive to negative the power is always positive (except for when the current and voltage are 0, when the power is also 0). When current and voltage are out of phase it gets a bit more complicated (and is one situation where i, er j, is necessary)! This is a pretty weak analogy, because the voltage/current isn't really going negative, it's just changing direction.
With most of these kinds of things there are a lot of “depends”. For example Ben and I obviously prefer mucking around with hypothetical mathematics that has no tangible reality attached to them, like number theory. Others, who are so inclined either by profession of just how they think, prefer to have a realistic, tactile base for their thinking. Neither way is better or worse. I had a physics professor who once claimed that Americans were better experimentalists and Russians better mathematicians because the Americans had heaps of money to spend on experiments and the poorer Russians had to do everything in their heads. [If you are Russian and take offence, I apologise, I think it was just an example].
And now onto our last commentator; the one who made me a real blogger. He (or she) was anonymous and said that I sucked. My first flaming! I have now entered that hallowed hall of Blogdom.
Anonymous said...The first thing I would like to say is: You represent negative numbers in an easy manner for a living? You can make a living from that? Like an actual living or do you just sit around the house and create real world examples with your friends?
You guys suck you really do (I'm pretending I'm on talk back radio, and flaming when a well reasoned argument would do).
Firstly: i is not a pure as the driven snow kind of concept. We only use it cause it's
applicable. Do students love learning about i? No, cause i sucks. The graph for square roots has NO solutions in the negative area, ergo no i plural.
As for real world examples of things Surely we only talk about voltage cause we've
never seen these electrons (or fundamental particles). So the maths tells us what voltage is. And we are going to use this maths (which is not represented by a thing we can ever have a meaningful relationship with) to represent the negative times negative problem? Sounds fishy. Step back (I do this for a living):
If your friend takes 4 strawberries (something you can have a meaningful relationship with), then you can say you have lost 4 strawberries. Hence, negative 4 strawberries. Well, if you then find out the person was going to steal 4 strawberries off you twice, but never did, you'll be eight strawberries better off. Eight positive strawberries. Actually, bank accounts are better for illustrating this kind of thing.
And now to continue: Yep, there is no such thing as i. It’s called an imaginary number for a reason. My point was that it is hard to teach kids about a concept that really doesn’t exist, as you said. It does have a use though (not when it was first thought of mind you). In fact a lot of uses outside of engineering (although I can’t remember one off the top of my head, it was in a class called Complex Analysis, the title didn’t lie [heehee] and so I’m a little shady on the details).
Infinitesimals were cool too. They really don’t exist either along with numbers larger than infinity like “Aleph-0” and “Aleph-1” (insert symbol didn’t work with blogger). But you can use both as tools to make valid proofs of otherwise very hard theorems.
Now your example. It takes quite a large logical step, which might go over many kids’ heads (although I never mentioned kids in the question). I think you’ve actually just gone 2x4, but claimed it was negative at the start. Though to be honest this seems like the type of thing most economies are run on (“there was no stock collapse so you doubled your money). What you have said, and please correct me if I’m wrong I’m just blogging out loud here, is that he was going to steal from you twice but didn’t. You say that is -2 lots of stealing? [In order to get (-2)x(-4)] Isn’t it actually 0 lots of stealing?
Anonymous (can I call you Nonny?) maybe you were right; this makes more sense in terms of money. I’m thinking gambling debts.
Anyway if you’ve got something good send it in.