Imaginary numbers challenge

26 10 2008

I have a challenge for people who understand imaginary numbers (if that is indeed possible).

Now, I have seen how imaginary numbers can be useful. Just as negative numbers can.

For example, what is 4-6+9?  7. Easy. But your working memory may well have stored ‘-2′ in its mind’s eye during that calculation. But we cannot have -2 oranges. Or travel -2 metres. Oh sure, you can claim 2 metres backwards is -2 metres. I say its +2 metres, the other way (the norm of the vector).

What about a negative bank balance? I say that’s still platonic, a concept. In the real world it means I should hand you some (positive) bank notes.

We use negative numbers as the “left” to the positive’s “right”. Really they are both positive, just in different directions.

Now for imaginary numbers. I have seen how they allow us to solve engineering problems, how the equations for waves seem to rely on them, how the solution of the differential equations in feedback control loops seem to require them.

But I argue that they are just glorified negative numbers. The logarithmic version of the negative number.

So what is my challenge?

Well, the history of mathematics is intertwined with the history of physics. Maths has made predictions that have subsequently helped us to understand things in the real world. Maths models the world well, such as the motion of the planets, or the forces sufferred by current carrying wires in magnetic fields.

But the question is: is there any basis in reality for imaginary numbers? Or the lesser challenge, negative numbers?

Is there a real world correlation to “i” ? Or is it a mere placeholding convenience?

Or perhaps positive numbers also lack this correlation?

The speed of time

26 10 2008

I want to talk about something very close to my heart.

It has been an obsession for some time now, and I have probably thought about it a little too much, and gone a little too far without checking with some peers. Alas, I don’t know too many physicists down here in Cornwall, and if I wrote papers, they would probably be too disconnected, and not do me any favours. Besides, I suspect the academic world would not really take a shine to someone like me sending in papers without affiliation to any university or research group.

Anyway, my present subject of study (call it a do-it-yourself dissertation) is “the speed of time”. What controls it? How do we measure and sense it? Is there an absolute? That sort of thing.

My thoughts have gone to some interesting places, and some propositions I would like to test provide some interesting implications.

But let me start with my first problem. It relates to how people seem to constantly ignore the implications of special relativity. Take for example, the age of the universe…

Have you ever noticed how people will, one moment, make declarations about the age of the universe, and then in the next agree that time is relative? Isn’t this a contradiction?

I mean, on the one hand, Katie Melua was informed that her estimate was too low (12 Billion years). She actually recorded a gag version of her song after a respected academic (Simon Singh) chided her for getting it ‘wrong’, and also for calling it a guess, which, he said was an insult to a century of astronomical progress.

Then, if you read a bit about special relativity, it explains that time is relative and can ‘dilate’. For my readers who don’t know what that means, it means that how much time passes depends on how fast you are moving. This theory has some well known implications, such as the “twin paradox” in which a space travelling twin returns from his travels younger than his brother.

Now how are we supposed to square these two well-accepted bricks in the foundations of modern physics? The universe is ‘strictly 13.7 billion years old by current estimates’, but never mind, because time is relative, so if you happened to be travelling at 99% of the speed of light during that time, your clock will only have ticked away ~0.3 billion years (according to the Lorentz Transformation). To make matters worse, light waves (/particles) that set off at the start, travelling at the speed of light of course would have yet to see their watch tick at all, making the universe brand-new as far as they are concerned.

Doesn’t this make a nonsense of the whole concept of age? Or should we say: “for objects in our inertial frame, the universe appears to be 13.7 billions years old”?

That’s pretty wishy-washy – and besides, who is to say that our inertial frame is superior to any other?

Please someone help me sort this out, as I can think of some pretty serious implications if we can’t.

If you would also do me a favour, pass on this challenge to your nerdiest friends.

PS. This one is just the start. I have others, and perhaps like this one, all they need is a reality check!