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?

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3 responses

27 10 2008
Re fuzziness

Absolutely.

It’s just because we are humans. To us the natural numbers seem the most “natural” because we’ve been counting things for a million years.

We don’t understand the rational numbers (or negative integers) at all until we show are shown – as kids – how we can use them to describe some aspects of reality (ratios of things, or comparisons of amounts).

And so eventually some of us come to understand complex numbers while investigating those aspects of reality which can be described as quadratic equations. Imaginary numbers are not just a placeholder, they’re the foundation of an extension to real numbers, which is necessary to describe certain things. Quite a lot of things.

The distinction between natural, rational, real, and complex number systems only exists in the minds of homo sapiens – not in reality.

27 10 2008
jarrodhart

Mmm. Are you saying that all numbering systems are firmly platonic?

I suppose our platonic constructions (such as a circle) are ‘based’ on things we see in the real world (like ripples), and so the fact that the manipulations we do with them allow us to predict things (the tidal “sinusoid”) is not surprising.

But maths does not have any clear distinction between 3-dimensional space and 4-dimensional space. Does than mean n-dimensional space is predicted to be real?

Does it mean that the fluctuating electromagnetic fields in a light beam really do spiral through complex space, or complex space a purely platonic construct?

What I am really trying to do (and I think I need to understand complex numbers well to do it) is to try to find a metaphor for EM radiation that the human brain can understand (living in a regular 3-D world as it does!) I think our metaphor for light waves as ripples in the surface of water, or in a slinky, is limiting us…

28 10 2008
Re fuzziness

Well – this is really just a discussion of what “real” means, isn’t it? (and I can’t go there, I’m afraid). Good luck!

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