Complex numbers

Sometimes the divide between physics and mathematics is a thin one. Particularly between theoretical physics (which is what I do most of) and mathematics. The difference is that physicists have to keep one foot loosely planted in reality. It’s true that sometimes it just the tip of a little toe that’s behind the reality-line, but look hard enough and you’ll find realism in all that a physicist does.

One example is a physicist’s love of complex numbers. What’s a complex number? Well, think of your square numbers: One times one is one, two times two is four, three times three is nine, etc.  Now turn the problem around – for example, what number when squared is 36? Answer, six. But let’s make it awkward. What number squared is minus one?   Well, it’s not one (one squared is one), and neither is it minus one (because minus one squared is also one). Put minus one in a calculator and hit the square-root button and you’ll get (if your calculator is like mine) an error. Why? There is no real number that when squared equals minus one.

 But this needn’t stop the mathematician. if there isn’t a real solution, he just makes one up, and calls it an ‘imaginary’ number. He calls it by the letter ‘i’.  So i squared equals minus one. Beyond reality, yes, but maths is not limited by reality. The mathematician now proceeds to construct a whole algebra (complex algebra) around this concept. For example, you can add, multiply, square, squareroot, take the exponential of etc complex numbers.

Now this last one is why a physicist gets interested. It turns out that the exponential of an imaginary number is related to the cosine and sine functions. And the cosine and sine come almost everywhere in physics. But problems using exponentials are much easier than using sine and cosine. So the physicist turns his problem from a real-world problem to one that is in the ‘imaginary’  (or ‘complex’) world, does his complex algebra on it, then transforms it back into the real world, to get the result. It’s a neater way of doing things. Note that reality remains, because at the end of the journey he’s back in the real world. (It’s only a toe that needs to be behind the line, the rest of the body can well and truly be beyond it.)

If that has confused you, it’s rather like journeying from Auckland to Wellington. You can travel overland, which will take you a long time and you’ll have to cross rivers and go halfway up mountains on the way, but you’ll get there. Or you can go to the airport, leave the ground, travel  there quickly, and land again (don’t forget the landing bit – or you’ll be a mathematician). You end up in the reality of Wellington by either route; one just happens to be quicker and simpler, and the fact that it’s off the ground doesn’t matter.

2 thoughts on “Complex numbers”

  • Hi
    Liked the big picture stepped back view of the reason physicists use complex numbers – am wondering if I can ask some slightly more in-depth questions:
    1. What is the link between them and sins/cosines?
    2. What is the complex log of say -1000?
    Sounds like its a bit like using a slide rule, dropping into exponents to handle the hard multiplications and divisions by adding and subtracting…
    PS: while I’m writing to someone in the physics area – would it be correct to say that everyday objects do not collide at all – they simply get very close and then the nuclear forces between atoms cause repulsion like two poles of a magnet coming close together? With the possible exception of high energy collisions or black holes where maybe stuff does get jammed together?
    Many thanks
    Robert Taylor
    PPS: Sorry – I’ve asked questions and some not even on the topic! Not exactly comments! Prepare the vat of boiling oil!

  • Lots of questions there…. So, in order:
    1. ‘e’ (2.71828…) to the power of ‘i times x’ equals cos x + i sin x. So the ‘real’ part of e to the power of (i x) is cos x, and the ‘imaginary’ part is sin x.
    2. Natural log of -1 is i times pi. (No wonder my computer couldn’t do it…)
    3. Log (base 10) of -1000 is 3 + 1.36436 i (the 1.36436 is pi diveided by the natural log of 10). To do the log (base 10) of a negative number, do the log of the absolute value of the number (the 3 in this case) and add 1.36436 times i.
    Yes – slide rules (do they still exist?) are based on logs
    4. I guess it depends what you mean by collide. For practical collisions, it will be the repulsion between electrons in the atoms that cause the objects not touch. But at high enough energy (e.g. the Large Hadron Collider) nuclear forces are imporatant, and there will be interactions there. In a black hole, of course, matter is packed as closely as it could be – I guess you would say that is touching.

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