Physical meaning of imaginary numbers

[bbbeard]

Can someone give a physical meaning for imaginary numbers?

As Ken G has indicated, complex numbers are "rotational" numbers. They are handy whenever a physical problem involves a rotation or a phase. Consider, for example, a case where we have two unit vectors which make angles "a" and "b" with the x axis, and want to know how adding the angles changes the projections onto the x and y axes. In other words, we want to know how the sines and cosines of a and b relate to the sines and cosines of the total (a+b). We could look up our trig identities

sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)
cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b)

or we could just use the identity

exp(i*x) = cos(x) + i*sin(x)

to learn

exp(i*(a+b)) = exp(i*a)*exp(i*b)
=(cos(a) + i*sin(a))*(cos(b) + i*sin(b))
=(cos(a)*cos(b) + i²*sin(a)*sin(b)) + i*(sin(a)*cos(b) + cos(a)*sin(b))
= cos(a+b) + i*sin(a+b)

... which only works because i² = -1. In other words, the algebra of rotations is embedded within the complex numbers by virtue of the definition i² = -1.

This has many uses. For example, if we have two arbitrary vectors, if we want to add them to find a resultant, then ordinarily we can just add them component-by-component. So converting them to complex notation (x,y) => x + i*y doesn't buy you much. But if you want to rotate a vector, you can simply multiply (x+i*y) by exp(i*theta) to find a new rotated vector.

Because waves are for the most part oscillatory phenomena, complex numbers help to denote the time evolution, exp(i*omega*t), or the spatial dependence, exp(i*k*x). We could write the same thing in terms of sines and cosines, of course, but the complex exponential is much simpler to manipulate. This "rotational" aspect of complex numbers is why they are so ubiquitous in quantum mechanics.

BBB

 

[HallsofIvy]

But the OP is right- imaginary number are imaginary- in the same sense that real numbers, rational numbers, and even integers are imaginary- products of the human mind. I had a friend who argued that integers, at least, are objective concepts- after all there is a difference between "one elephant" and "two elephants". My point is that they make concrete sense only if you can always distinguish "one" from "two". Yes, with elephants that is easy- but can you distinguish one slime mold from two?

(Or, as just occurred to me sitting on the porch- how in the world do you count how many humming birds there are around the feeder? Those little devils are so hard to distinguish one from another, I'm not at all sure it is an integer number!)

 

[Ken G]

I had a friend who argued that integers, at least, are objective concepts- after all there is a difference between "one elephant" and "two elephants". My point is that they make concrete sense only if you can always distinguish "one" from "two". Yes, with elephants that is easy- but can you distinguish one slime mold from two?

You are definitely an empiricist, yes? I've had similar arguments with mathematicians. One great one once said that god gave us the integers, and the rest are human creations, because the axioms of Peano arithmetic are about as close to "self-evidently true" as you can get in mathematics. But for me, that's still not close enough-- I think we made those too.

 

[Studiot]

But the OP is right- imaginary number are imaginary- in the same sense that real numbers, rational numbers, and even integers are imaginary- products of the human mind.

Please correct me if I'm wrong but strictly are we not all discussing the wrong animal?

The OP says "imaginary numbers", not as I posted about earlier, complex numbers.

Are "imaginary numbers", not strictly just single entities involving the square root of a negative number?