Physical meaning of imaginary numbers

[Studiot]

So what are you 'measuring' in my voltage examples and how do you use this 'measurement' ?

I am perfectly happy using complex numbers as representations of some physical property that does not possesses the ordering structure. But surely that is another way of putting the original poster's question?

 

[BobG]

So what are you 'measuring' in my voltage examples and how do you use this 'measurement' ?

I am perfectly happy using complex numbers as representations of some physical property that does not possesses the ordering structure. But surely that is another way of putting the original poster's question?

You're measuring two separate things - their magnitude and their direction. I guess that does present some ordering problems. You can't say a vector with a magnitude of 5 and a direction of 0 deg is 'greater' than a vector of 5 and a direction of 90 degrees, but you can definitely say they're not equal.

Or in your example, you're measuring their magnitude and their phase, which presents the same ordering problem.

 

[Studiot]

You can't say a vector with a magnitude of 5 and a direction of 0 deg is 'greater' than a vector of 5 and a direction of 90 degrees, but you can definitely say they're not equal.

That's exactly it.

To say 5 miles is greater than 3 miles is meaningful.

To compare 5mph south from Stockholm with 3mph west from Rome is meaningless.

By themselves both are valid statements and can be represented (although this would be unusual) in complex format. However they cannot be combined in any meaningful manner.

I think the OP was seeking better physical examples than mine, rather than platitudes.

 

[AlephZero]

-3: can't think of physical meaning, but maybe there is one. If you interpret -3 as going backwards 3 steps, then -3 becomes only an aid to do calculations.
This is the problem I am having with i. Is i simply a tool, or is it real?

If you can accept that "-3" is "only an aid to do calculations", then you can just think of "i" the same way if you like.

As another post said, in many "practical" applications of complex numbers in science and engneering, you don't really NEED to use n complex numbers. You could replace them by 2n real numbers, but the complex numbers are a neat way to automatically keeping track of the "n" similar relationships between each pair of real numbers. In those situations, often there is no phyiscal difference between what the "real" and "imaginary" parts of the numbers mean, and it would make no difference if you swapped over the the real and imaginary parts of all the complex numbers, except that you would alos have to flip the sign of one of the real numbers in each pair. But that goes back to the reason for using complex numbers in the first place - they keep track of details like that without you having to think about them all the time.

For some purposes, numbers that are "even more imaginary" than complex number are useful. Google quaternions and octonions, for example.

The reason that the imaginary parts of complex numbers are call "imaginary" is just a historical accident. The different meanings of "real" and "imaginary" in ordinary English don't have any mathematical significance, and neither do the different meanings of "rational" and "irrational" in ordinary English.

 

[bbbeard]

The different meanings of "real" and "imaginary" in ordinary English don't have any mathematical significance, and neither do the different meanings of "rational" and "irrational" in ordinary English.

"Rational" means that the number can be formed by the ratio of integers. "Irrational" means that it cannot be formed by the ratio of integers.

"Imaginary", though, does not mean what it sounds like. ;-)

BBB

 

[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?