What purpose do "imaginary numbers" serve?

[Payton's Servant]

Today was the first day of my summer school math class, and the teacher gave a bit of a description of imaginary numbers, such as the square root of -4. But he didn't say what you would use imaginary numbers for. Well, he did, but all he said was that engineering students would use them.

So, can a Math Doper fill me in a bit?

[Otto]

I haven't been able to access the boards for two days!

[mojave66]

Here's another site for an explanation of real-world uses of complex numbers:

http://mathforum.org/library/drmath/view/53879.html

[ultrafilter]

Multiplying by i is the same as rotating by 90 degrees around the origin. So they're used in graphics too.

[g8rguy]

And of course physics would be all but hopelessly intractable without the darned things. Basically, any time something rotates or oscillates, imaginary numbers provide a clean and compact representation.

[bbeaty]

Cellphone circuit design. Most audio & video equipment. Radio transmitters. If the electronics is not all digital, then it was probably made using imaginary number math. Remember Ohm's law? When dealing with capacitors and coils, Ohm's law expands into the imaginary number realm.

[Desmostylus]

ultrafilter and g8rguy: they're a neat trick for 2-D work, and that comes up often enough for them to be useful. But, as I'm sure you know, they aren't extensible to higher dimensions. Maxwell's system is, and it's just an accident of history that we still use complex numbers for 2-D work.

[MC Master of Ceremonies]

They are absolutely essiential to quantum mechanics.

[Desmostylus]

Originally posted by MC Master of Cermonies
They are absolutely essiential to quantum mechanics.

No, they aren't. You can always rewrite as vector equations or real equations.

[Desmostylus]

Just thought of something else.

You often encounter situations where an equation is non-analytic, i.e. non-differentiable in complex form, and so you're forced to re-write as two real equations and proceed from there.

[psychonaut]

And naturally, they figure pretty big in Christian theology.

[Grey]

3 Phase power. Don't forget, imaginary current can kill you.

[Francis E Dec, Esq]

Originally posted by Desmostylus
ultrafilter and g8rguy: they're a neat trick for 2-D work, and that comes up often enough for them to be useful. But, as I'm sure you know, they aren't extensible to higher dimensions. Maxwell's system is, and it's just an accident of history that we still use complex numbers for 2-D work.

But for 4-D you can go to quaternions, with i² = j² = k² = -1. And I think there is a 16-D version too. "Hyperquaternions" maybe?

(Nowadays quaternions are usually encountered as the Pauli spin matrices, and few people even use the term "quaternion." The 16-D space is the space of the γ matrices of Dirac.)

[Orbifold]

Francis: are you thinking of octonions?

Demostylus: quaternions can also be used to represent rotations of three-dimensional space, so quaternions have applications in computer graphics.

Also, complex numbers come up all the time in three-dimensional hyperbolic geometry. But sadly not everyone would consider that to be an "application" as such...

[Desmostylus]

Originally posted by Francis E Dec, Esq
But for 4-D you can go to quaternions, with i² = j² = k² = -1. And I think there is a 16-D version too. "Hyperquaternions" maybe?

For 4-D you use what I called "Maxwell's system". It isn't entirely due to Maxwell. Gauss, Green, Stokes, et al contributed. I guess it was Maxwell that finally said: "fuck it, that's the way to go."

The term "quaternion", as I understand it, was originally coined for the 3-D analogue of a complex number, but no satisfactory artifact was ever found.

[Francis E Dec, Esq]

Octonians? Nope, but I'll google for more on that.

The space I was talking about is the one spanned by the 16 Dirac matrices described here: http://mathworld.wolfram.com/DiracMatrices.html

They are 16 matrices whose square is equal to the unit matrix. If you multiply some of them by i so that their there square is -1, you get a 16-d space that is analogous to the quaternions, with similar commutation rules. (Similarly, you have to multiply the Pauli spin matrices by i to get quaternions.)

Originally posted by Desmostylus
The term "quaternion", as I understand it, was originally coined for the 3-D analogue of a complex number, but no satisfactory artifact was ever found.

Yes, it was. Check this out: http://world.std.com/~sweetser/quate...ex/qindex.html

[Jinx]

Imaginary numbers is a misnomer...obviously, they have real applications. You might say it's more like the rules of math we've invented to model the physical world around us has a slight flaw!
- Jinx

[ChordedZither]

Jinx has the right of it. I would state it even more firmly.

Imaginary numbers are, themselves, not useful for anything.

In none of the applications mentioned so far is there any real significance to the fact that one of the components of these numbers happens to be the square root of -1. The "i" could be replaced with any kind of marker at all, and the marker doesn't have to have an underlying meaning. It's just a notational trick for denoting an axis in a two-dimensional system.

Consider a 2-dimensional cartesion coordinate system. We usually denote points (or vectors, which amount to the same thing) in that system as (x,y) tuples: e.g., (2.5, 7). But that notation does not lead to "natural" or "expressive" manipulation in many contexts. Another way to write the same thing is to introduce markers for the unit-length vectors along the x and y axes. These are usually indicated by writing ^ over the x and y, but I'll just write them as x^ and y^ in this posting. So (2.5,7) can be written as 2.5x^ + 7y^. In that form, a lot of "ordinary" mathematical manipulation becomes natural. Addition of vectors, multiplication, etc. all work out the way you would expect if you simply treat x^ and y^ as arbitrary symbols that can't be replaced or substituted for.

Now, take any mathematical formulation in terms of x^ and y^. Replace x^ by 1 and y^ by i. So (2.5,7) becomes 2.5x^ + 7y^ becomes 2.5+7i. Whatever mathematical steps you apply to the x^ y^ notation will work equally well with the complex number notation and the reverse is also true. The "i" is simply filling the role of an arbitrary symbol that can't be replaced or substituted for. But sometimes the complex number approach leads to a more compact representation.

Nevertheless, there's a subtle danger in using complex number notation in some circumstances. Too many people believe that the imaginary component either means something in the systems being described is imaginary/mysterious and maybe not really valid. Others may forget that the "i" direction often denotes something very "real". For example, when complex numbers are used in graphics, thee "i" is just the "y" direction. When complex numbers are used to describe electromagnetic waves, the "real" part of the complex number is the electrical field and the imaginary part is the magnetic field.

Imaginary and complex numbers are a convenient notational device. But the way they get introduced in many math/science courses often seems to me to reflect a mistake in emphasis. A physical system does not become any more or less "real" simply because we choose to describe it using "imaginary" numbers. I suspect that many people would understand the underlying math of many systems better if it were presented in vector (x^, y^) form.

[bup]

Payton's Servant - In math, you use imaginary numbers to solve problems that couldn't otherwise be solved, e.g.
x² = -1.

Mathemeticians, who are the group you are specifically asking your OP of, don't care if there's a real world problem it's used for.

So the answer to your question is mu.

[Whack-a-Mole]

Originally posted by ChordedZither
Nevertheless, there's a subtle danger in using complex number notation in some circumstances. Too many people believe that the imaginary component either means something in the systems being described is imaginary/mysterious and maybe not really valid.

You can say that again.

When I was introduced to imaginary numbers in high school they gave no explanation as to their use or purpose. It was just, "This is how to handle it and do it because we told you to."

My mind utterly balked at this point. To me they were admitting they were making crap up to do things that were impossible. I never learned well by simple memorization...I needed to understand what was happening, the why of it as it were. Not that I was ever great at math and considered advanced mathematics important to what I thought I'd like to do when I grew-up but I can definitively say that imaginary numbers finished me on math for good.

I would have hoped by now schools would be better at teaching and explaining concepts but given the OP that sadly doesn't seem to be the case.

[Chronos]

Mathematicians use them just because they like completeness, in that they vastly increase the number of questions you can answer. To elaborate: With the natural numbers (positive whole numbers), you can answer questions like "What is 2+3?", or "What is 10 / 2?". If you want to be able to answer all subtraction problems, though, that's not enough, so you introduce negative numbers. For most division problems, you need rational numbers. And for infinite series, you need all of the real numbers. But that's still not enough. What's the logarithm of -5? What's the square root of -8? What's the arcsine of 2? It turns out that all of these questions can be answered with complex numbers. Even if you try to take the arcsine or square root or log of a complex number, you still get a complex number. So the complex numbers are, in some sense, "complete".

In real-world applications, they're never actually necessary, since you could always just use two numbers for each "complex" quantity, and invent a brand new set of rules for dealing with those pairs of numbers each time. But if you did that, you'd find that you were re-inventing the same rules every time, so it makes sense to just call them complex numbers.

I'll disagree with ChordedZither's analysis, though, that using complex numbers is equivalent to using 2-d vectors. That works just fine as long as you're just adding numbers/vectors, but not if you're multiplying them. There isn't any natural way to multiply two vectors to get another vector, but there is a natural way to multiply two complex numbers to get another complex number. And the use of complex numbers in graphics, for example, is not restricted to just using them as vectors. Much image processing is based on Fourier transforms, and to do Fourier transforms properly, each pixel in your image should have a complex value. Now, in your original and final images, those complex numbers will probably happen to be real, but in some of the intermediate steps, they're part imaginary.

[andymurph64]

When my students doubted the existance of Complex (or Imaginary) numbers, I would ask them if they thought -2 was a legitimate number.

When they said yes, I would ask them to point to -2 of something

Complex numbers is a two-dimensional number system. Sometimes are usual one-dimensional system doesn't cut it

[gypsymoth3]

[Q]When they said yes, I would ask them to point to -2 of something [/Q]

you can't "see" -2, but we understand the concept of having $0 and owing someone $2.

2i isn't quite that easy to understand.

[ElJeffe]

In none of the applications mentioned so far is there any real significance to the fact that one of the components of these numbers happens to be the square root of -1. The "i" could be replaced with any kind of marker at all, and the marker doesn't have to have an underlying meaning. It's just a notational trick for denoting an axis in a two-dimensional system.

I appreciate the sentiment of this, but I don't completely agree. It's true that all of the fields that use complex numbers could get by without them, but there is significance in the fact that "i^2 = -1". In developing the equations that arise in circuit analysis, for example, you typically start with a bunch of real numbers. When you go to solve these equations, you'll wind up with complex numbers, where both the real and imaginary components signify different (and very important) things. While these systems could certainly be analyzed without complex numbers, the use of complex numbers in this case is fairly intuitive. It's just a matter of knowing what the complex numbers mean.


Jeff

[andymurph64]

Yes it is gypsymoth3.

It's just that you are not used to it because it is not needed for routine everyday things.

[Whack-a-Mole]

Originally posted by andymurph64
Yes it is gypsymoth3.

It's just that you are not used to it because it is not needed for routine everyday things.

You're suggeting that -2 is as easy to understand as 2i? I seriously hope you don't take that stance with your students. I can follow the concept of -2 easily. Even after math in high school and now this thread I still have a problem grasping imaginary numbers (although my understanding is now better than it was...unfortunately the internet and the SDMB weren't around when I was learning this stuff).

[chillihead]

Imaginary & complex numbers?

The stuff tax returns are made of!

Grins Runs Ducks

[ChordedZither]

Originally posted by ElJeffe
I appreciate the sentiment of this, but I don't completely agree. It's true that all of the fields that use complex numbers could get by without them, but there is significance in the fact that "i^2 = -1". In developing the equations that arise in circuit analysis, for example, you typically start with a bunch of real numbers. When you go to solve these equations, you'll wind up with complex numbers, where both the real and imaginary components signify different (and very important) things. While these systems could certainly be analyzed without complex numbers, the use of complex numbers in this case is fairly intuitive. It's just a matter of knowing what the complex numbers mean.


Jeff

Yes, I did slightly over-simplify. In most of the applications being mentioned, multiplication of complex numbers is fairly rare because maultiplication of the quantities being modelled is not a meaningful or useful concept.

The underlying operation denoted by multiplication of complex numbers certainly can be defined in 2-d vector terms, but i will grant you that the complex notation is more convenient for this instance. Other common operations in the 2D plane (e.g., vector dot products) have a more natural and compact representation when expressed in vector notation.

I don't mean to denigrate a notation by suggesting it's "merely" more expressive or compact. The choice of an appropriate notation is important in practical situations, leading both to less work and, in many cases, better insight into the behavior of a modelled system. I would no more object to using complex numbers where they simplify matters than I would object to using polar coordinates rather than Cartesian.

What I really want to get across is that people should not get hung up on the fact that these numbers aren't "real" (pun intended) and should not infer that therefore they can't be meaningful. It's a notational device. I can formulate Fourier transforms in 2D (to answer Chronos' similar objection), but it wouldn't be elegant. And elegance is not a luxury in mathematics - it does have a real, practical value.

So my answer to the OP's question of what you would use imagnary numbers for is really, "you use them for whenever they are convenient, and you abandom them whenever they aren't". But, as in any situation where you use a mathematical system to model something, you want to be sure you understand the mapping from model to notation.

As an aside, I'll admit I have a certain bias towards believing the unit vector notation should get more "air time" than it does in current applied mathematics curricula. As it happens, my PhD thesis involved a vector-space model of a system in N dimensions (N being a variable quantity and often quite large) and was taken aback by hte number of mathematicaly literate people (in CS, Physics, and Engineering) I encountered who knew Cartesian coordinates, knew basic complex analysis (not quote the oxymoron that it sounds like), but had never seen the same ideas expressed in terms of orthogonal unit vectors and had a great deal of trouble understanding why I would employ that approach for manipulating N-dimensional quantities.

[andymurph64]

Whack-a-mole,

Complex numbers (two-dimensional numbers) are inherently more complicated than the Real (one-dimensional) number system. It does take more effort to work with them. However, they are no less 'real' (no pun intended) than the number -2. They can be understood intuitively just like the number -2. It's just that in day-to-day routine life you are much more likely to confront the number -2 than 3+2i.

An analogy:

Negative numbers vs irrational numbers. Someone more used to math 'intuitively' understands e/4 just as well as -2. Another person (like my parents and siblings) e/4 is some complicated, voodoo made-up thing that requires much effort to understand.

If you were to work with things that require a 2-dimensional number system, you would become used to it.

[Francis E Dec, Esq]

One application in which imaginary numbers are necessary and not merely a convenient notation is in the evaluation of certain integrals. Many integrals on the real line would be perhaps impossible to solve analytically, yet when extended to a contour integral on the complex plane become easy. The real integral is then a certain limit of the complex version.

That said, I suppose one could still object with the observation that the complex plane is simply R² with an appropriate "complex" multiplication substituted for ordinary multiplication, and that therefore the integrals and theorems used could be restated using strictly real quantities. However, I would maintain that "R² with an appropriate "complex" multiplication" is identically the complex plane, and you are in fact using complex numbers whether you use the notation or not.

An analogy was drawn earlier with regard to negative numbers. And I would agree. Negative numbers can be regarded as purely a notational device to indicate the operation of subtraction on positive numbers.

But how about irrational numbers? Show me a √2 or π or an e in the real world. Every measurement that can in principle be made can only result in fractions. So aren't irrational numbers as hypothetical as imaginary numbers?

[sailor]

Some people lack imagination

[MC Master of Ceremonies]

Complex numbers aren't that difficult to understand, it's just like working with algebra, but with a few special relations.

[JasonFin]

I'll add that complex numbers make it a heck of a lot easier to solve certain types of differential equations. Complex eigenvalues of a linear system correspond to periodic solutions through Euler's formula e^(Pi*i) = -1. This shows up all over the place in physics, engineering, etc.