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.
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
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).
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.
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.
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 R[sup]2[/sup] 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 “R[sup]2[/sup] 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?
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.
