[scorpiomover]

Originally Posted by jndiii To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

Yeah, but you're missing the point, here. Imagine (pun intended), that no one else in the world understands complex numbers, but you do. Now, try to explain i = sqrt(-1), and how this has anything to do with anything they might care about. Explaining it as a "rotation" is step one. "Those areas of science where rotations are part of the equations" includes all wave equations and all but the simplest differential equations ... and allows those simplest differential equation solutions to be smoothly mapped (e^kx -> e^-ikx) to their sinusoidal aspects. Once it's applied to rotations, then other uses become acceptable, rather than just regarded as weird number theory. Yes, it's more than "just rotations", just as positive integers do so much more than count apples, but never underestimate the value of the concrete explanations.

I suppose that is one way of teaching it.

The way my maths teacher taught us it, was as part of Algebra. He taught us solving basic linear equations, and then solving quadratics. Then he explained how for any real number x, x^2 must be >= 0. Then he followed on with teaching us about equations where that wasn't possible, such as x^2 + 1 = 0. We accepted this implicitly, because it extended our the ability to solve algebraic equations, to ANY real-number equations.

Of course, there is also the practical aspect. But then, he'd been teaching us lots of examples of solving algebraic equations that are applicable to real life, such as how to maximise one's resources, ballistics, ball-throwing, examples from playing pool and snooker, and examples involving friction. He also taught us about how to use solving algebraic equations in solving differential equations, with examples that apply to real life, such as oscillating functions. Many times, such equations do resolve into complex solutions. The imaginary parts are not relevant to real life. But the real parts are, and the exact real parts that are real-life solutions, require the full complex solution, to calculated the real parts.

By the time we left high school, it was obvious to us that complex numbers were as real and as useful as real numbers are.

[Monte314]

Much of the richness of mathematics arises from the fact that objects can be viewed from very different theoretical perspectives.

In the article of the OP, a purely algebraic stance was taken, because that's what the intended audience could grasp. High school kids do not have a functioning knowledge of geometry, topology, or analysis sufficient to serve as an *intuitive* basis for understanding the complex numbers; but they know the Quadratic Formula!

The fact that these different views of an object exist is how almost all advanced mathematical work proceeds. When considering problems in one domain, it is common practice to "map" them to a domain where a corresponding theory provides a solution; this is then carried back to the original domain by means of the "mapping". A simple example is using a "change of variables" in calculus to do an integral: by applying an appropriate transform to a messy integrand, it is morphed into a function that is easy to integrate; after this simple integration is completed, the result is carried back by means of the inverse transform.

This approach is so powerful, in fact, that much of mathematics has shifted in emphasis to studying such mappings in abstract. The entire fields of Topology and Algebra are now mostly about mappings between objects; the mappings themselves are the new objects! The "bridge" field of Algebraic Topology is this in Spades.

This process can be repeated hierarchically: collections of mappings can constitute spaces having their own geometries, and mappings between these are studied... etc. In Analysis, this is subsumed under the field of Functional Analysis, where the objects of study are algebra-preserving mappings between a space of functions and a base field. This is of special interest to me, because most of my theoretical work is in this area. Here is a simple example:

Consider the Cartesian Plane, R2. Each point is an ordered pair of real numbers. Consider the point (a,b). I could identify this point with the line that has intercept a and slope b: the mapping L(x)=a + bx from R to R. I have identified each point with a function.

Given a function, L(x)= a +bx, I could pull out its coefficients or order 0 and 1, and put them in an ordered pair, (a,b), which I can think of as a point in R2. I have identified a function with a point.

Now I can ask all kinds of interesting questions. Notice that a line in R2 consists of infinitely many points. Regarding each of these points as a function (using the coordinates of each point as coefficients to define a function as above), I see that the original line characterizes an infinite collection of lines (as functions) in another space, called the "Dual Space". This allows me to geometrize, topologize, and analyze spaces where the "points" are functions: function spaces. This process can be repeated hierarchically, forever.

This approach has resulted in the emergence of entirely new areas of mathematics. A very recent one is "Information Geometry", where each point in the space is a probability distribution, and the goal is to find the "best path" in this space from one probability distribution to another. Believe it or not, this is very practical, having significant applications in machine intelligence particularly in the areas of document understanding, and certain kinds of optimization.

Quite a few of the problems I have posed here on the Forum have this flavor. Keep your eyes open and you will notice this.

[scorpiomover]

Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

Much of the richness of mathematics arises from the fact that objects can be viewed from very different theoretical perspectives. In the article of the OP, a purely algebraic stance was taken, because that's what the intended audience could grasp. High school kids do not have a functioning knowledge of geometry, topology, or analysis sufficient to serve as an *intuitive* basis for understanding the complex numbers; but they know the Quadratic Formula! The fact that these different views of an object exist is how almost all advanced mathematical work proceeds. When considering problems in one domain, it is common practice to "map" them to a domain where a corresponding theory provides a solution; this is then carried back to the original domain by means of the "mapping". A simple example is using a "change of variables" in calculus to do an integral: by applying an appropriate transform to a messy integrand, it is morphed into a function that is easy to integrate; after this simple integration is completed, the result is carried back by means of the inverse transform. This approach is so powerful, in fact, that much of mathematics has shifted in emphasis to studying such mappings in abstract. The entire fields of Topology and Algebra are now mostly about mappings between objects; the mappings themselves are the new objects! The "bridge" field of Algebraic Topology is this in Spades.

I noticed this myself when in university, so much so, that I was thinking of doing a dissertation on Lattice Theory in my final year, even though for my degree in maths, we didn't need to do a dissertation.

Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

This approach has resulted in the emergence of entirely new areas of mathematics. A very recent one is "Information Geometry", where each point in the space is a probability distribution, and the goal is to find the "best path" in this space from one probability distribution to another. Believe it or not, this is very practical, having significant applications in machine intelligence particularly in the areas of document understanding, and certain kinds of optimization. Quite a few of the problems I have posed here on the Forum have this flavor. Keep your eyes open and you will notice this.

Excellent points. I'll keep this in mind in the future.