Real and/or Complex?

I have no idea why I've been looking into quaternions lately, or what I would ever use them for. I've been trying to use vectors to simulate elliptical orbits of planets in Geogebra, but it's been a lot of work. I don't even know if Geogebra handles quaternions. Reminds me of a Lisa Simpson line: "Bart, I'm NOT going to learn Ancient Hebrew!"

One positive thing it's led me to is the book Visual Complex Analysis by USF math prof Tristan Needham. He definitely sums up my opinion about how most math classes lose students: there's no attempt to link the abstractions with our experience of the world, and this applies to the disdain mathematicians feel for geometric reasoning, and worse:

More likely than not, when one opens a random modern mathematics text on a random subject, one is confronted by abstract symbolic reasoning that is divorced from one’s sensory experience of the world, despite the fact that the very phenomena one is studying were often discovered by appealing to geometric (and perhaps physical) intuition.

It's tough for a math nut like me to admit, but when I open most Dover math texts or those yellow ones in brainy bookstores, the ridiculous symbols and complete lack of visual appeal make my heart sink and I wouldn't carry the book home if it were free.

Another aspect of my Calculus Program Needham anticipates in the book is my lack of rigor. After giving credit to those who believe rigor strengthens the "precarious" structure of mental constructs, he suggests:

...that our mathematical theories are attempting to capture aspects of a robust Platonic world that is not of our making. I would then contend that an initial lack of rigour is a small price to pay if it allows the reader to see into this world more directly and pleasurably than would otherwise be possible.

Small price to pay, Prof.

The Fun Calculus Program

The first Fun Calculus Program will be starting in San Mateo on Monday night! We'll see how much Calculus we can learn in only SIX classes, held once a week.

After years of tutoring otherwise capable math students who were stopped in their tracks by the pointlessly abstract and obscure way Calculus is taught in high school and college classes, I finally created a program to teach the fun stuff.

Yes, I said the fun stuff.

Mathematics has been called the Queen of the Sciences. If so, Calculus is the Crown Jewel of Mathematics. The years of algebra and geometry I endured was all leading up to getting the keys to the strange and marvelous kingdom of Calculus. I found the instantaneous velocity and acceleration of falling, flying, spinning or oscillating objects, and the areas inside weird shapes and under intricate curves. I was able to find the largest rectangle that could be inscribed in a certain triangle or ellipse, and the level of production of widgets that maximized the profit or minimized the costs for a company.

The discovery of this magic kingdom (oops, queendom) was spoiled only by the new and pointless introduction of the topic of limits. Long story short: zooming in on curves as Newton and Leibniz did seemed to mean dividing by zero, a no-no in math. This made mathematicians uneasy, and it took 150 years or so before the artifice called limits was invented to shut everybody up. In the meantime science had embraced Calculus and its power to model the behavior of everything from fluids to heat to planets. Apparently you don't need limits to actually use Calculus.

I figured I would mention limits in the Fun Calculus Program, but see how much useful stuff I could get to in 6 class meetings without it. in every calculus textbook limits are introduced in chapter 2 (after a short review in chapter 1), but I have an email from a professor of applied math who agreed engineers never use limits and I could probably ignore the whole subject.

Another obstacle to fun in calculus is all the calculating you have to do. Good news: computers can do it more quickly and with a lot less complaining than we humans can. I use a powerful and FREE graphing program called Geogebra that can take derivatives and integrals, leaving me to the human part of analyzing the results.

Forget doing weeks of quotient rule problems using pencil and paper; plug the function into WolframAlpha.com and get your derivative in 2 seconds! To me, the important part of math in the future will be the ability to set up the right differential equations to model the system you're analyzing. Technology can help you crunch the numbers, leaving you to do higher level thinking. Don't take my word for it, listen to Conrad Wolfram's TED talk on this very topic.

Check back for updates on the Fun Calculus Program, happening every Monday night from January 3 - February 7!