The world for the most part offers niche occupations for the average citizen, who among all others finds the most contentment living upon it. I live in a country which rewards a university education with a public service position. Now, is that actually counterproductive or not? It turns out, as recent research indicates, that only fifteen percent of the founders of the Fortune 400 companies have an MBA. Face it, life is geared to benefit the average. Many people never really grasp a deeper mathematics beyond the simple concepts involving operations on fractions. That is the point at which they begin to acquire the traits of skepticism. That's grade three arithmetic, by the way.

I am convinced that the roots of skepticism lie for the most part in the inability to grasp concepts that are essentially over one's head. Mention the concept of the imaginary number i for example, and most people will immediately close their minds, cross their legs and fold their arms in a harrumph of incredulity. The square root of minus one, indeed! And yet, the imaginary number exists! It is a real thing after all. To prove it, I will explain it to you. Reading comprehension skills are mandatory if you wish to continue with this blog.

In certain scientific applications, in my example I will use electronics since it encompasses all sciences, it is found useful to use imaginary numbers, particularly with respect to solving electronic circuits. I will begin by discussing the j operator (it is the same as i in non-electronic applications but in that discipline i denotes electric current (amperes), so we use **j**). The **j** operator is used to denote rotation of 90° in the counterclockwise direction on the 2-dimensional X-Y matrix. For example, on the X-Y graph a line **a** units long can be operated on by the operator **j** to become **ja**, a line of the same length as before but rotated 90° in the counterclockwise direction to lie on the Y axis. Any quantity operated on by **-j** will rotate through 90° in the clockwise direction. The quantity **j(ja)** is written **j²a**, and **j(j(ja))** is written as **j³a**. So **j²a** becomes **-a** (2 90° counterclockwise rotations). Things become interesting when we analyse the situation where **j**-ing **a** twice in succession brings it to the same point as a single operation with a minus sign therefore **j²=-1** and we can therefore conclude that **j=√-1**. **j³** must equal **j(-1)** or **-j**, and **j^4** must equal **j²•j²** = (-1)(-1) = +1. Remember, this is all about rotating around the X-Y axis.

So far, so good. In mathematics, the square root of a negative number is known as an imaginary number. Its terminology is misleading because in dealing with some scientific applications imaginary numbers become real. In order to avoid difficulty in dealing with square roots of negative numbers we consider that every imaginary number can be expressed as the product of a positive number and the square root of -1, for example the √-25 is √-1•√25 = √-1•5. We can then write this expression as **j**5. The term complex number refers to an expression wherein an imaginary number is united to a real number by a plus or minus sign. 3**-j**4 is a complex number.

Even if you have understood all of this, you are still inclined to say "so what?" If you haven't understood any of this you are a happy camper and no doubt a skeptic. Go in peace. And if you, the former, wish to dispel any trace of ambiguity with respect to the practical purpose of the imaginary number and are interested in knowing more then continue by reading this short .PDF document (*temporarily unavailable*). The Adobe Reader is required. The paper was composed using the LaTex engineering and mathematics markup language, used for publishing those clever vector diagrams and formulae with all those neat symbols. Study it, understand it, and go to the head of the class.

I am writing this not merely to impress you, or to acquire a modicum of credibility, although I do hope to have accomplished some of both, but to point out that a lack of understanding of anything should not be allowed to permit the introduction of doubt into that which is real. Oh, and also just because somebody has a university degree doesn't mean that they're better than you.

## Friday, January 18, 2008

### Trust not the skeptic..

Posted by S.W. Lussing at 12:55 PM

Labels: imaginary number, j operator, mathematics, skepticism

Subscribe to:
Post Comments (Atom)

## No comments:

Post a Comment