[baudrunner]

I will briefly try to explain imaginary and complex numbers.

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). The j operator is used to denote rotation of 90° in the counterclockwise direction. 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 counter clockwise 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. 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=sqr.rt -1. j³ must equal j(-1) or -j, and j^4 must equal j²•j² = (-1)(-1) = +1.

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 sqr.rt of -1, for example the sqr.rt of -25 is sqr.rt -1•sqr.rt 25 = sqr.rt -1•5. We can then write this expression as j5.

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-j4 is a complex number.

All operations can be performed on complex numbers. It is not within the scope of this discussion to give examples of their actual application in electronic circuit problem solving as that would be going too far, but trust me, these imaginary numbers are quite real when used practically.