Complex Numbers

In my blog on numbers and operators we introduced complex numbers that involved i. I said i was an operator, not a number. As an operator it rotated a numerical scale through 90 degrees. We said that applying the operation twice as in i 2 was a rotation of the scale through 180 degrees thereby reversing the scale and equivalent to a minus (-) operation.

A complex number delivers a point on a flat 2 dimensional plane. It can also be regarded as a vector to that point from point 0, 0 in the coordinate system. Let’s say we have a point 6 + i 3 at which we construct an object. Out object points are relative to this point and we construct our object on what is a local copy of our global coordinate system but crossing at this point as shown in red.

Suppose we remove the constructed object from this location and rebuild it at say a point – 2 + i 7. It would be as if we had moved our local red coordinate system -8 horizontally and + i 4 vertically to that new point and constructed our object on that coordinate system there. We call this – 8 + i 4 a translation and it contains the elements we have to ad to the original point elements to to get the elements of our new point.

After translating the local coordinate system we use it rebuild our object at this new location. Alternatively, we might rebuild the object at points progressing along the blue line. By making the points close enough and the time intervals between constructions small enough our repeated rebuilding of the object we would see as its linear motion

If we multiply (6 + i 3) by (a + i 0) using the FOIL (first, outer, inner, last) method and then simplify we get a(6 + i 3i). We are effectively scaling the vector length from 0,0 to our point 6 + i 3 by whatever a is.

If, on the other hand, we multiply (6 + i 3) by (0 + i 1) we get 0 + i 6 + 0 + i2 3 = -3 + i 6 (as i2 = – (minus). It is as if the new point has been created by rotating its vector around the global 0,0 point some 90 degrees anti clockwise. In our above diagram we show this 90 degree rotation of 6 + i 3 in green. Multiplying any point, for example the points on our object or the points that make up the lines of its local coordinate system, by (0 + i 1) will do likewise. It is as if we have taken the local coordinate system at 6 + i 3, translated it to point -3 + i 6, rotated that system by 90 degrees anti clockwise and then rebuilt our object on that rotated system. If we multiply by (0 + i a) we still rotate by 90 degrees but scale all points multiplied by a.

If we multiply (6 + i 3) by (3 – i 2) as an example we get 18 ( a scaled up scalar), we get – i 12 ( a scaled up vertically down (- i) component), we get + i 9 ( a scaled up vertical (i) component) and finally we get – i2 6 (equal to a positive scalar 6). The whole simplifies to the vector or point 24 – i 3.

The complex conjugate of a complex number z = x + i y is defined as z* = x – i y. So the product of a complex number and its conjugate z z* which is also known as the quadrance of z Q(z)= (x + i y)(x – i y) = x2 + y2 which is the square of the vector’s length or the square of its conjugate’s length.

Here we show how the operation x i can act on other operators like plus and minus to change a linear direction. But such an operator only has effect when applied to numbers. So we also show how on a unit circle it can change + 1 through i1, -1 and -i 1 before returnng to +1. However it is essential to realise that x i does not just operate on these 4 select points on the unit circle. It can operate on any vector, including any unit length vector ending at a point on the circle as represented by a unit length complex number. It will rotate such vectors clockwise 90 degrees around an imaginary axis coming toward us out of the centre of the circle .

The Irish mathematician Sir William Rowan Hamilton spent much of his life truing to apply complex numbers to the triple numbers of 3 dimensions. In 1843 whist walking over what is now called the Broom Bridge in Dublin he had a flash of inspiration that involved basic rules for multiplication of quadruple numbers, later called quaternions. He carved into the stone of the bridge i2 = j2 = k2 = ijk = -1. I wonder would he have been better leaving out the 1 but retaining the minus ( a reverse operation).