## A Conversation for Imaginary Numbers

### Imaginary numbers

Stephen Started conversation Aug 14, 2003

When I was doing A level maths back inthe 15th century when Iwas just a lad, I understtod that imaginary numbers could be though of as numbers located in a plane and real numbers as numbers located on a line. The imaginary operator, i, could then be though of as an operator that turned The maginiude of ther real number by which it was multiplied trough 90 deg...I seem to remember anticlockwise but I may be wrong about that.....although thinking about it in terms of ordinary graphing conventions that would make sense.

eg to reach the point described by the number x+iy, you start at the origin (0,0), move x units along the x axis, turn 90 deg anti-clockwise and then move y units parallel to the y axis.

It has always struck me that we could add another operator, call it j, which would turn us through 90 deg anticlockwise(?) so that we are the facing parallel with the z axis and add another imaginary element to locate our number in a 3 dimensional space.

Having done that we could then add another to move into 4 dimensions and so on...

Of course it all gets more and more difficult to conceptualise as we get into more and more dimensions but the reasoning seems sound to the humble holder of an, admittedly poor, A level in pure maths now some 27 years old (The A level not me! Im at least twice that age! Then some...)

I can't convince myself, much as I'd like to, that this thinking is original. Nor have I worked out any of the implications of it.

Any thoughts?

### Imaginary numbers

MuseSusan Posted Nov 3, 2003

Very good observation, and very intriguing. In my introduction to my personal space, I was joking around about the same thing, saying that the "recipriversexclusion" (a number made up by Douglas Adams in his passage about Bistromathics in Life, the Universe, and Everything; it is defined as "any number other than itself") is equal to the cross product of any two vectors in the real-imaginary plane (or "complex plane"), thereby forming a new vector perpendicular to both, or going in the z direction. Obviously I was just joking around, but it would be interesting. Now, if "real" numbers are square roots of some positive number and "imaginary" numbers are square roots of some negative number, what would that third axis be? Square roots of an imaginary? (Actually, no, the square root of an imaginary would just be another complex number.) Anyone have any ideas?

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