When I learned that the product of two imaginary numbers became the negative of the real-factor equivalent product, I did not get into why, but jsut found it to seem to make sense, without getting it. Does there exist a theory behind this definition? Or is it simply emirically found that it makes practical sense?
If you want to be confused, watch our "president" shovel his way out of the mess he has made in only 6 months. He is on almost every network, 24/7 the last few days. I can't avoid him nor can I figure out what he's doing?
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What a refreshing change to find someone who wants to try to understand instead of just getting an easy answer to homework. I wish we had more questions like that.
Like "happiness" or "friendship" the square root of minus one, i, (or j for engineers, to avoid confusion with current), can certainly be said to exist as a concept. But it is more than just that, because its square and fourth powers are real values. Cardano was able to use it in the form of complex numbers as an intermediate step to finding real solutions to the cubic equation
If we embrace the square root of minus one as a useful entity and define the complex number as the most general form of number, a satisfying consistency in the resulting set of broad and unifying ideas, (such as the fundamental theorem of algebra, or the periodicity of exponentials), starts to appear.
The use of complex numbers fills in missing pieces of the mathematical jigsaw, making connections between disparate mathematical fields, and resolving "dead-ends" caused by the limitations of the real domain.
So it is a concept that can be put to use to find answers about real things.
How can we begin to get a "feel" for interpreting the mysterious i ?
All the real numbers that we have dealt with before this, (positive, zero and negative), are associated with the one-dimensional geometry of the number line. Using the rules of algebra it is possible to show that the square root of minus one, (here written as i), is neither more than zero, equal to zero or less than zero; so one might argue that it does not exist at all. But this apparent "non-answer" is in fact a clue to the answer we need.
It turns out to be useful to take the view that this only shows that it does not exist anywhere on the one-dimensional number line, leaving its existence elsewhere as possible, (if only conceptually); but we must wonder where else it could "live"? (This is where I wish it was easy to post diagrams here).
The big clue is that -1 is equivalent to a rotation of 180°.
Let multiplication by r represent the operation of a counterclockwise rotation by pi/2 radians.
Double multiplication by r, i.e. r^2 results in pi radians or 180°.
Triple multiplication by r, i.e. r^3 results in 3pi/2 radians or 270°.
Quadruple multiplication by r, i.e. r^4 results in 2pi radians or 360°, back where we started.
Now we take these geometric results and try to find algebraic interpretations.
The easiest one is r^4 = 1, back where we started (with a connection to 360 degrees).
Also r^2 = -1, (makes sense because squaring both sides we get r^4 =1 back).
It follows that r corresponds to the square root(-1) = i
The rotation by pi/2 radians or 90 degrees must take us off the x-axis, and explains why i cannot be expressed in terms of the one-dimensional line representing real numbers.
It is of course not a real number solution, and we need to widen our definition of "numbers" to include a more general kind z = a + bi, called a complex number as you may have heard. We also need to illustrate such numbers using a special plane, (called an Argand diagram or simply the complex plane), which copes with the "imaginary" component in a similar way to the representation of y-values in a normal graphical plane.
What I have described above is just a connection between rotations of 90 degrees and the powers of r, which are i, -1, -i and +1.
Further into the subject it can be formally proved that
e^[i(pi/2)] = i
e^[i(pi)] = -1
e^[i(3pi/2)] = -i
e^[i(2pi] = +1
and much more wonderful stuff. But that is enough for today.
Regards - Ian
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