What complex numbers are and what they are used for.
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Here’s a slightly different take from other explanations. In school, we have been taught about how plus, minus, multiplication, reciprocals etc are computed between numbers.
But it turns out, mathematicians have taken that basic idea and generalized it to different types of ‘objects’ besides numbers.Objects include ‘real’ numbers (the numbers you are famliliar with), integers (whole numbers without a decimal point), vectors (a sequence of numbers), matrices (a two dimensional table of numbers), rational numbers (like 3/4) and so on.
For each of these object types, mathematicians have type-specific rules for how to add and multiply and subtract and take reciprocals. When you say a + b = c, if a and b are matrices, then the operation will yield the matrix c. Similarly, there are rules for multiplication, division, reciprocals etc. There’s a lot more that can be said on this topic, but it is not germane to your question.
Now to your question. A complex number is one such type of object; it is merely a pair of real numbers. Forget the terrible terms complex and imaginary.Suppose complex number p is a pair of the form (a,b) where a and b are real numbers, and q is (c,d), then the rule for addition is _defined_ as follows: p + q = (a,b) + (c,d) = (a+c, b+d). That is, the result is a pair where each is a pairwise summation of the first and second parts of p and q. Say, p = (10,20) and q = (30, 40).Verify that p + q = (40, 60).
Likewise, multipication p * q *is* *defined* *as* (a*c – b*d, a*d + b*c).
Verify that p * q = (-500, 1000).
This looks like an arbitrary rule, and there’s nothing intuitive about it on the face of it. But it turns out complex numbers map directly to 2D geometry. A point in 2D space can be represented by a complex number since it is a pair of numbers. Imagine a point p = (10,20), and imagine a line joining the origin (0,0) to p. Without any confusion, we can refer to the line also as p. Each complex number has a corresponding radial line leading to it.
Verify that p * (0, 1) = (-20, 10). Call this point (and corresponding radial line) q.
If you plot the lines p and q, you will find that the act of multiplying p by (0,1) has the effect of rotating p counter clockwise by 90 degrees, yielding the line q. Check it out. Similarly, p + (10, 0) has the effect of shifting p 10 units to the right on the x axis, and p + (0, 20) has the effect of shifting the point 20 units vertically.
Now, If you have a shape defined by a bunch of points, then multiplying each vertex by (0,1) would end up rotating the shape 90 degrees, since all points rotate counterclockwise the same way.
More generally, multiplying by arbitrary complex numbers ends up rotating and shearing shapes in different ways. I’ll stop here, except to add in passing that since we are speaking of rotations (and hence angles), you can imagine deep connections to trigonometry.
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