Visualizing Complex Functions (opens in new tab)

Students these days are very lucky.

More accurately: a complex number is not a point in the plane (in the same way as a real number is not a point in a straight line)

Given a line and an arbitrary selection of a 0-point and positive direction, you can map between real numbers and points on the line, and you can do a similar thing with complex numbers and points on a plane (which requires separate selection of positive directions for the real and complex axes.)

But real numbers aren't points on a line, and complex numbers aren't points on a plane.

The correct view is that the real numbers, also known as scalars, are quotients of 1-dimensional vectors, while the complex numbers are quotients of 2-dimensional vectors.

This means that given two 1-dimensional vectors, the second being non-null, there is always a real number by which you can multiply the second vector to obtain the first vector.

The same for two 2-dimensional vectors and a complex number. In this case the magnitude of the complex number changes the magnitude of the vector, while the phase rotates the vector.

When you understand this fact about complex numbers, than it becomes obvious that the imaginary unit is not imaginary but just a rotation with a right angle and it is trivial that its square, i.e. a rotation with 180 degrees is equivalent with a multiplication by -1.

The point are a third, different kind of mathematical entities, distinct from both real & complex numbers and from vectors.

In fact points (i.e. members of 1-dimensional, 2-dimensional and so on affine spaces) are the primitive objects.

From points you can define the vectors as differences of points (i,e, translations). Then from vectors you can define real numbers, complex numbers, quaternions and higher-dimensional matrices as quotients of vectors.

(Such definitions define e.g. a 2-dimensional vector as a class of equivalence of pairs of points in plane that are transformed into each other by a translation, and a complex number as a class of equivalence of pairs of 2-dimensional vectors which are transformed into each other by a proportional change in magnitude and a rotation with a fixed angle.)

Obviously, my explanation here is very simplified, but it is useful to be aware that e.g. a point in plane, a 2-dimensional vector and a complex number are 3 very different kinds of mathematical objects, even if all 3 are determined by a pair of real numbers.

They are very different because the set of operations that can be applied to each is different. For example only the complex numbers are members of a field. You can add 2-dimensional vectors (i.e. compose 2 translations), but you cannot add 2 points from a plane.