In Part 1, we investigated the origins of the imaginary number i and the complex numbers formed when an imaginary number pairs with a real number, learning about the centuries-long developments required to bring the concept of i into mathematical vernacular. Here, we will further explore its meaning and application.
To begin, let us look at the complex plane introduced at the end of Part 1.
The horizontal line in the diagram, which would be the x-axis in a Cartesian coordinate plane, represents the range of real numbers; the vertical line, which would be the y-axis in a Cartesian coordinate plane, represents the range of imaginary numbers.
Say we have the complex number 2 + 2i. To plot this number, we would move two units to the right on the real number axis and two units up on the imaginary number axis, placing us, looking at the figure again, at the upper right corner of the rectangle.
We can also think of complex numbers as vectors. The black line bisecting the red rectangle above is a vector representing a + bi, where the distance from the origin to the point is the modulus, or magnitude, and the angle from the real number line to the vector is the argument, or direction.
To view this in another light, complex numbers are the rotation of a vector lying on the real number line. From a visual geometric perspective, imaginary numbers cause rotations. More specifically, the number i is a rotation of 90 degrees. This concept helps us better understand why i squared equals -1: A 90 degree counterclockwise turn, applied twice, flips 1 to -1.
(Note that though i is a rotation of 90 degrees, a complex number does not make the full rotation because it has one part clinging to the real side of mathematics. For example, the complex number 1 + i is a vector with a modulus of 1 and an argument of 45 degrees.)
Now, this mathematical reasoning is all well and good, but inevitably we will encounter those mockers who exclaim, “Well, good fellow, where does the world have any use for such nonsense?” Alas, it is the struggle of mathematics teachers worldwide, assuring students that the abstractions in the classroom have applications in the real world.
In everyday life, the average Joe not involved in mathematical modeling has no need for imaginary numbers. However, several fields still use them, both directly and indirectly, including quantum mechanics, engineering, and physics.
For a more concrete example: Circuit analysis employs imaginary and complex numbers to simplify the measurement of oscillating sinusoidal currents, allowing easier calculation to combine such currents; and quadrature signals, responsible for digital communication and radar systems, are defined as complex numbers. (For more information on quadrature signals, see this site.)
Furthermore, for the artistically-inclined, complex numbers can create stunning fractal art.