berational_large_2
(From crushedtees.com)

Origins

Through elementary school and usually most if not all of middle school, mathematics in America consists of real numbers and the basic mathematical functions we can apply to them (addition, subtraction, etc.). Then, algebra steps on stage, bringing with it the familiar numbers and symbols of elementary mathematics… and the alphabet.

In most cases in algebra, the letters stand for variables (e.g. the x in y = 2x). However, two noteworthy letters do not – e and i. These letters serve not as place-holders for numbers, but as numbers themselves! For the present, we will concern ourselves with i, also called the imaginary number, and, by extension, complex numbers.

First, for those unfamiliar with the number i and complex numbers, definitions are in order.

Basically, i is the square root of -1, or the radical of -1.

i = √-1

Paired with a real number in the form a + bi, we have a complex number, where a is the real part and b is the complex part.

This post has two parts. In this first part, we will explore the origin of imaginary and complex numbers.

Mathematics begins with the geometric constructions of the ancient Greeks, the ever-familiar length, width, area, and volume the enabled construction. As a negative measurement could not possibly exist, none meddled in the idea of negative numbers except to deem their existence absurd and, according to particularly vehement opponents of the concept, a stain upon mathematics.

45dd31b2b772bab0a19703050ffa048f
Image found on Pinterest

Mathematicians in the East had toyed for centuries with these funny negatives though, like European scholars, did not regard them as numbers in themselves. As late the 18th and even 19th centuries, mathematicians held this ancient opinion of negative numbers.  Still, in the 15th century they came more into use, and calculations for 3rd- and 4th- degree polynomials revealed a need for negative radicals.

Thus the imaginary number, named as an insult to its ludicrousy, emerged at the the same time that mathematicians were just acclimating to negative numbers.

185479-004-38ff6399
Gerolamo Cardano (Photo from Encyclopedia Britannica)

Italian intellectual (and gambler – mathematicians are human, too) Gerolamo Cardano, a Renaissance thinker who contributed to the emerging study of probability, published his Ars Magna, which included solutions to the cubic and quartic equations. One calculation of his yielded a negative radical which, though acknowledged as the answer, he did not accept as useful or meaningful mathematical constructs.

Some brave hearts after Cardano promoted the legitimacy of imaginary numbers. In his 1573 treatise L’Algebra, Rafael Bombelli (1526-72) established rules for the properties of imaginary numbers, namely the multiplication of them, such as that i squared equals -1. However, not until Leonhard Euler (1707-83), who dabbled much in complex numbers, and Carl Friedrich Gauss (1777-1855), who presented a clear investigation of complex numbers and of the connection between imaginary numbers and real numbers, did mathematicians really start to accept the concept.

gausseulernewton-782260
Gauss and Euler (Image from Gene Expression)

Gauss popularized the graphical explanation of complex numbers that others before him, Jean Robert Argand in particular, discovered. Now call an Argand diagram, this graph plots complex numbers as points on a complex plane, where the x-axis represents real numbers and the y-axis represents imaginary numbers.

argand-diagram-real
If you don’t understand the diagram now, no worries. The next part will explain it.

Interpretation and Application

argand-diagram-real

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.

220px-rotations_on_the_complex_plane-svg
(Photo from Wikimedia Commons)

(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.)

fractal-dark
Fractal art created using complex numbers (Photo from Interactive Math)

Furthermore, for the artistically-inclined, complex numbers can create stunning fractal art.