"The shortest path between two truths in the real domain passes through the complex domain."
~Jacques Hadamard
Historical Background and Development
Before we can talk about imaginary numbers and their history, we need to understand The Fundamental
Theorem of Algebra which was first proved by the mathematician, Gauss. The theorem states that,
Every polynomial equation of degree n has exactly n roots.
For example, f(x)=x2 will have two roots and g(x)=x3+2x2+4
will have three roots. Typically, the easiest way to understand and find roots for an equation would be to
look at a graph and check to see where it crosses the x-axis. The problem with this comes with equations like
f(x)=x2+1. This equation will never cross the x-axis, but according to The Fundamental Theorem
of Algebra, it has to have two roots. The x,y-plane that most of us are familiar with only deals with what have
been labeled as “real numbers.” Because of this, this plane that we all know and love is missing numbers that
would help us to solve problems like this. These numbers are known as “imaginary numbers.” They are just as real
as real numbers, as in they do indeed exist. They have just been given the unfortunate, misleading name of “imaginary.”
To gain a better understanding of the theorem above, I have included a video that goes into greater
detail about The Fundamental Theorem of Algebra.
So what is an imaginary number? An imaginary number is written in the form bi, where b is any
real number and i is the square root of -1. Euler was the mathematician who first decided that the square root of -1 was a number. Although, he did refer to them as
imaginary numbers, he also insisted that they can indeed be used to solve equation that were once believed to be unsolvable.
He was also the one who first represented this number as “i,” the imaginary unit.
There is a series on YouTube all about imaginary numbers from Welch Labs. It’s a
thirteen part series. I’ve included the first video in the series. The video shows ways
to visualize the complex number roots. One problem students often have with complex numbers
is that when you graph a polynomial with complex roots, you can’t see the roots. This video
shows a way to think about this very idea.
Students often have trouble with imaginary numbers. They don’t understand how something "imaginary" could exist.
The word "imaginary" is what students tend to focus on. Gauss, a mathematician who lived from 1777-1855, said the following:
“That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity,
is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1
had been called direct, inverse, and lateral units, instead of positive, negative and imaginary (
or even impossible), such an obscurity would have been out of the question.”
Gauss is suggesting here that if imaginary numbers had been called "lateral numbers" instead, there wouldn't be any
confusion. Unfortunately, the name stuck around.
"It’s called the Imaginary axis not because it isn't there, it's just as real as the real axis, but the numbers on it are
the pure imaginary numbers, the ones without any real part."
~Professor David Eisenbud
Explanation of Mathematics
Complex Number Arithmetic
Despite the name, complex number arithmetic isn’t all that different than what arithmetic
with the real numbers. For addition and subtraction, you add.subtract like terms,
which in this case means you add/subtract the real parts together and to imaginary
parts together. Multiplying complex numbers is similar to the idea of multiplying
factors of a polynomial. I’ve included an image of examples of these to the right.
Complex number division is also not as complicated as it sounds, but it does have
an additional aspect to keep in mind. Before explaining the process, it’s important to
explain this aspect, which is how to find the complex conjugate of a complex number.
To do this, you change sign of the imaginary part of the number. For example,
the complex conjugate of 3+2i would be 3-2i. We use the complex conjugate in order to
get rid of any imaginary component in the denominator. Now we can follow the steps
below to use complex number division.
The Complex Plane
The complex plane is a way to visualize the complex numbers. The horizontal axis
represents the purely real numbers and the vertical axis represents the purely imaginary
numbers. Everywhere else represents a number that has an imaginary and a real part to it.
One of the advantages of using the complex plane is that is can be helpful when doing
arithmetic with complex numbers. The examples I’ve included are fairly simple and using
the complex plane may not make the process any faster, but when you have more complicated
problems, it can be an easier way to do arithmetic with complex numbers.
Often, vectors are used to represent a complex number on the plane. The length of this
vector is known as the point’s modulus. The angle from the real axis to the vector is known
as the argument. These two measurement are what will be used in order to be able to do
arithmetic on the complex plane.
Visualizing Arithmetic on the Complex Plane
Multiplication and division on the complex plane are next. If you want to multiply
two numbers on the complex plane, you take the arguments (angles) and add them together.
Then you take the modulus (length) and multiply those together. For example, we’ll
multiply the numbers that are represented below with the red and blue vectors. The
red vector represents the number 2+2i and the blue represents the number -2i. The blue
vector is more difficult to see in the picture because it overlaps part of the y-axis.
The arguments for red and blue vectors are 45 and 270, respectively. This means that
the argument for the product (represented with the purple vector) will be
45 + 270 = 315. We know the angle, now we just need to know the length.
The length of the red vector is 2.828 units and the length of the blue vector is
2 units, so the length of the purple product vector is 5.656.
Two-Dimensional Numbers
The video below gave a new way of thinking of the complex numbers. This was to think of
complex numbers is as two-dimensional numbers. Real numbers are one dimensional and complex
numbers add an additional dimension: imaginary numbers. This idea helps when it comes to
understanding the complex plane. This now begs the question: are there three-dimensional
numbers? Four-dimensional? To learn more about this idea, watch the video below:
Fantastic Quaternions.
"This is the deeper meaning beneath imaginary numbers. They aren't just some random extra number or hack - they are the natural
extension of our number system from 1 dimension to 2."
~Stephen Welch
Significance and Applications
Mandelbrot Set
The Mandelbrot Set is one of the most well-recognized fractals, but how is it created? How do you know which points on the complex plane are included or discluded? Say we choose a point on the complex plane and call it z0. Now, we need the following recursive equation: zn+1=zn2+C, where C is equal to z0. Below I have listed the steps to follow to find out if any given point is a part of the Mandelbrot Set.
Julia Sets
For Julia Sets, we use the same recursive equation as before: zn+1=zn2+C. The difference now is that the value of c does not have to be the same as the value of z0.
Want more specifics? Want to see some examples? Want to create your own? Try out the applet below: Complex Numbers and Julia Sets.
If you want to learn more about the mathematics of Julia Sets, click here.
“That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity,
is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1
had been called direct, inverse, and lateral units, instead of positive, negative and imaginary (
or even impossible), such an obscurity would have been out of the question.”
~Carl Friedrich Gauss (1777-1855)
References
Andreescu, T., & Andrica, D. (2014). Complex numbers from A to ... Z. New York: Birkhäuser.
https://link.springer.com/content/pdf/10.1007/978-0-8176-8415-0.pdf
Mandelbrot Set. (n.d.). Retrieved from http://mathworld.wolfram.com/MandelbrotSet.html
[Numberphile]. 2016, January 18. Fantastic Quaternions - Numberphile. Retrieved from:
https://www.youtube.com/watch?v=3BR8tK-LuB0
[Numberphile]. 2014, July 9. Fundamental Theorem of Algebra - Numberphile. Retrieved from
https://www.youtube.com/watch?v=shEk8sz1oOw
Welch, S. (n.d.). Imaginary Numbers are Real: Workbook. Welsh Labs, 1-96. Retrieved December 7, 2018,
from https://static1.squarespace.com/static/54b90461e4b0ad6fb5e05581/t/5a6e7bd341920260ccd693cf/
1517190204747/imaginary_numbers_are_real_rev2_for_screen.pdf.
Wittens, S. (2013, January 5). How to Fold a Julia Fractal: A tale of numbers that like to turn.
Retrieved from http://acko.net/blog/how-to-fold-a-julia-fractal/?second
Untitled Artile: http://www.mesacc.edu/~scotz47781/mat120/notes/complex/dividing/dividing_complex.html
For example, f(x)=x2 will have two roots and g(x)=x3+2x2+4
will have three roots. Typically, the easiest way to understand and find roots for an equation would be to
look at a graph and check to see where it crosses the x-axis. The problem with this comes with equations like
f(x)=x2+1. This equation will never cross the x-axis, but according to The Fundamental Theorem
of Algebra, it has to have two roots. The x,y-plane that most of us are familiar with only deals with what have
been labeled as “real numbers.” Because of this, this plane that we all know and love is missing numbers that
would help us to solve problems like this. These numbers are known as “imaginary numbers.” They are just as real
as real numbers, as in they do indeed exist. They have just been given the unfortunate, misleading name of “imaginary.”
