Tag Archives: complex numbers

Applications of Imaginary Numbers

The concept of imaginary numbers has always been a fascinating one. The Greek mathematician Heron of Alexandria, born around 10 AD, is noted as being the first person to have come up with the idea of imaginary numbers. It wasn’t until the 1500’s, though, that rules for arithmetic and notation for complex numbers really came to fruition. Of course, at the time most people thought imaginary numbers were just stupid and pointless. Heck, today I’m pretty sure most people still think imaginary and complex numbers are stupid and pointless. Surely they can be used for more than just generating pretty looking fractals (like the Mandelbrot set), right?

Yes, because of imaginary numbers there is a solution to any type of polynomial equation… but there has to be more use to them than that, right? The topic I wish to present in this article is about some of the other applications of imaginary numbers. Imaginary numbers are really useful and they can be used to do all sorts of awesome things!

While presenting this information, I do not claim to list every single practical use of imaginary numbers. There are many useful applications that involve some crazy complicated mathematics and are admittedly beyond the scope of my understanding at the present time. Rather, I wish to share a few of my favorite applications of imaginary numbers. It is my hope that the reader will learn more about why mathematicians have studied so much about imaginary and complex numbers.

First, complex numbers have a remarkable application in triangular geometry. There is a fascinating theorem called “Marden’s theorem”. I read about this theorem in an article written by Dan Kalman, a doctor of mathematics who works in the Department of Mathematics and Statistics at American University. He claims that this theorem is “the Most Marvelous Theorem in Mathematics.”

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A visualization of a Steiner inellipse with its foci. The ellipse is based on the polynomial p(z)=z3-(9+9i)z2+(3+52i)z+(33-39i). The black dots are the zeros of p(z), and the red dots are the zeroes of p'(z) and the foci of the inellipse. Uploaded by User Kmhkmh for Wikipedia on 2/6/2010. Creative Commons license. Reuse permitted.

Basically, this theorem can help one find the foci of a Steiner inelipse. A Steiner inellipse is simply an ellipse that is inside of a triangle and is tangent to the midpoints of the three sides of the triangle. Such an ellipse is shown in the following diagram.

The foci of a Steiner inellipse can be found by using complex numbers! The triangle’s vertices can be written as points in the complex plane as follows: a = xA + yAi, b = xB + yBi, and c = xC + yCi. Marden’s theorem states that if you take the derivative of the cubic equation (x-a)(x-b)(x-c) = 0 and set it to zero, then the solutions of this equation will be the two foci of the Steiner inellipse in complex numbers. Isn’t that a really bizarre theorem? If you think about it, though, it makes some intuitive sense. When you take the derivative of an equation and set it equal to zero, the solutions of that equation give you the maximum and minimum values found on the arcs in the equation. A regular cubic equation could have up to two arcs, so it’s natural that there would be two max/min values. The fact that these two values are the two foci of the inellipse is really interesting.

As it turns out, using complex numbers here gives us a very amazing and useful geometric tool to use. There are also a few generalizations of this theorem that apply to different types of polynomials and other geometric shapes!

So it seems that first we have geometric applications for complex numbers. Now I would like to present a second category of applications. These are related to phasor calculus. A phasor is a complex number that represents a sinusoidal function. Thanks to the amazing Euler’s formula (e= cosx + sinx), sinusoidal functions can be rewritten as complex numbers. This allows for easier problem solving and analysis for many types of problems.

For instance, in electrical engineering alternating currents can be a pain to analyze sometimes. After all, they have voltages that exhibit sinusoidal behavior. With the use of phasors, one can analyze aspects of AC circuits more easily. Analysis of resistors, capacitors, and inductors can be combined into a single complex number, which is called the impedance. Phasors are comparatively easy to interpret, so it’s a lot easier to study AC circuits when studying them in the complex plane! In addition to AC circuits, complex numbers are similarly useful when studying electromagnetic fields, where the quantities of electric and magnetic field strength are combined into a single complex number.

The last application I wish to bring up involves the usage of imaginary numbers to solve integration problems. As it turns out, we can use the aforementioned Euler’s formula to simplify real integration problems and help us find real answers. This is done by using a base integral that has a complex solution. An example of a base integral would be∫ e(1+i)xdx. Using simple u substitution, we can find the answer to this integral, which is ((1-i)/2)e(1+i)x + c1 + ic2. With this known imaginary answer, we can compute the answer to a real integral.

Consider, for example,∫ excosxdx. First, we rewrite the previously mentioned base integral as: ∫ exeixdx. Then we can use Euler’s formula to alter this integral further:∫ exeixdx = ∫ ex(cosx + isinx)dx. This will further simplify to∫ excosxdx + i∫ exsinxdx. We can set the known solution of the base integral equal to this complex integral and solve for ∫ excosxdx , which is the real integral we are trying to compute. We will see that the imaginary parts must be equal and the real parts must also be equal. Solving in this manner will show us that ∫ excosxdx = .5ex(cosx + sinx) + c. Hopefully I don’t have to explain how useful integrals are! The fact that complex numbers can help us solve integrals alone means they are really useful.

I think in general it seems that whenever there’s an oscillatory phenomenon of any kind then complex numbers are naturally helpful in describing said phenomenon. Complex numbers have multiple substantial applications in a multitude of scientific problems. In addition to the few I’ve mentioned, complex numbers are also used in: quantum mechanics, control theory, signal processing, vibration studies, cartography, and fluid dynamics. Dang. Since a long time ago complex numbers have been thought of as trivial and inconsequential. Descartes himself (who coined the term “imaginary”) called these types of numbers imaginary because he meant for this to be derogatory. However, as we have learned more about math throughout the ages we have found many a useful application for imaginary numbers.

The aforementioned Mandelbrot set. This is a fractal involving a set of complex numbers. Uploaded by User Localhost00 on 10/13/2013. Creative Commons license. Reuse permitted.

Sources:

-Imaginary Number. (n.d.). Retrieved February 24, 2015 from Wikipedia: http://en.wikipedia.org/wiki/Imaginary_number#cite_ref-1

-Complex Number. (n.d.). Retrieved February 24, 2015 from Wikipedia:

http://en.wikipedia.org/wiki/Complex_number#Applications

-Dan Kalman, “The Most Marvelous Theorem in Mathematics,” Loci (March 2008)

– P. Ceperley. 8/28/2007. Phasors. Retrieved from:

http://resonanceswavesandfields.blogspot.com/2007/08/phasors.html

-Integration Involving Complex Numbers. Retrieved February 24, 2015 from:

http://www.mathnotes.org/?pid=109#?pid=109

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Where do numbers come from, anyways?

A short history of imaginary numbers

Mathematicians first came up against imaginary numbers in the mid 16th century and it wasn’t until the mid 19th century that they saw how awesome complex numbers could be. Before we look at how imaginary numbers came to be, let’s look at some other familiar number systems.

Number Systems Solve Problems

The first, most obvious, number system is the integers, or counting numbers. We have integers to answer really useful question that we see all the time in day-to-day life like, how many grapes can I really stuff into my mouth at a time? (About 9)

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The next number system we might think about is the rational numbers, or fractions. These also serve to answer really useful day-to-day questions like that involve division like, if I have 6 roommates but only 1 pint of ice cream, what portion of the tub can I eat?

This assumes that I’m a fair roommate who would never eat more than her share of the communal ice cream- which leads us to our next number system. Negative numbers are used to measure debt; like how much ice cream I might owe my other roommates.

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With these two systems we can count and divide stuff, but we also might have other sorts of problems like how to measure things. Like, for instance, I might need to walk 1 block south and 2 blocks east around a park to get to school, but since I’m inherently lazy (a good quality for all mathematicians), I cut straight through the park, and find that I’ve walked √5 blocks to get to school, which is totally irrational. We have to deal with irrational numbers when we measure distances because it turns out (to the Greeks’ great sorrow) that not all distances can be measured with rational numbers.

So what about imaginary numbers?

Where did they come from, and what are they good for?

We’ve got Real Problems: Imaginary Numbers give Real Results

In the mid 16th century a mathematician named Tartaligia came up with a general solution for finding the roots of 3rd degree polynomial, but he held his method as a closely-guarded secret. Another mathematician named Cardano eventually managed to convinced the reluctant Tartaligia to tell him the method, on the condition that he would never ever tell anyone else. Well, I think they should make a soap opera about 16th century mathematics because in 1545 Cardano completely betrayed Tartaligia by publishing the solution in his book ‘Ars Magna’.

Tartaligia’s method is really important in the history of imaginary numbers because there are some perfectly good 3rd degree polynomials with perfectly good real roots that this method doesn’t make sense for. When you use Tartaligia’s method for these certain polynomials, you get a nonsense step in the middle of the calculation where you have to take the square root of a negative number.

Consider for example the equation:

x3 = 15x + 4.

 This cubic has a real root x = 4, but when we apply Cardano’s formula we get:

x = ∛[ 2 + √(-121) ] + ∛[ 2 – √(-121) ]

The real problem (pun intended) was that even though everyone knew that taking the square root of -121 was totally ridiculous, they also knew that the root x=4 was a totally reasonable real solution. There was this breakdown in what the equation was trying to communicate.

The first mathematician to really break through this mold was Rafael Bombelli, who got around this problem with the crazy proposition that, well let’s just imagine that there’s some number that’s negative when we square it. With this assumption he was able to manipulate Tartaligia’s equation, for instance the example above becomes:

 ∛[ 2 + (√-121) ] + ∛[ 2 – (√-121) ] = (2 +(√-1) ) + (2 –  (√-1))   (**!)

= 4 – 2(√-1) + 2(√-1) – (√-1)2

= 4 (!)

Conveniently, the ‘imaginary’ numbers cancel out, leaving good real roots! Way to take a leap of faith, Bombelli!

** Okay, hold on, what just happened there? Well it turns out (2 +(√-1))3 is :

(2 +(√-1))3 = (2 +(√-1))*(3 + 4i) = (2 + 11i) = 2 + (√-121)

 Same goes for ( 2 – (√-1)). Neat.

About a half-century later in 1637, Descartes coined the term “imaginary” when he wrote about roots of nth degree polynomials in his book ‘La Geometrie’. He wrote that these polynomials might have as many as n solutions, but sometimes they have fewer, as some of the solutions are ‘impossible’, ‘improbable’ and ‘imaginary’. He meant it in a demeaning way- like we should be doing ‘real math’, not ‘day-dream math’.

At this point in history mathematicians swept imaginary numbers under the rug; they cautiously imagined that they might exist but only for long enough to cancel out and yield real solutions. It wasn’t conceivable that they might be useful by themselves.

It’s sort of a complex story

 The complex number system was really first understood as the incredibly powerful mathematically tool that it is in the 19th century when Gauss took an interest in imaginary numbers. He came up with a geometric interpretation for complex numbers (which, to be fair, was also independently discovered by the Norweigan mathematician Wessel and the French bookstore manager and amateur mathematician Argand). Gauss’ interpretation was that the imaginary number line is just like the real number line, so a complex number (a number with a real and an imaginary part) is actually a coordinate in a plane, like in the image below. We just say that real numbers lay on the horizontal axis, while imaginary numbers lie of the vertical axis.

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The really amazing and exciting thing about this description is that it’s totally consistent with operations we might like to do on complex numbers, like addition and multiplication. Consider what happens when we multiply by i, for instance 1*i. We rotate 90 degrees, from the coordinate (1, 0) to the coordinate (0, 1), so we can say that multiplying by i is the same as rotating by 90 degrees. Then consider i*i (which is i2): we rotate another 90 degrees and end up at -1! Neat!

Later in the 19th century complex numbers got a lot of traction because they turned out to be very good at describing waves. At this point in history, physicists were developing ways to describe electricity and magnetism, and complex numbers enabled them to really understand these phenomena.

The neat thing about complex numbers is they show up everywhere in our day-to-day lives. Anything you have that uses electricity only works because some engineer somewhere knew how to build it using their understanding of imaginary numbers. Can you even imagine your life if you couldn’t send your mom photos of other people’s dogs? Any time you snap a photo or make a phone call your phone does a Fast-Fourier-Transform, which is a method based on complex numbers, to compress the data into just tiny amounts of storage.

So do imaginary numbers really exist?

Complex numbers are great representations for lots of natural phenomena, like electricity. Remember how we used our other number systems- like how we used integers to count how many grapes I could fit in my mouth? In some sense, it’s just the grapes that exist, not that the integers. The integers exist mathematically- they’re only there to describe the real world, and this is true for every number system. In this sense, not only do imaginary numbers ‘exist’ mathematically, but they’re first-class citizens because they describe so many awesome things that we use every day.

Sources:

http://www.bbc.co.uk/programmes/b00tt6b2

http://en.wikipedia.org/wiki/Imaginary_number

http://www.storyofmathematics.com/16th_tartaglia.html

http://en.wikipedia.org/wiki/Niccol%C3%B2_Fontana_Tartaglia

http://mathfaculty.fullerton.edu/mathews/c2003/ComplexNumberOrigin.html

http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

http://en.wikipedia.org/wiki/Caspar_Wessel

http://www.ms.uky.edu/~sohum/ma330/files/eqns_4.pdf

http://www.und.edu/instruct/lgeller/complex.html