Use Your i-M-A-G-i-N-A-T-i-O-N

It’s kind of embarrassing to admit this as a 24-year-old college student, but I have an imaginary friend. His name is √(-1). He doesn’t keep it as real as some of my other friends, but he’s always my main dude in some of the more complex friendships I get involved in. He’s got a pretty interesting life story, and he’s exceptionally old and wise (imaginary friends never die). He was birthed by the Greek mathematician Heron of Alexandria in the first century A.D. He had a rough childhood (neglected and completely ignored by his father), partially because no one understood him. People ignored him, calling him useless. His name was even meant to be derogatory at one point. √(-1) met Gerolamo Cardano (coined the term “imaginary”), his first true friend, in 1637. After Cardano passed away, √(-1) was lonely until he was accepted into a new circle of friends during the 1700 and 1800s: Leonhard Euler, Carl Gauss, and Caspar Wessel. Although his new friends were a little on the nerdy side, they helped restore his personal image – eventually helping him to become a internationally known, and well respected celebrity.

imaginaryFriends

Image: Matheepan Panchalingam, via Flickr.

Okay, enough with my weird metaphorical stories. I’m here to tell you all about imaginary numbers. Now although the previous paragraph was strange and pretty corny, a lot of truth resides within. Heron of Alexandria was the first known person to have encountered imaginary numbers, back in the first century A.D.¹ He has been called a Greek mathematician, but interestingly enough, we now believe he may have actually been Egyptian.¹ So, how did Heron, the father of imaginary numbers, make such an important discovery? Well, he was trying to find the volume of a frustum of a pyramid with a square base. Specifically a pyramid with a side of the lower base is 28, of the upper 4, and the edge 15. Now the formula to solve this problem, as noted in the Stereometria of Heron of Alexandria is:

equation 1

Using a = 28, b = 4, and c = 15, Heron ultimately reduced this problem to equal √(81 – 144). However instead of writing h = √(-63), he wrote h = √(63).¹  Now whether Heron did this intentionally or not, we’ll never know. What we do know is this: No other scholar even bothered to take notice of these imaginary numbers for at least a thousand years. Believe it or not, from here it took another five hundred years for scholars to start taking imaginary numbers seriously.¹

The Italian engineer and architect Rafael Bombelli started searching for solutions to cubic equations in 1572, and during that time, he started defining some guidelines for using imaginary numbers.5 At this point, most mathematicians didn’t want to believe these numbers were significant, or that they even existed. Bombelli’s “wild idea” – that you could multiply imaginary numbers and end up with a real solution, didn’t sit well with fellow scholars. In fact, Bombelli didn’t even think imaginary numbers could ever have a true purpose, he just viewed them as a useful artifact to solve his equations. Later on, along came Leonhard Euler, a math rockstar in many people’s eyes. Euler really takes the cake here for noticing one of math’s more bizarre imaginary (in a math sense) equations. He showed the world how closely i, e, and π are related and further convinced the math community that i was relevant:Today, this is known as Euler’s identity.

e = cosπ +i(sinπ) 

Which reduces to:

e = -1 + 0

In simpler terms:

e = -1

In 1842 Carl Gauss did the impossible. He actually defined imaginary numbers (although Cardano coined the term “imaginary”.). Later on, once the reality of i was more widely accepted, our understanding mathematics expanded greatly. Euler, Gauss, and Wessel (1700s and 1800s) utilized imaginary numbers and led to the creation of complex numbers, which have a vital importance in math/science today.

Complex Numbers:

Instead of describing numbers and algebraic equations as points on a line, complex numbers and equations become points on a plane. Numbers are two dimensional – just like the integer “1″ is the unit distance on the axis of the “real” numbers, “” is the unit distance on the axis of the “imaginary” numbers. This results (in general) in complex numbers: They consist of two components, defining their position relative to those two axes. We generally write them as “a + bi ” where “a” is the real component, and “b” is the imaginary component.4

imaginary_Real_Graph

A basic graph showing real and imaginary points making up a complex line. Image: Public Domain.

Let me simplify this for you – What this means is that every polynomial equation has roots. Specifically, a polynomial equation in “x” with maximum exponent “n” will always have exactly “n” complex roots.

Me, myself, and i :

Imaginary numbers are more prevalent in the world than you’d think. Without the electronic circuit theories conceived from imaginary numbers, I wouldn’t be typing this blog post, and more importantly, you wouldn’t be reading it. When we plug something into an electrical outlet, we just expect our electronic devices to receive power. In fact, I bet you plugged in your laptop charger today and didn’t even think twice about it. What does exactly does this AC power have to do with imaginary numbers? Everything.

Voltage can be viewed as the amount of force pushing the current through your laptop charger and ultimately into your laptop’s battery. Voltage is a complex number. In fact, if you have a voltage of 110 volts AC at 60 hz (US Standard), that means is that the voltage is a number of magnitude 110. If you were to plot the “real” voltage on a graph with time on the X axis and voltage of the Y, you’d see a basic sine wave. Take the graph below for example:

sin_electricity

Basic sine wave showing only real voltage. Image: Public Domain.

If you decided to put your key in the outlet when the voltage was supposedly zero on the real portion of that graph, you’d still get electrocuted. That’s right, even though real voltage is zero here, “imaginary” electricity can give you quite a shock.3 Take the moment marked t1 on the above graph for example. The voltage at time t1 on the complex plane is a point at 110 on the real axis. At time t2, the voltage on the “real” axis is zero – but on the imaginary axis it’s 110. In fact, the magnitude of the voltage is always constant at 110 volts, and this is an important fact to know.5  All our modern day electronics account for these weird voltage patterns, and strangely enough this imaginary unit is represented with a  “j ” rather than an i (“i ” is used to represent current)!3

All in all, you can thank Bombelli and Euler for introducing imaginary numbers. Not only do they make math more confusing, but they also have many crucial applications in the world around us today.

Sources:

  1. http://press.princeton.edu/chapters/i9259.pdf 
  2. http://www.mathsisfun.com/numbers/imaginary-numbers.html
  3. http://en.wikipedia.org/wiki/Electrical_impedance
  4. http://en.wikipedia.org/wiki/Imaginary_number
  5. http://scienceblogs.com/goodmath/2006/08/01/i/
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