The Power of “i”
I Think, Therefore I Am?
Actually we got it backwards,
“I am, therefore I think!”
Let us use our “I am-ness” to ponder on the power of “i.”
Not I, as in yourself. You are already an expert in that, unless you are like me and that’s debatable.
Yes, the title can be considered a little misleading, but I guarentee that if you read on, you’ll learn something new. Not only that, but what follows can readily be understood by anyone.
I am talking about the mysterious mathematical creature known as “i,” the imaginary unit. I want to relay a little knowledge, and creativity about the nature of “i” and how it can be used to understand the nature of “you.”
If you have attempted and failed to understand it before, know that you are not alone. Actually, there really is no understanding it. Even the best of the best mathematicians don’t even really understand it, but they acknowledge its behavior. Sadly, much like how men and women perceive each other.
You have to see where it comes from to really appreciate its power.
At its core math is heavily reliant on a few fundamental definitions. One being the idea of “zero.” Quite philosophical actually. I am sure you can recall that ol phrase your math teacher tried to pass off as a little jingle,
“pos times a pos is a pos, neg times a neg is pos, pos times neg is a neg.”
Have you ever wondered why “they” tried to relentlessly and successfully drill that into our heads? I mean the first one is a given, but how the hell do you get a pos from two negs? This may be obvious to a narcissist, but to the average lay person this is not very intuitive.
How do we even imagine that?
Well, you kinda don’t. Instead, you utilize the idea of zero. Zero is always zero, regardless of whatever you try to multiply it with. We assert this symbolically as (c x 0)=0. Where “c,” represents any constant. The constant can be anything. It can be positive, it can be negative, and a hand full of other fancy labels that mathematicians have conveniently created for our suffering.
It is pretty straightforward to show that a neg. x pos.= neg., so I will omit the “seemingly” obvious. However, I have to admit though, that even when I look at some of the proofs for this seemingly simple fact that we often take for granted, my gut is left unsatisfied. We are so used to seeing numbers in their natural form expressed as simple symbols, yet that’s not all they are. Each one of them has a sign applied to them. For example, “3” is not just “3." It’s actually “+3.” In general, any integer “n” is actually “+n.”
I apologize if I have made things even more confusing, but bare with me. Things are about to smooth out.
In this sense, it is a very valid question to ask, “well what happens when “+” intermingles with “-”? Can’t we just say that -(n)=-n? Well, not really, because that equation is actually -(+n)=(-)(+)n, and we’re not sure what (-)(+) becomes. Do you see the hang up?
So without further ado, lets pick out two boogers with one finger with the following proof.
The only way that last expression can equal zero is if the term in the parenthesis is zero. Meaning, (-)(+) must equal a “-”, and (-)(-) must equal a “+”. If you take the time to think about all the different scenarios that can possibly occur within the parenthesis(which I recommend), then you’ll see that the little jingle your math teacher successfully brainwashed into your mind, is indeed the only possible conclusion.
Now that we are satisfied, we can just go back to acknowledging traditional positive numbers as symbols without sign.
You may be wondering what this has to do with the power of “i.” Well everything actually, because the definition of “i” is dependent on how we understand signs(+,-) that we apply to numbers. In particular, the square root of numbers.
For example, can you tell me what the square root of 1 is?
Of course, it’s 1, but there is another solution. (-1)(-1) also equals 1, so -1 is also a solution. What about (-1)(1)? That’s two different numbers, so not a square, and it equals “-1” so that is not a solution. In fact in general, the square root of any positive number has two possible solutions.
Now see if you can figure out what the square root of “-1” is.
Using what you now know about the property of signs and how they interact with one another, can you come up with a solution? What when multiplied by itself equals “-1”?
As easy as it looks, it’s not so easy is it? That is because there is no conceivable solution with the known properties of positive and negative numbers. It is literally in a league of its own, which we call the imaginary realm or “i” for short.
It is the only number, that when squared, gives “-1”.
These imaginary numbers were first considered when trying to come up with solutions to solve certain equations involving squared variables, commonly referred to as quadratics. Back in the day, they were highly ignored, because no one could make sense of them. It wasn’t until later on that a man named, Girolamo Cardano, and several others in the 16th century started giving legitimacy to these numbers.
Here is a list of known properties of “i”.
That last one I think is underrated, and is really just a special case of the one above it when we consider values of “n” that are negative, in this case, n=-1.
Many individuals have contributed to showing just how relevant “i” is in understanding physical phenomenon, and its application in a broad spectrum of mathematics.
One individual in particular really paved the way for demonstrating its power.
, is widely considered the greatest mathematician of all time, and for good reason. Any student in math comes to appreciate this as they continuously stumble upon his work as they progress. He laid the foundation for many areas in math, including “complex analysis,” which is the study of functions that involve imaginary numbers. They posses curious properties, especially when it comes to differentiation.
One identity in particular for which he is well known for, and perhaps you have seen before, is
This simple, yet extraordinary identity really captures the beauty, and awe inspiring power of mathematics and its use of the imaginary number. Three seemingly unrelated numbers, each with their own deep unique insights, are tied together with the concept of the negative integer.
To end this little journey on the power of “i”, I want to share some other fascinating identities that involve the number. Some of these I have come up with on my own and are uncertain if any are published, but they are all fairly simple to derive with math no further than calculus, and if you are intrigued enough I suggest seeing if you can prove them.
For that last one, taking “i” to the power of “1/i”, actually yields an infinite number of solutions, and the one displayed is the simplest. Similar to how square roots gives two solutions. It is what they call “periodic” in nature, same with Euler’s expression above.
For further reading on “i” and its application, I suggest researching the topic of complex numbers.
If you have any questions, or additional insights regarding anything discussed in this article, please comment below. I will do my best to clarify any confusion.
*For the mathematical equations “displayed” in this article the author used LaTeX converted to PNG.
