In the realm of intellectual achievements, mathematics is the GOAT. It’s also the most hated, which is really too bad. Consequently, I suspect that this will be my least read blog post. People who refuse to learn about mathematics are cutting themselves off from a beauty that can only be felt by the mind. Imagine taking the fluttering of butterflies, the racing of hearts, the heaviness of breathing, the focus of eyes, and the redness of cheeks and stuffing all of that into the head with the added benefit of improved perception and greater understanding. There’s no rush like it. Love is the closest comparison. So again, it really is sad that so few experience it. I will demonstrate a single, stunning beauty which if you do not appreciate, perhaps there really is no hope for you. You should admit that your mind is as cold as your heart. Move to an isolated desert island where your dispassion will never douse the spirit of another human being!
So if you’ve made it this far, let’s begin by painting the background. It involves computing square roots. √4 = 2, right? Yes, because 2² = 4. Okay, we’re off to a great start. Now what is √-4? It can’t be -2 because (-2)² = +4, not -4. It can’t be 2, again because 2² = +4, not -4. It can’t be zero, nor any positive or negative number. So it doesn’t exist, right? Well, that depends on what “exists” means. No, I’m not playing word games like President Clinton did when his response before the grand jury was that, “it depends upon what the meaning of the word ‘is’ is.” As a mental exercise, let’s suppose that such an exotic number does exist with the property that when you square it, you get -1. This number is nowhere on the number line because it can’t be negative, zero, or positive.
Early mathematicians ignored such a number because it didn’t satisfy their preconceived notions. Nevertheless, certain Renaissance mathematicians found this number to be very useful. They began to consider its validity when solving certain types of equations called cubics. The computation of these solutions required this special number, even though it would disappear in the final solutions. They couldn’t find any way to eliminate this mysterious quantity from their equations. It appeared to hold some important significance. Critics felt that these mathematicians were playing frivolous games and snubbed this new number as “imaginary.” The label stuck. Eventually everyone, advocates and detractors alike, called it i for imaginary. Hence, if such a number exists, then i ² = -1 and √-4 = 2i because (2i)² = -4.
The only problem with calling i imaginary, is that all numbers are imaginary. Even counting (or natural) numbers like 3 are imaginary. Hold up your pinky, ring, and middle fingers (not in the opposite order, please; we don’t want to start any fights). I’m holding a pen, a pencil, and a marker. The number 3 is the idea or notion that your fingers and my writing utensils, while vastly different from each other, have a sort of equivalence. You cannot show me the number 3 because it’s an idea, not something physical. The same goes for any number. A balance of -5 in my bank account corresponds to the notion that I owe someone 5 dollars. You can’t show me -5 dollars because it doesn’t exist.
Okay, If you’re still with me, I’m truly impressed by your endurance. Michael Phelps has nothing on you. Are you ready to go for the gold? Let’s give it a shot. Now that we’ve established that all numbers are imaginary, let’s proceed to see what’s so special about i. Let’s consider a simple sequence of equations. We’ll start with the counting (natural) numbers: 1, 2, 3, …, etc. Moses brought these down from the holy mount written upon tablets of stone by the finger of God himself, so they’re dependable, sound, and cozy. Right? If b is any counting number, then the solution to any equation like x – b = 0 is a counting number. Now tweak the equation ever so slightly like this: x + b = 0. To express solutions to this type of equation, we now need negative integers: -1, -2, -3, …, etc. A tiny modification of the equation yielded a massive expansion of the number system. Now glance over the table below. In each case, the coefficients a, b, or c represent numbers from the previous solution sets. Notice how minor adjustments to the basic equation yield massive expansions of the previously existing number system. (If you understand the quadratic formula then you can verify the solutions in #6 yourself.)
Basic Equation | Example | Solution(s) | Solution Set | |
1. | x – b = 0 | x – 2 = 0 | x = 2 | Naturals |
2. | x + b = 0 | x + 2 = 0 | x = –2 | Integers |
3. | ax ± b = 0 | 3x ± 2 = 0 | x = ±⅔ | Rationals |
4. | x² – b = 0 | x² – 2 = 0 | x = ±√2 | Irrationals |
5. | x² + b = 0 | x² + 4 = 0 | x = ±2i | Imaginary |
6. | x² + bx + c = 0 | x² + x + 1 = 0 | x = -½ ± √3i | Complex |
Continued… | ||||
Any polynomial with complex coefficients |
Now what? Dear God, save me. Right? |
The solution sets of #3 and #4 combined are called real numbers, not that they’re any more real than imaginary numbers, but you get my point. A complex number is a real number plus an imaginary number, like the example solutions in #6. (Technical note: For brevity and simplicity, I referred to solutions like those in #4 as the irrationals. In actuality, they’re a quadratic subset of those numbers, but that’s not a distinction that is important to the point that I ultimately want to make.)
Now this process could continue indefinitely. Imagine polynomial equations (like those in the table above) with ever increasing numbers of terms and exponents. Think in the thousands, the millions, the billions, or beyond. The solution sets would get extremely complicated, right? Well, this brings us to the stunning beauty of complex numbers; no matter how much more complicated the equations become, the solutions will never be more complicated than those in example #6. This means that the complex numbers are complete! Let that sink in. If your heart hasn’t grown as hard as your skull, the completeness of the complex numbers should at least feel warm and fuzzy. Mathematics enables us to reach out and touch eternity. It’s the understanding of the incomprehensible. In a world of uncertainty, it gives us something upon which we may rely. Where there are no answers, it whispers hope. As with love, or perhaps embedded within it, mathematics is one of God’s greatest gifts, the gift to think and to think powerfully.
Ariel Hammon
Author of JACK
One of my favorite posts so far!
Thanks!