It is difficult enough to explain the beauty of mathematics to those who are not aware of its intrinsic qualities and think math is just for balancing your checkbook, assuming anyone does that anymore.
I wrote paragraph after paragraph comparing math to music and the structure and symmetry and rhythm. That didn’t work out. Torn up and thrown in the “Mac Trash.”
My second attempt started with a history lesson as I reviewed the three greatest mathematicians of all time: Aristotle, Newton, and Gauss — the Prince of Mathematics. That took four pages and I wasn’t even to the point yet. Besides, the most beautiful formula in all of mathematical thought came from Leonhard Euler!
(Let’s get something straight right now. You pronounce “Euler” like the “Huston Oilers.” No “eeeuuulll” sound in there at all.)
No good. Second draft. Crumble, drag, empty trash.
The problem is that this most beautiful formula in all of mathematical thought involves some advanced mathematics. It contains “e,” the base of natural logarithms, “i,” the square root of minus one, “pi,” the ratio of a circle's circumference to its diameter, and exponentials — a mathematical operation most have not used since High School.
The crux of my problem is I can’t teach you readers all the math needed to really appreciate the equation in just a few pages. It could take years of study. Even simply understanding the notation and concepts like complex numbers and irrational numbers is pretty heady stuff.
Then as we delve deeper into the beauty of the most beautiful formula in all of mathematical thought, we encounter trigonometric identities, logarithms, and deep philosophical questions such as how can unlimited growth represented by exponentiation become repetitive like the sine and cosine functions. It is deep water indeed, and most readers would require a life jacket, life raft, lifeboat, and possibly even life insurance to get to the point.
It has been one of my life’s goals when studying subjects to thoroughly understand them. With math, it was never enough to just be able to do the homework and pass the tests. I wanted to know why. To this day I council Mark in his physics classes that he should learn to derive the formulas, not just memorize them. Memory fades, but understanding remains.
When I was teaching at Electronics Technical Institute, my boss was Joe Clark, the Technical Director of the school. He and I were and remain good friends. He was going to school while working, as was I. He was pursuing a degree in distributed studies. Basically he had three minors and that was equal to one major. He got his degree in Physics, Math, and Philosophy.
He related that, while studying Div, Grad, and Curl, important topics in differential equations and needed in physics to understand the equations of electromagnetic force developed by Maxwell, he was struggling with understanding.
He said he went to the professor and asked for help. The professor said, “You are doing fine on the homework and quizzes. What help do you need?” Joe responded that he was also a philosophy student, so he needed to know more than just how to manipulate the symbols. He needed to know why.
I so agree with that.
There’s always a better way to explain an idea. Insights are fluid, mutable, and work for different people. I’m sharing the insights that helped me, hoping they’ll help you too. Here’s my take on learning. Ideas start hard and finish simple.
Complicated ideas get easier. Multiplication, reading, and even tying your shoes seemed tough at first and are trivial now. Likewise, I believe (I know!) that math, science, business, technology or any topic can be understood at an intuitive level — after overcoming the initial complexity.
The best teacher is you — after you’ve learned the subject. You are your own perfect tutor. Think about it — you overcame the problems and can explain the solution in language that makes sense. Unfortunately, we can’t go back in time. But I can share the “Ah-ha!” moments that I’ve found.
And maybe you think like me, so the explanation works for you too. Or maybe you think a little differently, and a helpful comment gives an insight that works better. Our “ah-ha!” moments are different, and that’s OK. There’s no single way to explain an idea, yet we all have a single textbook.
Get a map, not directions. Memorization often masquerades as learning. Just follow the recipe, memorize the formula, get from A to B without asking why.
Sure, directions are easy — too easy. But what about wrong turns? New destinations? Detours and roadblocks? Helping a friend who’s starting at point C, not A? You need a map. Yes, it’s more work, but once understood a map gets you from any point to any other point. You don’t memorize, you derive.
These were ideas I had long before I slammed my head against the vector calculus books; insights eventually tumbled out. “Why wasn’t it explained like that in the first place?” I thought, furiously scribbling my thoughts before they escaped. But I’m never done.
“It is impossible for a man to learn what he thinks he already knows.” — Epictetus
So I have made this my mantra for learning. Learn the rules. Then learn the “whys.”
Let me explain my philosophy. Math is no more about equations than poetry is about spelling. Equations and spelling exist to convey an idea. Understand that idea.
A few other quotes that capture my attitude on learning:
On what we truly know:
- “I know nothing except the fact of my ignorance.” — Socrates
- “If I have seen a little further it is by standing on the shoulders of Giants.” — Isaac Newton
- “If you can’t explain it simply, you don’t understand it well enough” — Albert Einstein
- “Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in language comprehensible to everyone.” — Albert Einstein
- “The only real valuable thing is intuition.” — Albert Einstein
- “Everything should be made as simple as possible, but not simpler.” — Albert Einstein
- “You can know the name of a bird in all the languages of the world, but when you’re finished, you’ll know absolutely nothing whatever about the bird… So let’s look at the bird and see what it’s doing — that’s what counts. I learned very early the difference between knowing the name of something and knowing something.” — Richard Feynman
- “Imagination is more important than knowledge.” — Albert Einstein
- “Do not seek to follow in the footsteps of the wise. Seek what they sought.” — Matsuo Basho
- “If I’d listened to customers, I’d have given them a faster horse.” — Henry Ford
- “Debugging is twice as hard as writing the code in the first place. Therefore, if you write the code as cleverly as possible, you are, by definition, not smart enough to debug it.” — Brian Kernighan
Alas, I can’t do it in such a short space. My passion is now exhausted. All that is left for me to do is to simply show you the most beautiful formula in all of mathematical thought.