Math Formulae, and the Square Root of -1

by Paul J. Nahin

blog review

Shortly after reading a book on Fermat's last theorem, I convinced myself that I was the appropriate audience for pop mathematics books. You know the kind of book I mean: they're not text books, but pithy and friendly tales mixing history with whimsy with a brief peak in at the technical stuff. Just enough to make you feel smart without overwhelming you. I generally appreciate these kinds of books, as long as they announce themselves properly, and this was supposed to be one of those books. An Imaginary Tale: The Story of the Square Root of Minus One is sort of marketed as a gentle introduction to how and why imaginary numbers came to have been, well, imagined.

And for the most part, it is. It's a fascinating overview of a bunch of early mathematicians from ages and ages ago. The only problem is that the first page of the first chapter contains, for instance, this:


x3 + a1x2 + a2x + a3 = 0

That's fine, unless you have no idea what a depressed cubic and a coefficient are, or how they can express "unity without loss of generality" [p. 8]. And that's probably the most basic equation in the book. It only gets more complex from there.

I don't really know enough about publishing to know why or how a book gets labeled as a friendly introduction to a topic, or comes to be seen as "pop science", but I assume that author Paul Nahin assumed his audience would be familiar with maths. If I were writing a book about maths, I would probably make the same assumption. The only "problem" here is that there's no pre-book briefing for non-maths people, to prepare them for what's to come. So, here's a quick introduction to mathematical formulae.

I should mention here that I'm not a maths person myself. I failed most or all of my maths courses back in middle school, to the point that I don't even think I took a math course in high school before I dropped out entirely. However, I took some courses here and there in an attempt to get better at math, and while I'm still not good at it, there was one explanation at the very start of a course about matrix transforms that utterly changed the way I viewed mathematical formulae, and so I'm going to share that with you here. Once you understand this, you'll be able to pick up those "pop science" books and maybe not understand a formula you find staring back at you, but at least you won't be afraid of it any more.

Functions

To understand the concept of an advanced math formula, you first need to understand a little bit of computer science. That's backwards, of course, because math came before computers, but most of us grew up with computers and have had them explained, in one way or another, a heck of a lot better than any of my past teachers, at least, ever explained mathematics.

Say you want to add two nonthreatening numbers, like 1 and 1. You can express that in this way:


1 + 1 = 2

Simple. Now say you want to record for posterity how it is that you add numbers. It's a pretty basic concept, but you're a completionist and want it written down somewhere. Besides, if you write it down, then others can use it when they want to add two numbers. So you create something called a function:


function addition(x,y) {
x + y
}

Even if you've never written a line of code in your life, you can probably see what's going on here. You've created a sort of script or recipe that demonstrates how to add two numbers. Not just 1 and 1 but any two numbers. To represent the concept that this function works with any two numbers, the numbers in question are represented by x and y.

The function tells the reader that if they take some number (x) and another number (y), then they can perform a function called addition by combining the two together.

To make the function a little more precise, you can provide a third number to represent the sum that the function produces. This is different than the x and y values, though. Both the x and y are fed into the function, and this third number is what's coming out the other side. So it's clear to people that all they need to bring is the x and the y, different notation is used to represent output:


function addition(x,y) {
z = x + y
return z
}

And now you have a pretty good function, written in pseudo code, to represent the process of combining any two numbers in order to produce a third. You can test it:


function addition(1,1) {
z = 1 + 1
return z
}

z=2

It works! Does it work with other numbers? Well, you can probably guess that it does, but feel free to test it.

Formulae are functions

As it turns out, math formulae are just another notation method for functions. Look at how your pseudo code function can be rewritten to mimic those complex formulae. To make it look really mathy, I'll rename the function from addition to some cool Greek letter, like Σ, and I'll say that


z = Σ (x,y)

Well, that looks intimidating, but as long as you know what function a given symbol represents, and you know which other symbols expect to be replaced with numbers, you can use a formula. You know, because you've read how I invented this wacky formula, that the funky Σ letter represents our addition formula, and that you're expected to replace x and y with numbers. So you can use this formula to solve the old question of what 1 and 1 produce when added:

 
2 = Σ (1,1)

Or something new, like the sum of 10 and 13:

 
23 = Σ (10,13)

It works because you and I know mutually declared that Σ means add the numbers. (This is probably an appropriate time to warn you that Σ actually does already have a meaning in mathematical notation that is similar, but different, to how I'm using it here.)

Even if I come up with a new formula on my own, you can read and use it as long as I tell you the key to what each element represents. For instance, this is meaningless to you now:


z = Δ(x,y)

But if I tell you that Δ means subtract the larger number of either x or y from the smaller, then you can solve for z, given any two numbers:


3 = Δ(10,13)

2 = Δ(4,2)

0 = Δ(1,1)

Numbers are just the vehicle for ideas

So why are math formulae written in fancy notation, and why do so many of them leave out the one thing that mathematics is mostly all about (numbers)?

It's complicated.

First of all, math notation has a lot of history to it and existed long before advanced computational devices existed. Certainly modern computers don't use that notation, so there are obviously alternate ways of representing any mathematical process, but there's a real efficiency in how math formulae are written, so it sticks around.

Secondly, there's often an assumption that certain common symbols are know entities. For instance, you're not likely to see a math formula with a footnote explaining what Π stands for (or Σ or Δ for that matter), because they're really common symbols every mathematician is expected to understand. And sadly, if you're not a mathematician, you have to go look up symbols and notation before you can understand the functions they represent.

Finally, mathematical formulae are meant to scale. Basically nobody comes up with a math formula that is intended to do one thing, once, and then never be used again. The idea of a math formula is that you can run any variety of numbers through it and end up with a valid, mathematically correct, answer.

In other words, math Formulae aren't really about numbers, but the way numbers interact with one another.

Now you can read the book

So now you know how to look at math Formulae without being intimidated, and you know that you can look up fancy symbols to unlock secret math functions. This doesn't make you or me a mathematician, but at least you can go read pop science books about wizards of the Numbers school of magick.

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