Friday, September 6, 2019

Imaginary Numbers and Beyond

Everyone knows what a number is. It is an amount of something. We have the natural numbers, 1, 2, 3, … We add zero and negative numbers to get the integers. Between the integers, we have rational numbers, which can be written as fractions or ratios. And we have the real numbers, which include numbers like π and  2  that cannot be written as fractions. It would seem like that’s all the numbers, because if we try to fit anything else in, it counts as a real number.

But let’s do something weird. If we multiply a number by itself, we get its square. 42 = 4*4 = 16. We can do the opposite, what is called the square root.  16  = 4. But when we play around with this, we notice something. A negative times a negative equals a positive, so -4*-4 = 16. Therefore  16  = 4 and  16  = -4. It might seem weird that there are two answers, but that’s all right. If we find a square root in a calculation in physics, it just means there are two right answers. For instance when you calculate the moment in time when a cannonball will be a certain height after it is fired, you find it’s at that height twice, once as it goes up and once as it comes down.

But consider the following operation:  -1  . What is the answer? It’s not 1, because 1*1 = 1. It’s not -1, because -1*-1 = 1. So does it just not have an answer? No, it does. Everything in math has an answer. If the answer can’t be found in what we already know, we have discovered something new. We define  -1  as i, and see what happens from there. In official math, the set of all real numbers *i is called the imaginary numbers, and the set of all combinations of imaginary numbers plus real numbers (for example, 8 + 4i) is called the complex numbers.

Let’s play around with i. i*i = -1, so i*-i = 1 logically, 4i*i = -4, and the same holds true when you substitute 4 with any other number. But again, what happens when we get to  i  ? It’s not i or -i. It’s not anything that doesn’t include i. Do we have to postulate a new type of number? Something like j =  i  ? Miraculously, we don’t.  i  can be written as a number that doesn’t involve j or any other letter besides i. That number is:

 i  = 1/ 2  *(1 + i)

Hold on. That looks weird. Let’s do the calculation to make sure it actually is the answer. We start by squaring it:

(1/ 2  *(1 + i))2

By association, this is equal to

1/ 2  2*(1 + i)2

The left part is easy.

1/2*(1 + i)2

Next, we need to know the rules for how to square groups of numbers that are added together. (1 + i)2 does not equal 12 + i2. It equals, not forgetting the 1/2

1/2*(12 + 1*i + i*1 + i2)

We simplify this to

1/2*(1 + 2i + i2)

i2 = -1, so we have

1/2*(1 + 2i - 1)

The 1 and the -1 cancel each other out, so


1/2*2 = 1. So when we simplify it completely, we are left with


And there it is! We have just proven  i  = 1/ 2  *(1 + i). No new dimensions of numbers are required.

Are imaginary numbers just a math thing, or do they have applications to the real world? One significance of imaginary numbers is that they represent things that don't exist. For instance, you can calculate the moment in time when a cannonball in flight will be higher than its highest point, and you get an imaginary number. On the other hand, sometimes complex numbers are shortcuts we can take to make math easier. For instance, in the famous Schrodinger equation in quantum physics, momentum is represented by an imaginary number. We could represent it by another dimension of real numbers and put in more sines and cosines, but imaginary numbers make it a whole lot easier.

But we can do other things besides take the square root. Complex numbers obey a rule called commutativity, which means 4*5 = 5*4, and the same is true with any other pair of numbers. But what if it weren’t? In math, it’s perfectly okay to ask questions like that. For this one in particular, we get a new, 4-dimensional set of numbers called the quaternions. Their units, and their basic operations are

1, i, j, k

i2 = -1,     j2 = -1,     k2 = -1

i*j = k,     j*k = i,     k*i = j

j*i = -k,     k*j = -i,     i*k = -j

i*j*k = -1,     k*j*i = 1

Notice the differences between the 3rd and 4th rows. If we switch the order of multiplication, we get a minus sign. This is weird, and you may wonder if it even makes sense, or if the mathematicians who dreamed it up were smoking something. To assuage your fears, there is a more intuitive way to understand it, and that is to use matrices. Although the quaternion i can be thought of as the same as the complex i, it can also be written as a matrix, as can j and k, and even 1.

To multiply matrices, you take the first row in the first matrix and the first column in the second matrix, multiply each pair of numbers in the order they appear in the row and column, add the results, and put the answer into a new matrix in the place where the row and column cross. If that was as confusing to read as it was for me to write, here is a single step as an example. Suppose we want to calculate i*j in matrix form. Specifically, we want to know what the top right element will be. To do that, we choose the first row of i and the last column of j

The top right element of the result will be

0*0 + -1*1 + 0*0 + 0*0



Do that with every other combination of i rows and j columns, and you’ll find that you end up with k, just like we expected. And the same is true with every other product combination. I won’t prove it here, because that would be a lot of work and no one would read it, but you can work it out for yourself if you like, or you can take my word for it. In any case, hopefully you are convinced that the quaternions make sense, and aren’t just random gibberish spouted by people who want to be seen as smart.

Quaternions aren’t just a quirk of math. They are useful in modeling 3D rotations, and they are used in all kinds of simulations, movie special effects, and video games. This goes to show that math is everywhere, and even the weird, out-there mathematics can have a practical use.

Are quaternions the end, or are there larger sets of numbers still? There are. By a process called the Cayley-Dickson construction, which I know nothing about, you can get an infinite amount of them, increasing in size by powers of 2. Beyond the quaternions there are the 8-dimensional octonions, and then the 16-dimensional sedenions, and on and on. What are these number groups useful for? Heck if I know. But it sure is fun to know they exist!

Hopefully, this discussion has shown you a glimpse into the wild depths of mathematics. If not, then at least you will be able to wow your friends with imaginary numbers, and brag that you know what the square root of i is.

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