So the really reason for my last post was just to give a little background to set the stage for this post.
Almost as soon as the complex plane came to be accepted, mathematicians tried to extend it to the third dimension. Like I said in my last post, I tried to figure it out myself and gave up kind of quickly. Well, it turned out there actually isn’t an extension in 3d that is analogous to the change from the number line to the 2d complex plane. (Of course it would be possible to make your own coordinate system like x,y, and then zi, though it would behave in a way that doesn’t give it the properties we are looking for. Also, there is a fundamental difference in these new “numbers” and the Cartesian coordinate systems that is explained below. There is a great Wikipedia article on the Cayley-Dickinson Construction that shows the process of finding these new kinds of numbers, or what are really called algebras). hahaha I haven’t been able to find a proof of not being able to have a 3d algebra, but the Cayley-Dickinson construction does show the process of getting to the quaternions, which are the 4d algebra.
I want to stress that it is important to be careful when referring to the dimensionality of the algebra because quaternions are not just a 4 dimensional set of Cartesian coordinates. The dimensions in the quaternion system are not exactly the same as the ones in the Cartesian system. The Cartesian coordinates can be thought of as representing space and defining a position whereas the bases in the quaternion system are degrees of freedom. Another reason to take care is mathematical operations with quaternions do not function like you would expect in R4. That is because they are representing fundamentally different things. To make things simpler, let’s drop down two dimensions to two. Even the numbers you plot in a standard x,y plane are not actually two dimensional numbers. I mean, would you consider the point (3,4) a number on its own? That point represents a position, not an actual number. That point does not imply any function or operation or value like 3x + 4y. At most, you can infer a distance from the origin at (0,0) from the values of x and y *If you have learned about unit vectors, it may help keep those in mind and think about what they really represent* On the other hand, the point (3, 4) on the complex plane represents the complex number (one number, composed of a complex and a real part) that is 3+4i. Even vectors and their magnitudes, those are represented by a single number. A fun thing to do is to imagine what functions of complex numbers look like. Start with easy ones like f(z) = z + 4. What about f(z) = 2z? Can you see some of the potential complications you run into when trying to visualize these? However, I am sure nearly everyone has seen a picture of a complex function, they just didn’t realize it. What I am talking about is the famous Mandelbrot set (but that is for a whole different blog post)
So the quaternions are constructed with the real number line, a perpendicular i number line, a mutually perpendicular j number line, and another mutually perpendicular k number line. Rather than having just one imaginary axis, we now have 3! Luckily the math is fairly simple to work out. i, j, and k are all defined to be square roots of (-1). One oddity that makes the quaternions so fun (and sometimes a pain) to work is the fact that multiplication is no longer commutative, meaning order matters. For example, 6 * 5 = 5 * 6, but when you are dealing with quaternions, things are different. i*j = k, but j*I = (-k). All of the rules of quaternion multiplication are encapsulated in the formula: i*j*k = (-1). Wikipedia has a great chart and it was there I learned how to use that formula to figure out all the multiplication rules. For example(this part is taken straight from Wikipedia btw), starting with i*j*k = (-1), we can multiply both sides by k to get: i*j*k*k = (-1) * k. Here we must be thankful that 1 retains its role as the multiplicative identity, meaning that multiplying by it, regardless of the order, will leave you with what you started with. If it wasn’t that way, it would mean that multiplying 1 by a number would get you a different result than multiplying something by 1. Anyways, you end up with: i*j*(k^2) = -k = i*j*(-1). From there, you can simplify to ij =k. Knowing that identity, you can repeat the process to figure out the rest of the rules of multiplication. To figure out the little there is a little more work that needs to be done but I don’t want to spoil the fun that comes with figuring this out (I will post it in white so you can highlight it):
Wikipedia has a good chart with all the multiplication rules that I will copy here.

multiplication table for quaternions
For now I think that is a good introduction. I encourage anyone who wants to follow the next article to try to practice a little, maybe try to see how multiplying and adding two vectors of the form (a + bi + cj + dk) where b, c, and d are real constants. As I write this post, I realize I am going to have to do a bit more explaining before getting into their applicability for 3d purposes. Maybe in the future I will write articles where a lot of precision is needed in my wording a little earlier in the night




















