For nearly 20 years, Golem has been the machine on my desk. It’s been my mail server, web server, file server, … ; it’s run Mathematica and TeX and compiled software for me. Of course, it hasn’t been the same physical machine all these years. Like Doctor Who, it’s gone through several reincarnations.

Alas, word came down from the Provost that all “servers” must move (physically or virtually) to the University Data Center. And, bewilderingly, the machine on my desk counted as a “server.”

Obviously, a 27” iMac wasn’t going to make such a move. And, equally obvious, it would have been rather difficult to replace/migrate all of the stuff I have running on the current Golem. So we had to go out shopping for Golem V. The iMac stayed on my desk; the machine that moved to the Data Center is a new Mac Mini

- 2.3 GHz quad-core Intel Core i7 (8 logical cores, via hyperthreading)
- 16 GB RAM
- 480 GB SSD (main drive)
- 1 TB HD (Time Machine backup)
- 1 TB external HD (CCC clone of the main drive)
- Dual 1 Gigabit Ethernet Adapters, bonded via LACP

In addition to the dual network interface, it (along with, I gather, a rack full of other Mac Minis) is plugged into an ATS, to take advantage of the dual redundant power supply at the Data Center.

Not as convenient, for me, as having it on my desk, but I’m sure the new Golem will enjoy the austere hum of the Data Center much better than the messy cacophony of my office.

I did get a tour of the Data Center out of the deal. Two things stood out for me.

- Most UPSs involve large banks of lead-acid batteries. The UPSs at the University Data Center use flywheels. They comprise a long row of refrigerator-sized cabinets which give off a persistent hum due to the humongous flywheels rotating in
*vacuum*within. - The server cabinets are painted the standard generic white. But, for the networking cabinets, the University went to some expense to get them custom-painted … burnt orange.

Before we get on to talking about imaginary numbers and complex numbers, let's try and break down our preconceptions about numbers in general.

We look at the world around us and see many things which we categorise. We see a computer, a piece of paper, we see other people, we see our hands. These are labels that we use to categorise the world, but these objects seem very physical and very real. We rarely question their existence, though if one wants to take the Cartesian view, we should also question the reality we are in. We are not going to go that far, but let's try and ask about the existence of numbers.

I have definitely seen five pieces of paper, but I have never seen a five. I've seen the number written down, but I can write down anything I want and it doesn't necessarily mean that it exists. I can write down a erga[oeiave21 but that doesn't suddenly bring a erga[oeiave21 into existence. How about a -5? I've definitely never seen a -5 though I understand perfectly well what it means. The integers seem to be very good ways of describing, or more specifically counting objects and the negative numbers are a good way of keeping track the transfer of objects from one place to another. I can also ask you to give me 3 coffees, and here I am really asking you to apply 3 as an operation to the object coffee. 3 is acting almost more like a verb than it is a noun. When I describe that there are 30 people in a class, I am really thinking about this as a description, or an adjective. So in the real world, somehow numbers feel like verbs and adjectives. I certainly wouldn't say that 'heavy' exists, but certainly a book which has been described as heavy does.

However, there is a world in which numbers really do seem to be more like nouns than they do in the world around us, and that is in the abstract world of mathematics. In the universe of equations, numbers somehow feel much more concrete and I can manipulate them and transmogrify them from one form to another using a set of mathematical operations which become more and more finely tuned and specialised as we learn more and more mathematics. I can take a 5 and I can apply a the

Incidentally, I have here separated the real universe from the more abstract, platonic, mathematical one, but it is fair to say that we have found mathematics as the best language with which to accurately describe the real universe. All of our models and precise descriptions of the universe are built using mathematics, and it acts as an incredible way of describing the laws of nature. Which came first, the mathematics or the universe? That is not a question I am going to get onto here, but it's certainly a profound one!

OK, so we have a mathematical world of numbers and we can manipulate them. Thus, we should be perfectly happy to have some more ingredients in that world, that don't have such an obvious mapping to the things in the world around us. We will discover that actually they help us enormously in the things that we can do with the mathematical machinery. It's like having a powerful car but not the right fuel to really take it up to top speed. We are about to find out what that fuel is and push the limits of what our car can do!

Previously, if we set up a certain type of quadratic equation and plugged it into our machinery to find a solution, the machinery would jam and we wouldn't get an answer out. This was a real shame because it didn't seem to be that much more of a complicated equation than any other that we had studied. We are perfectly happy with solving an equation like:

x^2-1=0

You can plot the graph of y=x^2-1 and see that it equals 0 at two points x=1 and x=-1. That's fine, our mathematical machinery can deal with that fine, but when we ask to solve something so similar:

x^2+1=0

our traditional machinery comes juddering to a halt and we get an error message on the screen - you're not allowed to take the square root of a negative number, says our program. In fact when we plot the graph of y=x^2+1 it's clear that it doesn't cross the x axis, so it can't have a solution...can it? Maybe we're not looking hard enough. Maybe our machinery is fine, but we've just fed it the wrong fuel. In fact, we can find the solution just fine. The solutions are:

x=√-1 and -√-1

You might look at this and go "Absolutely not!" You can't take the square root of a negative number, but if you plug that into the equation, it works just fine and is a perfectly good solution. What is not true is that √-1 is like the normal numbers that we are used to using. In fact, let's give this solution a name. We'll call it

√-1=

(Note that we are actually being mathematically sloppy here, but for a first pass, this will do - we can explain the subtleties later - in particular the domain of the square root is only the positive real numbers and thus we have to say what we mean by this function separately to deal with non integer powers on negative numbers.)

What is

Once you have defined this new type of number - a number that squares to a negative number, we open up so many new possibilities. Things that previously would have driven our machinery to a halt are now very easily accessible. With this new number we've just upgraded our mathematical machinery so that it can handle so many more problems than it could before.

It might seem that

What we've shown here is that we can deal with √-1 but we can quite happily extend this to the square root of any negative number. From now on √-b where b is a positive number can simply be written as

How about adding a real number onto this? Well it turns out that adding together 3+4