 The universe is a large and complicated place, and humans have been trying to understand it ever since they've been around for millions of years. And for the last few millennia, we've been working particularly hard, building tools like mathematics and writing and education systems and philosophy and trying very much to understand the details of the world around us. But things changed a few centuries ago with the advent of science, and the difference between science and the other attempts to understand the universe is that in science we try and follow evidence. We don't follow authority what people think. We go out and look if we have an idea, we examine the evidence. And examining the evidence means we have to make a measurement. So central to all science is the idea of measurement. Most measurements, when you're trying to be precise, look at the world very carefully, come down to a number. So for example, we might ask, how tall are you, Joe? This is a picture of Joe. I'll write that there so that you can tell. And to answer that question, you do a measurement with a measuring tape or something similar, maybe a laser finder, whatever tool you have. And you might get the answer, six. Now if you're looking at answer six, there's an obvious, really, really important piece of information missing, the units. What are the units of this number? And anytime you make a measurement, it's going to have some kind of units. And those units are really important. So in this case, I was six times the height of a Scottish terrier. And that unit is called a rough. So six roughs, which is approximately equal to eight meows, which is the height of a Cheshire cat. Now I must admit that roughs and meows are fairly unusual units of length. In fact, I kind of just made them up. But at least they're units of length. I couldn't possibly have been six minutes tall. I couldn't possibly have been eight kilograms tall. If you get the number wrong, you can still be close. If I say I'm about a meter wide, that's not great if I'm trying to make a well-fitting suit. But it's fine if I'm trying to figure out how wide to make a doorway. But if I was to claim to be a vault-wide, that is so deeply wrong that I couldn't possibly use that to do any kind of design. And the reason for that is that a vault is not a unit of length. So the idea of length is more fundamental than the idea of say a meter. And this more fundamental idea is called the physical dimension. So a measurement is describing some physical quantity. And that physical quantity has a dimension. And then we also describe it in a particular unit. So for example, one inch is equal to a certain number of centimeters. So the quantity on the left is a length. And the quantity on the right is a length. And length is the dimension. The inch part is the unit. And the centimeter part is the unit. And the dimension is length. And this equation is all right because it has a length equal to a length. You can't have a length equal to a time. That just doesn't make any sense. All right. And so this idea that all relationships between physical quantities had to have the correct physical dimensions, although it does seem kind of obvious, on the scale of human endeavor, it's actually quite a new idea. It was only invented in 1822 by Joseph Fourier. And occasionally we still make mistakes along those lines. For example, in 2008, the Australian Capital Territory Government introduced the electricity feed-in bill. And the idea of this bill was to give people an incentive to put solar cells on the roofs of their houses by making sure that they got full price for the electricity that they generated. But in order to not distort the market commercially, they put a cap on how many solar panels you could put on your roof and said that households would only get the full price if they had up to 10 kilowatt hours of generating capacity on their roof. The problem? A kilowatt hour is not a unit of power. It's a unit of energy. So power is defined as a certain amount of energy per unit time. And for a bunch of solar cells on a roof, a kilowatt is a very reasonable choice for a unit of power. And a typical solar installation on a roof might have something between two to eight kilowatts of capacity. In other words, under full sunshine, you'd expect two to eight kilowatts of electricity to be generated. And so this legislation was probably meant to say that you get full price of electricity provided your total capacity was at or below 10 kilowatts. But it's not really the done thing to just kind of assume that when people write things in legislation, they probably meant something else and so we'll go with that. And so this is a real problem. Kilowatt hours does come up in electricity bills quite a lot. If we just rearrange this equation, we can see that if we multiply both sides by time, then the energy you get is just the power times the time. And just as kilowatts is a reasonable unit for power coming off a roof and hour is a reasonable unit of time, it turns out that the kilowatt hour is the unit of energy that electricity companies use to bill us. And so for example, you might spend 20 cents per kilowatt hour for your electricity.