 So here's a useful tool that shows up in many, many, many, many, many different applications, and it's known as dimensional analysis. And it's based on the following idea. Any time you measure a quantity, the amount of the quantity present is given in terms of a number of units. So for example, if you've talked about a mass of five kilograms, what you mean is that the mass that is present is the same as five masses, each of one kilogram, where our kilogram is the unit. Likewise, if I want to measure the duration of a movie, and I want to say it's 135 minutes, then it's the same duration as 135 one-minute intervals, where again, the one-minute is our unit. Likewise, I could talk about a volume of three gallons, and what I'm talking about is the same as having three one-gallon volumes. As a general rule, everything that you measure has to have a unit written down. If you don't write down the unit, the answer that you have written is meaningless. You can't talk about a length of seven, because it makes a difference whether you're talking about seven feet, seven yards, seven miles, or seven millimeters. This might seem to be adding an extra layer of complexity to any problem, but actually it makes it a lot easier, because units can be treated exactly like algebraic variables. And that's the basis for dimensional analysis. For example, if I multiply 2x times 3x, I know that I should get 6x squared. The 2 and the 3, the coefficients, multiply. The x and the x multiply, and there's my product. And if I treat the units just like algebraic variables, then, well, here's a simple calculation. Here's two feet. That's a length by three feet. That's another length. If I multiply these two together, I multiply the coefficients to get six. I multiply the units together to get feet squared, and my product six feet squared. And this actually helps us, because if I have unlike units, I can't combine them in any meaningful way. So I can't combine 2x plus 8y and get 10xy, and likewise I can't write two feet plus eight inches and get 10 anythings. Now, I can write the expression 2x plus 8y and just leave it that way. Likewise, I can write the expression 2 feet plus 8 inches and leave it that way, but I can't combine them without doing some extra work. So one of the things we might be able to do is, if I know the relationship between x and y, I can express the original expression in terms of one of the variables. So maybe I know that x is the same as 12y, and that means I can transform the expression 2x plus 8y into 2, here's something that is the same as x, 2 times 12y plus 8y, and that simplifies at the end of the day to 32y. Likewise, if we happen to know that one foot is 12 inches, because we have been brought up in a peculiar system of units that nobody else in the world uses, then I can transform the expression 2 feet plus 8 inches, that's 2 feet plus 8 inches, that's 24 plus 8, that's 32 inches. And the fundamental rule of dimensional analysis is the following. If the units of an equation don't match the units of the quantity sought, something is wrong, you're missing something important. So let's think about that. So to begin with, if you know the formula or definition of a quantity, you can figure out what the units are. So for example, in all but three nations in the entire world, those being Liberia, Myanmar, and there's some other places whose name escapes me right now, but in all but three nations of the world, we use the metric system. And so length is measured in meters or kilometers. Oh, that's right. In the United States, we use this peculiar adherent system that uses inches, feet, and miles. But now that we know that length is measured in meters or kilometers, inches, feet, miles, whatever, I do know how to find the area of a rectangle. It's the product of length and width, which means that the units of area could be meters times meters, and that, again, treating these like algebraic variables, that's meters squared, or we might read that as square meters. And that means that every area that we run into can be measured in units of square meters. Or if we're in one of these nations like Liberia or Myanmar, we might measure area in square feet or square inches or square miles. Also, since we know that volume is length times width times height, then my units of volume could be things like cubic inches or cubic feet or cubic centimeters if you're in most of the rest of the world. How about velocity? Well, the definition of velocity is its distance divided by time. So I can find its units. It's going to be a unit of distance, well, how about meters? Divided by a unit of time, how about seconds? So the units of velocity could be meters over seconds, which we generally write as m slash s, meters per second. Now, just as a simple example of how dimensional analysis works, let's say you're working through a problem, and all of a sudden, you can't remember how to find the area of a circle. And you remember it has something to do with pi, and there's a couple of formulas that are circulating in your head, maybe 2 pi r or pi r squared or 4 thirds pi r cubed. And you don't know which of these formulas is the correct one. So one of the things we can do is we can try to analyze the units to determine which one is not correct. So first important idea, numbers like 2 and pi and square root of 11 and all these other things that are pure numbers, they do not have any units. On the other hand, r is our radius, and that's a length. So whatever the units of r is, those will be units of length. So since I was brought up in this aberrant system of unit, the units of length of feet and inches come much more naturally to mind. So suppose r is one foot, then I can determine the units for each of the formulas. So let's consider that first formula, 2 pi r. Again, 2 and pi don't have any units associated with them. r has a value 1 foot, so the expression 2 pi r is going to give me something like 2 pi feet. And likewise, if I take that expression pi r squared, r is measured in feet. It's 1 foot, so when I square it, I get 1 squared. That's 1 foot squared, feet squared. And so this expression pi r squared becomes pi feet squared. And then that expression 4 thirds pi r cubed, again, r is 1 foot. So when I cube it, 1 foot cubed, 4 thirds and pi have no units. My units of this expression will be feet cubed. And now I'll take a look at each of these expressions. So feet is a unit of length. So whatever this first expression is, whatever this measures, is something that's measured in feet, and that's a length, not an area. So I know that first expression cannot possibly be the formula for the area of the circle. And I'll just get rid of it. Now this last expression, feet cubed, so I have cubic feet here. This is a measure of volume. So again, this last expression, whatever it tells me, is a volume. And again, that can't possibly be the area of a circle. So I'll get rid of that one as well. Now this expression here does have units feet squared. That is a unit of area, which means that this expression could potentially be the area of a circle. Now add one qualifier here. Remember that constants do not affect the unit. So if I threw a constant in front here, 2 pi r squared, the dimensional analysis would tell me that 2 pi r squared is something that is going to measure an area. What you have to watch for is that it might not be the area that you want. But at the very least, it will be some area. And given that our memory tends to be triggered more by associations than absolute recall, if we have at least an ability to remember that something 2 pi r, pi r squared, 4 thirds pi r cubed, if we happen to know that at least one of these happens to be the correct expression, the dimensional analysis will help us identify which ones can't be the correct expression.