 One of the most useful tools of algebra is something known as dimensional analysis. And this goes back to a basic idea. In general, a numerical value without units is completely useless. For example, the mass of the rock was 5. Well, do you mean grams, kilograms, or metric tons? We'll be there in 10. 10 seconds, 10 minutes, 10 years. The water's depth was 6 inches, feet, or miles. What's useful to recognize is that units act just like algebraic variables. For example, consider something like 12 feet. When you see something like 12x, you read this as 12 times x. So 12 feet can be read as 12 times feet. And likewise, 4 feet can be read as 4 times feet. So when you multiply them together, 12 feet times 4 feet, if you think about how the algebra would work, we can multiply the numbers, 12 and 4. We can multiply the variables, feet by feet, and get 48 feet squared. Now, sometimes in this process we'll use something called a pure number. And this is a quantity that doesn't have any units at all. And typically these things show up when we count something, as opposed to measure it. There are five people in the room. Or they might show up as part of a mathematical constant like pi. So 5 pi times 3 feet. Well, the 5 pi has no units. So this value is going to be 15 pi feet, which will have units of feet. And likewise, something like square root of 2 times 2 meters times 4 meters will multiply our numbers together. That's going to give us 8 square root 2 multiplied by the units together. Meters times meters gives us square meters. For example, let R and H be lengths measured in meters. Let's find the units for the expressions, 4 thirds pi R cubed, 4 pi R squared, 2 pi R squared plus 2 pi RH. So the first important idea is that pure numbers don't alter units. So in this expression, 4 thirds pi R cubed, the 4 thirds and the pi are pure numbers, and they don't have any units. So this expression has the same units as R cubed. And now we can find the units. If R is in meters, then R cubed is in meters cubed. And so is the expression for pi R cubed. By a similar argument, in the expression 4 pi R squared, we can ignore the 4 and the pi, and this expression will have the same units as R squared. And R squared is going to have units of meters squared, and so is 4 pi R squared. And finally, 2 pi R squared plus 2 pi RH. Again, 2 and pi are pure numbers, and they have no units. So this expression will have the same units as R squared plus RH. And R squared is going to be in meters squared, and RH is going to be in meters times meters, also in meters squared. And if you think about this, this expression is sort of like X squared plus X squared, and that will give you 2X squared, except 2 is a pure number and doesn't have a unit, so we can ignore it. R squared plus RH will also be in meters squared, and so will our original expression. It seems like this is sort of an added step that we're doing for no apparent reason, but it's actually very important, and we introduced the fundamental rule, and that's this. The only thing a formula can provide is measured in the units it produces. In other words, whatever units you get when you use a formula, those are the units you're stuck with. If nothing uses these units, then the formula is meaningless. Or, if what you actually want uses different units, the formula is irrelevant. For example, assume A, B, and C are measured in feet, and we want to find the area of a figure, but we can't quite remember which formula we have to use, and we know it's one of these three, but we don't know which. So let's try a little bit of dimensional analysis and see which one of these is actually relevant. So first of all, A, B, C will be measured in feet cubed. But feet cube, cubic feet, this is a unit of volume, and since we're looking for area, this expression is not going to be useful. How about this more complicated expression? Well, let's consider the units of this expression. So A, B, and C are all measured in feet, so if I add X, X, and X, I get 3X, and again I'm going to ignore the pure number 3, and that means this first factor is also going to be measured in feet. This second factor, some number of feet, minus some number of feet, is also going to be measured in feet, as are the other two factors. And so that means that we are going to get units of feet to the fourth power, except we're taking the square root of that, and that'll give us feet squared. And square feet is a unit of area, and so this formula might actually be relevant to the problem. And finally, let's take a look at our last expression. The units of A cubed plus B squared are going to be feet cubed plus feet squared, but because these are dissimilar units, I can't combine them in any way, and there's nothing that we measure in units of feet cubed plus feet squared. So this means that this particular expression is meaningless. Another useful application appears in conversions. Since units act like algebraic variables, we can use this approach to help us convert from one unit to another, and there are three things to remember. First, units act like algebraic variables. Next, any quantity multiplied by one remains unchanged. And finally, one is any quantity divided by its equal. For example, suppose we want to know how many square inches there are in three square feet. So for that, we need to know the relationship between these units. Now, for the people who live in 194 countries and for 95% of the world's population, you'll need to know that 12 inches are the same as one foot. Now, if you happen to live in that one holdout country and that small, small, small percentage of the world's population that refuses to learn the metric system, these are some of the things that you learn in school. Time spent that you could be learning other stuff. But I digress. Because 12 inches is the same as one foot, then one is going to be the same as one foot divided by 12 inches because they are the same thing. Or for that matter, one is also going to be 12 inches divided by one foot. And since both of these are equal to one, that means I can take this quantity three square feet and multiply it by either of them. Well, if you don't play, you can't win. So what happens if we multiply three square feet by this first one? The numerical value is going to be three-twelfths, but more important, our units are going to be feet squared times feet, that's feet cubed, over inches. And that doesn't look like we're heading toward square inches as a unit. If nothing else, because we have more feet. So let's try using the other factor, 12 inches over one foot. And if we do the multiplication here, which has fewer feet, so it looks like we're better off, but we still don't have square inches. Well, nothing in the rule says we only have to multiply once. Let's multiply by a second factor of 12 inches over one foot. And if we do that, the product of our numbers 36 and 12 give us 432, and the product of our units will give us square inches, which is what we want.