 A useful tool in mathematics, physics, chemistry, biology, economics, business, administration, and pretty much everything else that uses numbers is called dimensional analysis. Dimensional analysis is based on the following principles. Units can be treated like algebraic variables and, well, actually everything you know about algebra. Now, if you're going to be looking at the units, it helps if you know some basic units, so some basic units of length. For the metric units, the meter typically abbreviated m and the kilometer, and these are used everywhere except in a few backwards countries. These backwards countries use units like the foot, the mile, and the inch. We can combine these basic units to produce what are called derived units, so a simple set of the derived units, if our unit of length is the meter, then the unit of area is the square meter, which will write m squared. That corresponds to the area of a square one meter on a side. The unit of volume is the cubic meter, m cubed, which is the volume of the cube one meter on a side. And again, if you're in one of those backwards countries where you have to measure length in feet, in these cases, area is measured in acre, don't wait, square feet, feet squared, and volume is measured in cubic feet, feet cubed. Let's see how we can use this. So suppose w equals three meters and h equals two meters. First of all, let's find w, h, and then let's determine what it represents. So the key idea here is that units act like algebraic variables, and in particular we can treat the units as algebraic variable factors. So for example, three meters can be read as three m, where m is our symbol for meter. Two meters is two m. And so when I compute w, h, that's really three m times two m, and if we multiply that out we get six m squared. Now we get the numerical part of our answer, six, but we also get this unit part of our answer, m squared. And since the units are m squared, or square meters, then the quantity we find is an area. Now some quantities in a formula might be pure numbers. Countries have no units. For example, we might have something that actually is a pure number like one-half in the formula one-half b times h. Or we might have a constant like the pi in the formula pi r squared. And there are other things which we'll talk about later. So for example, suppose r equals five meters and h equals two meters. That quantity is measured by six pi r squared h. So we can compute that six pi r squared h, while six and pi are pure numbers so they don't have any units associated with them. Meanwhile, r is five meters, five m, and h is two meters, two m. So let's substitute those values into our formula and we can multiply. Sometimes we have the numerical part of our answer, 300 pi, but we also have the units m cubed cubic meters. And because the thing that cubic meters measures is a volume, whatever we found, it will be a volume. So here's a very useful idea from dimensional analysis. If the units don't measure what you want, the answer is guaranteed to be wrong. And this could be very helpful if we're not entirely sure how we want to proceed on a problem. So suppose the sphere has a radius of two meters and we want to find the area. Well, we sort of remember that the area for a sphere has a four and there's a pi in there and r has to play a role and so maybe it's four pi r cubed. Well, let's see if four pi r cubed gives us the area. And we can do that by finding the units. So if we compute with the units and four pi r cubed becomes, but our units are cubic meters and this is not a unit of area. And so our answer is guaranteed to be wrong. And even more dramatically, can the area of a racetrack be found by using the formula, I don't know, wh plus 2w plus 2h? Where w and h are lengths? Now we're not given the values of w and h, but for measuring them in meters. Well, let w be a meters and h equals b meters, then our formula gives us. And remember, these units are acting just like algebraic variables. So this is like having an x squared term and a couple of x terms. You can't combine unlike terms, you can't combine unlike units. So if you want to figure out what the units of the answer are, well, actually it's not clear what we've actually found, but we can definitely guarantee that it's not area because the units are not meters squared.