Most science textbooks start off with a relatively tedious chapter about measurements and math and units and dimensions and… blah blah… and most students read that stuff and their eyes get crossed because they have no context for it. My approach is to integrate units and math in context, so you will see these topics presented along with the physics that uses them.

I do think it’s important to have some definitions for reference. Please return to this page as needed.

Dimensions and Units:

• A dimension is something that we might measure. Length is a dimension. Time is a dimension. Mass is a dimension.

• A unit is a way to measure dimensions. We usually measure length using meters and time using seconds and mass using kilograms; however there are other options. We might measure length in inches or yards or miles or light years. We might measure time in minutes or hours or years.

Here’s the important rule: You can convert one unit to another unit. You can’t convert one dimension to another dimension. So, for example, 1 meter = 39.2 inches. Both are measures of length, so there is only one dimension involved. You can’t ask the question “how many kilograms equal one second” because the kilogram is a unit of mass and the second is a unit of time. They are different dimensions.

Fundamental and Derived Units:

Some dimensions — and their associated units — are fundamental. They cannot be made of other dimensions. The three that I have been discussing — length, time, and mass — are fundamental, and there are a few others that we will encounter later in the course.

Many more dimensions — and the units that go with them — are derived. Think about the speed of your car. It is measured in miles per hour, so the dimensions are length divided by time. The units might be miles per hour or meters per second or something else entirely, but the dimensions remain the same.

Units in Equations:

There are two big rules:

• The dimensions must be the same on both sides of the equation.

• The units you put into an equation will be the units you get out of the equation.

This will make more sense with an example or two. Let’s look at one that works that most people know:

v is velocity, which has dimensions of length over time. Delta x is displacement which has dimensions of length, and delta t is a time interval. Notice that if we put only the dimensions into the equation, we get:

You can see that the same dimensions are on both sides. Now consider some units:

• We might measure length in miles and time in hours, and we would get a velocity in miles per hour.

• We might measure length in centimeters and time in seconds, and we would get a velocity in centimeters per second.

• We might measure length in millimeters and time in years. This sounds silly, but millimeters per year is a good way to measure the movement of continents… or the growth of fingernails.

Now let’s look at an example that doesn’t work:

Try putting the dimensions above into this equation. You’ll get