1: Units and Dimensionality
- Page ID
- 56753
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If somebody asked me how tall I am, I might respond “1.78”. But what do I mean by that? 1.78 feet? 1.78 miles? In fact, my height is 1.78 meters. Most physical measurements have dimensionality to them. That is, they are meaningless unless you attach a unit to them. Dimensionality means the type of unit. For example, inches, meters, miles, and light-years are all length units; something measured in those units have dimensionality of length. Kilograms, grams, and solar masses are all units of dimensionality mass. Measurements of different dimensionalities cannot be meaningfully compared. How many kilograms are there in a meter? The question does not even make sense.
There are some dimensionless quantities. For example, ratios are nearly always dimensionless. How many times older than my nephew am I? I am seven times older; that seven doesn’t have any units on it, as it’s a ratio of two ages (42 years and 6 years, respectively). For any other number you report, it’s essential that you report the units of the number along with the number itself. Otherwise, you haven’t completely specified what you’re talking about.
- 1.1: SI Units
- There is a “standard international” system of units. You may ask, why does anybody ever use anything other than these? SI Units are a good set of units for everyday measurements. However, they are very clumsy when dealing with the very small or the very large. When talking about atoms, or about stars, it’s often convenient to use other units that are better matched to the scale of the system. What’s more, some places historically use other units; for instance, the United States still uses the Br
- 1.2: Arithmetic with Dimensional Quantities
- When you put together numbers that have dimensions on them, you have to keep track of the units as you are doing your arithmetic. You can do algebra with numbers that have dimensions on them. However, it is not a good idea in general to do algebra with numbers. Solve things symbolically first, and only put in the numbers at the end. When you do this, you will have various quantities with different units.
- 1.3: The Unit Factor Method
- Sometimes you will need to convert one unit to another unit. The trick for doing this: multiply by one as many times as necessary. You can always multiply a number by 1 without changing that number.
- 1.4: Significant Figures
- Suppose I tell you that one stick is 1.0 meters long, and that it is 4.7 times longer than another stick. How long is the second stick? Writing the words as equations (see previous section), you might write:
- 1.5: Dimensional Analysis
- You can sometimes figure out something about a physical quantity just by considering its dimensionality. If you know what sorts of things might affect that quantity, and you have good reason to believe that it is just powers of those things multiplied together to give you that quantity, you may be able to figure out (up to a dimensionless constant) the equation that relates that quantity to the things that might affect it just by figuring out what makes the units work.