Dimensions and Units

Physics is all about mathematically describing the world around us. In order to do this effectively, we need to carefully label the different quantities we measure. Units help us to do this.

What might we want to measure? Time is probably the most common example in your daily life. We often tell time in hours or minutes; these are two different units of time. Maybe instead we’re thinking about how long something physically is – say, how long of a walk the Cheyenne pup took. We could express the length of her walk in feet, but we could also express it in meters, inches, miles, etc. We get to choose which units to use, and some may make more sense than others depending on the scale of what we’re thinking about. (It makes more sense to talk about the length of Cheyenne’s walk in miles than in inches, for example.)

Units can build on each other, and a great example is the units for speed. Perhaps the speed limit is 65 mph – miles per hour – in which case we have units of length (miles) divided by (per) units of time (hour).

Now in physics we traditionally use the SI unit system (International System of Units), in which the standard (essentially “default”) unit for length is the meter. We have standard units for other quantities as well. For time, the standard unit is the second, and for mass, it’s the kilogram. As we already showed in describing speed, we can combine these units to form new units of interest, and it turns out that in the SI system, there are 7 “base” units from which all other units can be derived. (To learn more about base units, check out this Wikipedia article.)

Ok, so we’ve got a way to measure different quantities, but we also need a way to efficiently handle different size scales. This is where prefixes come in handy. There are many prefixes out there as you get bigger and bigger or smaller and smaller, but we’ll just focus on some of the main ones. In the table below is the name of the prefix, the symbol designating it, what it literally means, and an example of its use that you may be familiar with.

Prefix	Symbol	Literal Meaning	Example
nano-	n		nanometer
micro-			micrometer
milli-	m		millimeter
centi-	c		centimeter
kilo-	k		kilogram
mega-	M		megabyte
giga-	G		gigabyte
tera-	T		terabyte

When I say “literal meaning” I mean just that: 1 millimeter (mm) literally means meters. 1 megabyte (Mb) literally means bytes. This allows me to directly write any conversion factor I need. Let’s say I want to convert 13 mm into km (kilometers). I can do this by way of converting to meters first:

Notice that for the conversion factors, I use one (1) of the prefixed-unit in the denominator(numerator) and its literal meaning in the numerator(denominator). 1 mm is literally meters (the first conversion factor), and 1 km is literally meters (the second conversion factor). I can thus write down any conversion factor if I always take one (1) of the prefixed-version of the unit and then write what it literally means in the other part of the fraction.

When solving physics problems, it’s always important to pause and ask yourself if your answer makes sense. We know that 13 mm is a very small distance, whereas 1 km is many times greater. It thus makes sense that 13 mm is a tiny fraction of a kilometer – which is exactly what we see. 13 mm is on the order of* km. is a tiny number, as we said we expected.

Now you give it a try! How many nanoseconds are in a year? We’ll use 1 year = 365 days as our first conversion factor.

Does this make sense? Well, a nanosecond is a really really really small amount of time. A year is huge by comparison, so we’d expect to have many many nanoseconds in a year. And this is exactly what we found: is a huge number of nanoseconds!

By way of sensemaking, my closing remarks are on something called “dimensional analysis.” This is our fancy term for making sure that when all the math is said and done, we have the same units on each side of the equal sign. We will “perform dimensional analysis” (lofty language, eh?) all the time on this blog. Making sense of the scale and units of our answers is critically important in physics because at the end of the day, the results of our calculations need to accurately represent the phenomena that we’re studying. So: start now! Always ask yourself if your answer makes sense before deciding that you have completed a problem.

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*Here, “on the order of” means “at the scale of.” You’ll always see “on the order of” followed by a power of 10 (in this case, ), which represents the scale when the number under consideration is written in scientific notation**. We say that our answer has an order of magnitude of kilometers.

**As a reminder, “scientific notation” is a way of writing numbers such that you only have one non-zero numeral to the left of the decimal point and then write where is the power of 10 for that number. In calculating the number of km in 13 mm, the first answer I wrote down was km. This number is not written in scientific notation because it has two numerals to the left of the decimal point. The second version of it, km is written properly in scientific notation. This is as much as I’m going to say about scientific notation here; I’ll leave you to the Math is Fun page on scientific notation if you need to brush up.