Dimensional Analysis

Chemists are big on `dimensional analysis'. It keeps them from blowing up their labs. You should practice it too, even if whatever it is you are doing does not explode -- it will keep you from making silly errors.

Here is how it works. Whenever you write down a number, always include the units. In electricity, you get units like `volts' and `amps' and `watts' and `kWh' and so on. (Watts, of course, are really `joules per second'.) Then, whenever you work with two numbers, make sure the units all match up.

For instance, I was trying to convert from kWh (kilo-watt-hours) to J (joules). I knew this had to work out, but I was having a little trouble doing the arithmetic, because I failed to write down all the units. What happened was that I converted from hours to seconds the wrong way around. When I saw the tiny number I knew I had done something wrong -- each kWh is a lot of joules. So then I wrote it all down, like this:

3 kilo watt hour | 3600 seconds | 1 joule/second
-----------------+--------------+---------------
| 1 hour | 1 watt

Aside from the box on the left (with `3 kilo watt hour') in it, every other box is just the number one. That is, 3600 seconds and 1 hour are the same thing. 1 J/s and 1 W are also the same thing. As long as you multiply anything by 1, you still have the same thing. But by writing down all the `1's in various units, you can tell if you have the units the right way around. All you have to do is start crossing out things that match -- with a catch: when you go to cross two out, they have to be on opposite sides of the center horizontal dividing line.

When I was trying to do it in my head, I accidentally did this:

3 kilo watt hour |    1 hour    | 1 joule/second
-----------------+--------------+---------------   (note that this formula is wrong!)
                 | 3600 seconds | 1 watt

Now when I went to cross out `hour', I could see that something was wrong. Both `hour's were on the top! To fix it, I had to erase the middle part and write it all down as in the first (correct) formula above. Then I could cross out `hour's:

3 kilo watt hour | 3600 seconds | 1 joule/second
-----------------+--------------+--------------- (before crossing out)
| 1 hour | 1 watt

3 kilo watt      | 3600 seconds | 1 joule/second
-----------------+--------------+---------------        (after crossing out)
                 |    1         | 1 watt

The next bit is a little sneaky, but not too sneaky. In the last box, 1 joule/second over 1 watt, you can move the `/second' down to the lower half. (This makes sense because the 1 joule is being divided by 1 second, and the big boxes are just one thing divided by the other, same, thing -- 3600 seconds, divided by 1 hour, is 1.)

3 kilo watt | 3600 seconds | 1 joule
-----------------+--------------+---------------
| 1 | 1 watt second

Now you can cross out seconds again because they are on top and bottom as appropriate, and watts for the same reason:

3 kilo watt | 3600 seconds | 1 joule
-----------------+--------------+---------------
| 1 | 1 watt second

Removing the matching words in red and blue -- okay since they are dividing each other out -- gives this:

3 kilo | 3600 | 1 joule
-----------------+--------------+---------------
| 1 | 1

Now you can just multiply all the numbers and units on the top, and divide by all the numbers and units on the bottom. In this case, everything on the bottom is just `1' and can be ignored, so this leaves you with a number in kilo-joules -- 10800 kJ, to be exact. Voila!

more examples?