Here's a detailed example calculation (11min): Unit Conversion Example from Khan Academy on YouTube
Here's a fun video version of this topic and sig figs (12min): CrashCourse Chemistry: Unit Conversion and Significant Figures on YouTube
Dimension is like 1D, 2D, 3D. A normal movie is 2D; in a 3D movie you wear special glasses. It also means what unit you use to measure: a length in 1D is measured in meters, an area in 2D is measured in meters squared, m2, a volume in m3 or liters. Other types of quantities (time, mass, temperature) are measured in other units, because they have different dimensions. Analysis means thinking about something, often by thinking about one part at a time. In this case, dimensional analysis means thinking about units piece by piece. You can use dimensional analysis to correctly go between different types of units, to catch mistakes in your calculations, and to make many useful calculations in real life.
Essentially, dimensional analysis means multiplying by one. You collect a set of "conversion factors" or ratios that equal one, and then multiply a quantity that you are interested in by those "ones". For example, if you want to know how many seconds it would take to get from Seoul to Busan, you'd do it like this:
First, on the express train it takes 2.5 hours to get to Busan from Seoul Station. Then, we know that 1 hour = 60 minutes and 1 minute = 60 seconds, so (1h/60min) = 1, and (1min/60s) = 1. Now, all we have to do it multiply our starting number (2.5h) by one twice, making sure that the units cancel correctly so that we have only seconds at the end.
| 2.5h | 60min | 60s | = 9.0 x 103 s |
| 1h | 1min |
If we don't put each part in the right place, the units will come out wrong. For example:
| 1h | 1min | = 1.1 x 10-4 s-1 | |
| 2.5h | 60min | 60s |
In this case, we put the starting quantity on the bottom, so we got s-1 when we cancelled out the units. Here's an example of not being able to cancel the units correctly:
| 2.5h | 1h | 60s | = 2.5 s h2 min-2 |
| 60min | 1min |
The important part is that if you check the units to make sure they come out right, you can be pretty sure you set the calculation up right!
Here's an example of how dimensional analysis can help. I once saw a student who was calculating initial velocity (v0) using this equation: d = v0t + at2/2. But the student had solved the equation wrong, and had: v0 = d/t – at2/2. So the student had the wrong answer, but didn't know that, because he just put the numbers for d, t and a into his calculator following the wrong equation. If he always checked the units, he would have seen that d/t has units of v (m/s) while at2/2 has units of d (m).
I recommend that anytime you do a calculation, you write out all the quantities in the calculation with their units showing how they are combined. Before you enter the numbers into your calculator, check to make sure that the calculation you are planning gives you the right final unit. If you make this a habit, you will avoid many mistakes.
Dimensional Analysis is often useful when you want to estimate some quantity in the real world. For instance, maybe you want to know how much money you spend on coffee each month. If you spend 5000W/cup and have 2 cups/day, and there are approximately 30 days/month, than you can set up a calculation just like those above to calculate W/month spent on coffee. This works for many important, less obvious situations, for instance in business, to get an approximate idea of some quantity.