Simplifying Dimensional Analysis for Unit Conversions

[Beholder]

Hello I'm learning physics and also brushing up on some math (its been 10yrs) and I encountered difficulty in understanding Dimensional analysis. I don't really understand how they know what number goes on top or what goes on bottom. I've read some explaintations but they just confused me more, one for instance was using the example that 1/6th times 6 over 1 = 1, so what? what are they trying to show me here I'm not catching on. If they give me a fairly simple word problem where you have to convert different units I can do it in my head but I would like to understand D.A. in order to solve more complex problems with more ease. Can anyone offer some help?

 

[Zurtex]

It's not too difficult.

If you are trying to work some quantity out then the idea is to look at what units it would be measured in. For example if you have something where you know that it will be measured in length squared then you don't need to bother looking at time or energy as a variable.

Similarly it wouldn't make sense to add metres squared to metres cubed and things like that.

 

[EnumaElish]

... one for instance was using the example that 1/6th times 6 over 1 = 1, so what? what are they trying to show me here I'm not catching on. ...

Let's take meters. Meter times meter times meter = meter cubed.

If you invert meter cubed you'd get inverse meters cubed: $1/ \text{m}^3 = \text{m}^{-3}$

First of all, you may not add or subtract terms with different Powers. For ex., if A (for "area") is measured in meters squared (e.g. A = 1500$\text{m}^2$) and V (for "volume") is measured in meters cubed (e.g. V = 16000$\text{m}^3$), then A + V does not make any sense; it's like "adding apples and oranges."

Now, let's say you are multiplying a quantity (say V = 16000$\text{m}^3$) measured in meters cubed with another quantity measured in inverse meters cubed (say W = 9$\text{m}^{-3}$). Then, V times W = 144000 which is a unitless number (a "pure number"). It's as if the meters cubed and inverse meters cubed "cancel out." Just like 1/3 would cancel 3 out. That's what 1/6 times 6 = 1 is aiming to represent in my opinion.