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I had a pretty basic question about dimensional analysis. According to this site :

"Dimensional Analysis (also called Factor-Label Method or the Unit Factor Method) is a problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value. It is a useful technique. The only danger is that you may end up thinking that chemistry is simply a math problem - which it definitely is not."

There are a couple of things I don't understand about dimensional analysis. Firstly, how does "the fact that any number or expression can be multiplied by one without changing its value" actually tie in to the full concept of dimensional analysis?

In this image im1

In the actual setup of the problem you're solving. Why are you using multiplication?

In this image im2

What if you had something like 730 days is 2 years and so on? Are you taking the product of the numerators and dividing that by the product of the denominators or what? I don't actually get what's going on in this problem.

I'm aware I may be complicating certain aspects of this problem, but I have problems with understanding simple concepts when I don't really understand the root of the concept.

Apologies for the inexperience, please try to explain this as simply as possible. Thanks.

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1 Answer 1

Let's look at the first one in some detail. We know that something is $6$ inches long. How long is it in centimeters? That is the motivating question. The key idea is that $2.54$ centimeters is the same as $1$ inch. Thus $\dfrac{2.54 \, cm}{1 \, in.}$ acts like $1$ in the sense that multiplying by it doesn't change anything. (Just like $5 \cdot 1 = 5$ and whatnot - this is what they meant when they said that multiplying by one doesn't change anything).

Thus we can take $6\, in. \, \cdot \dfrac{2.54 \, cm}{1 \, in.} = 15.2 \, cm$, by cancelling the inches. Why is this the same, that is how do we know that it represents the same length? Because multiplying by $\dfrac{2.54 \, cm}{1 \, in.}$ is just like multiplying by $1$, so nothing changes.

The second problem answers the question: How many seconds are in two years? To do this, they first convert to days, then to hours, then to minutes, and the to seconds. Is this the only way to proceed? No! If you are a Rent fan, you will know there are $525600$ minutes in a year, for example (perhaps you should verify this). Or, as you mentioned, there are $730$ days in $2$ years. Do you get the same answer? (Why don't you try it and see? Then justify why.)

The underlying concept here is that sometimes we convert from one set of units to another, and these conversions can be done systematically. We can treat the units themselves, like centimeters or seconds, almost algebraically, so that they cancel themselves when they divide themselves and square when multiplied. This is a fundamental manipulation in the sciences, and it doesn't go away.

As an ending note, I wanted to point out that some of the greatest and most common daily numerical misconceptions comes from this sort of problem: we are often great at estimating one-dimensional things (like length), but terrible at estimating areas or volumes. When people say, for example, how many square feet are in a square yard, so many people would say $3$. But it's not - it's $9$. In fact $yd^2 \cdot \dfrac{3\,ft}{yd} \cdot \dfrac{3\,ft}{yd} = 9\, ft^2$. So when we ask ourselves how many gallons of tank are in a fish tank, or a swimming pool (gallons are fundamentally volume-measuring), we have a tendency to apply lengthwise or areawise estimates, and are often off by several orders of magnitude. Think to yourself: How many gallons would it take to fill up your bathtub, then look it up. Were you close?

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