**Annotation category:**

Chapter 1

Note: |

As an example of a dimensionless calculation (one in which you don't need to use units), add 6 + 2. It's 8. 8 what, you ask? 8 nothing. Just plain 8.

Now let's perform a calculation involving a physical scenario. Say we buy some tangerines: I buy 6 and you buy 2. What have we bought together? 8 tangerines. Not merely 8, but 8 *tangerines*. Tangerines are our **units**. Units indicate what the numeral 8 represents.

Here's a more complicated example. Compare the dimensionless calculation:

to this question: You run a mile in 8 minutes, and you run for a total of 16 minutes. How far did you run?(1/8) x 16 = 2

Let's write this word problem in a form that's easier to manipulate:

[(1 mi)/(8 min)] x (16 min)/1]

When you multiply fractions, the numerators multiply together, and the denominators multiply together. So our problem can be written as:

[(1 mi)x(16 min)]/[(8 min)x1]

where 1 is the implicit denominator of (16 minutes)

= (16 mi x min)/(8 min).

Now, to simplify this fraction, we are allowed to **cancel out **any unit that appears both in the numerator and in the denominator. Here, we see that "minutes" appears in both places. Thus we erase the "minutes" and are left with:

(16 mi)/8.

Dividing 16 by 8, we get 2. So our simplified answer is:

(2 mi) / 1, or (2 miles).

Finally, let's **check our units**, to be more confident that we have found the correct answer. Recall that the question asked us how far we ran. So our answer should be in units of distance -- and it is. This method of **dimensional analysis**, or unit analysis, is a useful guideline if you are unsure of your answer.

Let's try a few more examples.

**MULTIPLYING and DIVIDING WITH UNITS**:

Cancelling any unit that appears on both the top and the bottom:

2. (1 gram of gold) / [(196.97 grams / mole of gold) x (1.66 x 10^{-24}moles / atom)]

**ADDING or SUBTRACTING WITH UNITS**:

Here, the quantities involved and the sum they produce must all have the *same* units. You can appreciate this rule intuitively, by realizing that it is the only condition that makes physical sense. Consider, for example, adding 2 tangerines and 3 atoms of gold. What does that give you? 5? 5 what? No, you can only say that you still just have 2 tangerines and 3 atoms of gold. Simplification isn't possible.

ex: 3 seconds + 5 seconds = 8 seconds. ex: 5 meters - 12 meters = -7 meters. ex: 4 degrees Celcius + 18 lbs = ??