### Dimensionless numbers

Pure numbers have no dimension: 1, 5.2, 3.14159

(sometimes **constants** do, however!)

Alternately, sometime all units cancel out: 6 ft/1ft = 6

DEFINITION:

**Dimensionless quantities** can be either
pure numbers, or dimensionless **groups** (expressions where all
the units cancel).

### What are dimensionless groups good for?

For one thing, dimensionless groups are useful for "pure"
comparisons.

OUTCOME:

Compare two quantities using a dimensionless
group

TEST YOURSELF!

Which is further away, Cleveland (136 miles) or
State College (223 km)?

Must match units in order to compare (in miles or km?).

NOTE:

Dimensionless groups can be **any** expression
whose units cancel! Here we consider ratio of lengths and times,
but force, pressure, resistance, etc. all work as well! The only
thing you **can't** have is a ratio of things with different
dimensions! (Obviously, since they would no longer cancel.)