**Dimensional
analysis**,
also known as the **factor-label method**,
is a problem solving technique using units (labels) and conversion factors. **Units** tell us the type of measurement
being made; for example, "5.2 cm" has a unit (cm) which tells us the
measurement to be made is length. **Conversion
factors** (also known as "equalities") are fractions that relate
two kinds of units; for example, "6.0 cm / s" tells us that for every
1 second that passes (time), 6.0 cm are covered (length). Notice that "6.0
cm s^{-1}" is the same as "6.0 cm / s" in unit notation.

Most problems ask a question whose __answer__ is
a number with its unit. Problems also give __information__ that contains
numbers with their units. Multiply the information by conversion factors so
that all units cancel *except* the units needed in the answer. (A unit in the numerator may be canceled by
placing the same unit in the denominator of the neighboring conversion factor.
Conversely, a unit in the denominator may be canceled by placing the same unit
in the numerator of the neighboring conversion factor.)

- Numbers and units are considered separately.
- Multiply by as many conversion factors as necessary.
- Common conversion factors may or may not be supplied with the problem.

__ Examples__:

1. How many hours are in 6 days?

2. How many seconds are in 5 hours?

3. How many feet per second is 5 miles per hour?

A common **variation** occurs when more than one piece of information is provided with the problem. In these cases, start with the information that contains the same
type of units as the answer. (For example, if length units are needed in
the numerator of the answer, use the information that contains length units and
arrange it so that those length units are in the numerator.) Next, multiply by
conversion factors so that unwanted units cancel.

** Test Yourself**:

1. How many gallons of milk does a family drink in 5
days if they drink 4 pints per day? *Answer:* 2.5 gallons

2. How many minutes will it take an automobile
traveling 60 miles per hour to travel a distance of 400 miles? *Answer:* 400 minutes

**Scientific Notation** is used by scientists to express very large and very small numbers in a compact fashion.

To express a number in scientific notation, we *rewrite* the quantity as a number
(between 1 and 10) times 10 raised to a power (exponent) that tells us how we
moved the decimal point.

- Multiply the number by 10
^{0}(10^{0}= 1) - Move the decimal point to give a number between 1 and 10
- Every time we shift the decimal point to
the
__left__by one place, we__increase__the value of the exponent by one - Every time we shift the decimal point to
the
__right__by one place, we__reduce__the value of the exponent by one

__ Example__: Write 0.0000012 in scientific notation.

To express a number that is written in scientific notation as a non-exponential quantity:

- Move the decimal point the same number of places as the value of the exponent and eliminate the exponential part of the number.
- If the exponent is
__positive__, we move the decimal to the__right__to the same number of places as the value of the exponent. The result should be a number*greater*than 1the original number is negative.*unless* - If the exponent is
__negative__, we move the decimal to the__left__to the same number of places as the value of the exponent. The result should be a number*less*than 1the original number is negative.*unless*

__ Example__: Write 1.23 * 10

__ Example__: Write 1.11 * 10

2. A friend gives you 235 US dollars ($). How many US quarters does this equal?

3. Using the relationships of 128 fluid ounces = 3.7856 L and 128 fluid ounces = 4 quarts, convert 9.90 quarts into Liters (L).

4. A spacecraft has been orbiting Saturn for 3.75 days. Convert this number into seconds.

5. Gas costs $3.05 a gallon, and your car travels at 27 miles for each gallon of gas. How far can you travel in your car with $95 in your pocket?

6. Express 3100 in scientific notation.

7. Express 0.0210 in scientific notation.

8. Convert 5.1 x 10

9. Convert 2.7 x 10

10. I have been walking 4.22 x 10