Units give numbers and variables a real world value.
By understanding and knowing the units of a problem you are a large step on the way of being able to solve the problem.
A number itself does not convey much information. For example, the length of a wood board is 6. The number “6” does not tell us how long it is. Is it 6 inches or 6 feet -- a big difference.
Units fall into two categories, fundamental and derived.
Fundamental units include length, time, mass, and temperature. Examples include feet, meters, seconds, hours, kilograms, and degrees Celsius or degrees Fahrenheit.
Angle measures are unitless -- defined by a ratio or relationship -- they do not have a dimension. Angle degrees and radians do however have a relationship to each other: 180 degrees = pi radians. Use of degrees and radians is often like a unit or dimension -- you do need to be consistent in a problem and make sure you are doing your calculations with the right measure, degrees or radians.
Derived units are a combination of fundamental units. For example velocity is distance divided by time, and force is acceleration times mass. All derived units are from the application of an equation that uses fundamental or other derived units. If you manipulate the units properly the answer you get will have the right units. If the units of the answer are wrong you know you have made a mistake. You can often find a mistake by analyzing the units you carried in the problem.
Always carry your units through the problem solving process -- they will guide you to the right answer.
Unit conversion
There are three types of unit conversions: magnitude changes (centimeters to meters); conversion between definitions (feet/seconds to miles/hour); and conversions between measurement systems (feet to meters). A conversion is always a ratio that equals one. You multiply this ratio (equal to one) by the number you want to convert to give you the new units.
The Earth is 144 x 10^11 cm from the sun. How far is this in miles?
144x10^11 cm x (1 m / 100 cm) = 144x10^9 m [order of magnitude change]
144x10^9 m x (1 ft./0.3048 m) = 4.72x10^11 ft. [system conversion]
4.72x10^11 ft x (1 mile/5280 ft) = 89.48 x 10^6 mi [definition conversion]
Rules to remember
-- Unit conversion is done by multiplying by a ratio equal to one (eg. 1 = 1 ft./12 in.)
-- Combine conversions into one big equation for clarity.
-- Cancel units that appear in both the numerator and denominator.
-- Always carry the units -- they will guide you to the answer.
-- Simplify units at each problem step to avoid mistakes.
