Dimensional analysis uses conversion factors to change the unit in an amount into an equivalent quantity expressed with a different unit.

The "given" unit in the problem, which will be associated with a number, must be determined.  In the example above, the given number is 3.55, and its unit is meters.   The "desired" unit, which is the unit that the given quantity should be changed "to" or "into," must be determined.  In the example above, the given quantity should be changed to centimeters.   Determine which equality or equalities relate the given and desired units.  In the simplest dimensional analysis problems, only a single equality is needed.  However, more complex problems will require multiple equalities.  This step, which can also be referred to as "unit tracking," is generally the most challenging step in the dimensional analysis process.  Meters and centimeters can be related by the prefix modifier equality 100 cm=m100 cm=m { \text{100 cm}} = { \text{m}}.   Use the appropriate conversion factor derived from this equality to achieve unit cancelation.  Remember that the equality given above can be represented as two conversion factors: 100 cmm100 cmm \dfrac{ \text{100 cm}}{\text{m}} and m100 cmm100 cm \dfrac{ \text{m}}{\text{100 cm}} However, only one of these conversion factors will allow for the cancelation of the given unit.  Specifically, the unit to be canceled must be written in the denominator of the conversion factor.  This will cause the given unit, which appears in a numerator, to be divided by itself, since the same unit appears in the denominator of the conversion factor.  Since any quantity that is divided by itself "cancels," orienting the conversation factor in this way results in the elimination of the undesirable unit.  Therefore, since the intent of this problem is to eliminate the unit "meters," the conversion factor on the left must be used. 3.55m×100cmm3.55m×100cmm {3.55 \; \cancel{\rm{m}}} \times \dfrac{100 \; \rm{cm}}{\cancel{\rm{m}}} Why does this process work?  In the example above, 100 cm equals (1) m, so equivalent quantities appear in both the numerator and the denominator of the fraction, even though those quantities are expressed in different units.  Since the quantities in the numerator and denominator are equivalent, this conversion factor effectively divides a value by itself, and the entire process is equivalent to multiplying the given number by 1.  Therefore, while the given quantity does not change, the unit does.   Perform the calculation that remains once the units have been canceled.  The given number should be multiplied by the value in each numerator and then divided by the value in each denominator.  When using a calculator, each conversion factor should be entered in parentheses, or the "=" key should be used after each division.  In this case,  3.55×100 cm=355 cm3.55×100 cm=355 cm {3.55} \times {\text {100 cm}} = {\text {355 cm}} Note that the unit that remains uncanceled becomes the unit on the calculated quantity.   Apply the correct number of significant figures to the calculated quantity.  Since the math involved in dimensional analysis is multiplication and division, the number of significant figures in each number being multiplied or divided must be counted, and the answer must be limited to the lesser count of significant figures.  Remember that the equalities developed in the previous section are exact values, meaning that they are considered to have infinitely-many significant figures and will never limit the number of significant figures in a calculated answer.

Dimensional Analysis

Practice Problems on Dimensional Analysis