Dimensional Analysis
• Definitions (from the web)
– The analysis of the relationships between different physical
quantities by identifying their fundamental dimensions (such
as length, mass, time, and electric charge) and units of
measure (such as miles vs. kilometers, or pounds vs. kilograms vs.
grams) and tracking these dimensions as calculations or
comparisons are performed (Wikipedia)
– A method of using the known units in a problem to help deduce
the process of arriving at a solution (about.com)
– A technique of calculating with physical quantities in which
units are included and treated in the same way as numbers
(dictionary.com)
– A problem-solving method that uses the fact that any number or
expression can be multiplied by one without changing its value
(chem.tamu.edu)
– is by far the most useful math trick you'll ever learn (alysion.org)
• Limitations
– Useful for measured things
Dimensional Analysis
• Based on 5 Properties of Multiplication & 1 of Division
– Commutative Property
• 4x2=2x4
– Associative Property
• (2 x 3) x 4 = 2 x (3 x 4)
– Distributive Property
• 2 x (3 + 4) = (2 x 3) + (2 x 4)
– Multiplicative Identity Property
• 4x1=4
– Divisive Identity Property
• 4 / 1 = 4 or 4/4 = 1
– Multiplicative Inverse Property
• 4 / 2 = 4 x 1/2
Dimensional Analysis
• Straightforward Example
– How many pennies in 4 dollars?
• know:
– 4 dollars
– Answer should be in pennies
– 100 pennies = 1 dollar
100 pennies
1
1 dollar
4 dollar
4 dollar 
1
4 dollar
 1 = 4 dollar = ? pennies
1
4 dollar
100 pennies

 ? pennies
1
1 dollar
4 dollar
100 pennies 4×100 pennies 400 pennies



= 400 pennies
1
1 dollar
1 1
1
Dimensional Analysis
• Another Straightforward Example
– How many pennies in 4 dollars?
• know:
– 400 pennies
– Answer should be in dollars
– 100 pennies = 1 dollar
1 dollar
1
400 pennies
400 pennies
 1 = 400 pennies
400 pennies
1 dollar

100 pennies
400 pennies
1 dollar
400 dollar 4 dollar



= 4 dollars
100 pennies
100
1

Dimensional Analysis
• Yet Another Straightforward Example
– How many eggs in 2 dozen?
• know:
– 2 dozen
– Answer should be in eggs
– 12 eggs = 1 dozen
12 eggs
=1
1 dozen
2 dozen
12 eggs


1 dozen
2 dozen
12 eggs 2×12 eggs


 24 eggs
1 dozen
1
Dimensional Analysis
• Beyond Another Straightforward Example
– How many pizzas do I need to order?
• know:
– 29 students
– 1.5 slices/student
– 12 slices = 1 pizza
29 students
1.5 slices
1 pizza
×
=
1 student 12 slices
29 students
1.5 slices
1 pizza
×
×
=
1 student 12 slices
29 1.5 1 pizza 43.5 pizza

 3.625 pizzas  3.6 pizzas  4 pizzas
112
12
×