Beginning Algebra Made Useful

Subsection 1.4.1 The Order of Operations

The Order of Operations follows the historical introduction of operations. Addition happened first. Subtraction is the operation that undoes addition. Multiplication naturally occurred as people needed to determine multiple sums of the same number; multiplication is repeated addition. The operation that undoes multiplication is division; division can also be thought of as repeated subtraction. Exponentiation occurred even later signifying repeated multiplication. Taking a whole number root, the inverse operation of raising a number to the same whole number exponent, can also be thought of as repeated division. When you encounter an expression to be simplified, consider the historical introduction of arithmetic operations and perform the most complex operation first, reducing each subsequent expression in the opposite order that the operations arose historically. Of course, if parentheses are included to alter the order of operations further, expressions inside them must be simplified first!

Let's consider some examples. To simplify the expression, \(3 + 2 \times 7\text{,}\) recall that the multiplication sign represents repeated addition. An equivalent expression written entirely in terms of addition is \(3 + 7 + 7 = 17\text{.}\) Try this with your calculator; do you get this result? Some students want to add 3 and 2 then multiply the result by 7 because the expression is written that way. If you add first to get 5 then multiply by 7, you get 35. This is not equivalent to the original expression. Notice that \(35 = (3 + 2) \times 7\text{.}\) In the original expression, only 2 is multiplied by 7. This example helps us understand why multiplication is completed before addition when simplifying \(3 + 2 \times 7\text{,}\) and in general.

Now consider the expression, \(5 \times 3^2\text{.}\) As indicated above, terms with exponents represent repeated multiplication. The expression, \(5 \times 3^2\text{,}\) can be rewritten as \(5 \times 3 \times 3\text{.}\) To multiply \(5 \times 3\) then square the result changes the meaning of the original expression to \((5 \times 3)^2\text{.}\) Try both versions on your calculator. Because no parentheses are included in the original expression, \(5 \times 32 \ne (5 \times 3)^2\text{.}\)

Try to simplify this expression to see why the Order of Operations is helpful. Change all exponentiation to repeated multiplication then change all multiplication to repeated addition before simplifying.

Look at the simplification below. Explain how each expression arises from the one before it:

You can see from this simplification and your explanation that simplifying an expression by replacing exponentiation with repeated multiplication and multiplication with repeated addition will quickly get cumbersome.

The rules for simplifying expressions using the Order of Operations serve as a reminder. To recall which operation to complete first it is helpful to think about what each operation represents. There are cautions, however. Sometimes the Order of Operations can lead us to believe that the order in which we perform operations is rigid. We know that expressions like \(85 + 27 + 15\) and \(25 \times 11 \times 4\) can be simplified more easily if we rearrange the terms. We see that \(85 + 15 = 100\) so the sum \(85 + 27 + 15 = 85 + 15 + 27 = 100 + 27 = 127\text{.}\) We also see that \(25 \times 4 = 100\) so \(25 \times 11 \times 4 = 25 \times 4 \times 11 = 100 \times 11 = 1100\text{.}\) Addition and multiplication are commutative (the order can be changed without changing the result). We also know that we can group numbers differently when they are all combined with addition or with multiplication; this is the associative property: \(14 + (16 + 18) = (14 + 16) + 18\text{,}\) \(15 \times (6 \times 17) = (15 \times 6) \times 17\text{.}\) Both of these properties allow us to group and rearrange numbers combined with one of the operations + or x (not both together, though) to put friendlier, easier to combine, numbers together to aid in simplification.

We would expect that the Order of Operations is a universal convention, agreed upon and used everywhere in the world in the same way. This is not the case, as it happens. Consider the case of multiplication and division. The Order of Operations convention used in USA textbooks tells us to complete multiplication and division in the order they appear in an expression, left to right. In Kenya, students are told to complete division first, in the order division arises, then complete multiplication. Simplify the following expressions with each convention. What do you notice?

What do you think is happening here? Can you explain why the result is the same regardless of which version of the Order of Operations you use? Think about how multiplication and division are related then rewrite each expression to use only multiplication. Now what do you notice?

Which Order of Operations convention do you think is easier and more reliable to implement? Use that one! The following reminders in Figure 1.4.1.1 of Order of Operations might be helpful. Use whichever one makes more sense to you. Notice that the order you simplify expressions, in both conventions, is the opposite order that each operation was introduced historically; more complex operations are simplified first.

P	Simplify Parentheses (and other grouping symbols)
E	Simplify Exponents
M D	Complete Multiplication and Division in the order found, left to right.
A S	Complete Addition and Subtraction in the order found, left to right.