Section 4: The Order of Operations
The arithmetic rules and definitions discussed are applicable to all real numbers. However, in an algebra setting, not all real numbers are represented by the numerals we would expect. We've already seen that real numbers can also be represented by variables. But they can also be represented by expressions. For example, if x is a real number and y is a real number, then (x + y) is also a real number. That means that all rules can be equally applied to x, to y, and to (x + y).
Before we apply these rules to variables and expressions, we need to review what is commonly called the "algebraic order of operations" or just the "order of operations". This is a set of rules used to determine what operations are done first in an expression containing multiple operations. If you were asked to evaluate 3 + 4 × 8 - 1 without these rules, you wouldn't know whether to start by adding 3 and 4, multiplying 4 and 8, or subtracting 1 from 8.
1. Perform all operations inside symbols of inclusion (parentheses, etc.) starting with the innermost set of symbols and working outward. Within inclusion symbols, follow the other rules.
2. Treat fraction numerators and denominators as implied symbols of inclusion. In other words, perform any operations above and below a fraction bar before performing the division associated with the bar itself.
3. Perform all multiplications and divisions in the order they occur reading from left to right.
4. Perform all additions and subtractions in the order they occur reading from left to right.
The rules relating to exponents and roots will be added when we discuss exponents and roots later. For now, let's do some examples.
Example 7 3 + 4 × 8 - 1. There are no parentheses and no fraction bars so we can jump right to rule 3 and get 3 + 32 -1. Then we progress to rule 4 and get 35-1 and finally 34.
Example 8 20 ÷ 2 × (6 - 3) + 18
20 ÷ 2 × 3+ 18
10×3+ 18
30 + 18
48
Example 9 There are actually several
steps that could be done "first" because some of the items in parentheses are
independent of each other. Watch the following steps:
The Distributive Property Revisited
The distributive property of multiplication over addition that was introduced in Section 2 appears to be a violation of the order of operations just
discussed. The distributive property states a(b + c) = ab + ac. This means
that any term multiplied on the outside of parentheses can be "distributed"
across any added terms on the inside of the parentheses.
Try it yourself with 2(3 +4). The order of operations says to perform the addition in the parentheses first. That would give us 2(7). Then we would perform the multiplication and get 14. Using the distributive property you get 2(3+ 4)=2(3)+ 2(4) =6+8 =14.
In addition, since division and subtraction can be defined by multiplication and addition of inverses, the distributive property can be applied to division and subtraction problems.
Example 10 (4 - 8) ÷ 2 by the order of operations is (-4) ÷ 2= -2. However (4 - 8) ÷ 2= (4 + (-8)) × (1/2) = (1/2) × (4 + (-8)). Written in this form, it's a multiplication and addition problem where the distributive property can be applied: (1/2) × (4) + (1/2) × (-8) = 2 - 4 = -2.
Calculators and the Order of Operations
Most non-graphing calculators either automatically follow the order of operations or
give the user the option of setting the order followed. However, some old calculators
and some very inexpensive calculators ignore the order of operations and simply calculate items in whatever order they are entered.
You can do a quick test to see how your calculator works. Enter the following problem: 3 + 2 × 4 and check the answer. If your calculator automatically follows the order of operations you will get the correct answer of 11, (2 × 4 = 8, 3 +8 = 11). If your calculator does not follow the order of operations you will get the incorrect answer of 20, (3 +2= 5,5×4= 20).
If your calculator found the correct answer, then you have little to worry about. You can enter problems as they come throughout the semester. However, if your calculator found the wrong answer you will have to be very careful when entering problems. You will have to determine the correct order yourself and enter them into your calculator accordingly.