Reducing ambiguity by agreement
In general, nobody wants to be misunderstood. In mathematics, it is so important that readers understand expressions exactly the way the writer intended that mathematics establishes conventions, agreed-upon rules, for interpreting mathematical expressions.
Does 10 − 5 − 3 mean that we start with 10, subtract 5, and then subtract 3 more leaving 2? Or does it mean that we are subtracting 5 − 3 from 10?
Does 2 + 3 × 10 equal 50 because 2 + 3
is 5 and then we multiply by 10, or does the writer intend that we add 2 to the result of 3 × 10?
To avoid these and other possible ambiguities, mathematics has established conventions (agreements) for the way we interpret mathematical expressions. One of these conventions states that when all of the operations are the same, we proceed left to right, so 10 − 5 − 3 = 2, so a writer who wanted the other interpretation would have to write the expression differently: 10 − (5 − 2). When the
operations are not the same, as in 2 + 3 × 10, some may be given preference over others. In particular, multiplication is performed before addition regardless of which appears first when reading left to right. For example, in 2 + 3 × 10, the multiplication must be performed first, even though it appears to the right of the addition, and the expression means 2 + 30.
See full rules for order of operations below.
Conventions for reading and writing mathematical expressions
The basic principle: “more powerful” operations have priority over “less powerful” ones.
Using a number as an exponent (e.g., 58 = 390625) has, in general, the “most powerful” effect; using the same number as a multiplier (e.g., 5 ×8 = 40) has a weaker effect; addition has, in general, the “weakest” effect (e.g., 5 + 8 = 13). Although these terms (powerful, weak) are not used in mathematics, the sense is preserved in the language of “raising 5 to the 8th power.” Exponentiation is “powerful” and so it comes first! Addition/subtraction are “weak,” so they come last. Multiplication/division come in between.
When it is important to specify a different order, as it sometimes is, we use parentheses to package the numbers and a weaker operation as if they represented a single number.
For example, while 2 + 3 × 8 means the same as 2 + 24 (because the multiplication takes priority and is done first), (2 + 3) × 8 means 5 × 8, because the (2 + 3) is a package deal, a quantity that must be figured out before using it. In fact (2 + 3) × 8 is often pronounced “two plus three, the quantity, times eight” (or “the quantity two plus three all times eight”).
Summary of the rules:
- Parentheses first. Referring to these as “packages” often helps children remember their purpose and role.
- Exponents next.
- Multiplication and division next. (Neither takes priority, and when there is a consecutive string of them, they are performed left to right.)
- Addition and subtraction last. (Again, neither takes priority and a consecutive string of them are performed left to right.)
Common Misconceptions
Many students learn the order of operations using PEMDAS (Parentheses, Exponents, Multiplication, Division…) as a memory aid. This very often leads to the misconception that multiplication comes before division and that addition comes before subtraction. Understanding the principle is probably the best memory aid.
The Order of Operations and Variables:
Tip:
A good idea when working with many operations at a time is to do a little portion of the equation at a time, rewriting frequently. For example, do the portion within the parentheses and then rewrite the equation. Trying to do the entire equation at once can often lead to mistakes. Break it down into parts using the order of operations and do a little at a time.
What is the order of operations?
Operations are things like addition, subtraction, multiplication, and division. When you add two numbers together, you are performing the operation of addition on them. Similarly, when you multiply numbers together, you are performing the operation of multiplication.
The order of operations is the rule for what operations should be done first when there are several operations within the same equation.
The order of operations is like grammar rules for the language of math. It explains how to interpret an equation to mean what it is supposed to mean.
Applying the Order of Operations (PEMDAS)
The order of operations says that operations must be done in the following order: parentheses, exponents, multiplication, division, addition, and subtraction.
Parentheses
When there are parentheses, whatever is inside must be done first. The stuff inside the parentheses may also need to be broken down according to the order of operations as well. It is even possible to have parentheses within parentheses. In cases like this, work from the inside out.
Exponents
If there are exponents in the equation, these would be done next.
Multiplication and Division
Multiplication and division can be done together. In other words, it doesn’t matter if you do division or multiplication first, but they must be done after parentheses and exponents and before addition and subtraction.
Addition and Subtraction
Addition and subtraction also work together. You can do subtraction first, or you can do addition first. They are part of the same step, however, they can only be done after items in parentheses, exponents, and any multiplication and division.
PEMDAS
A frequently used expression in English to help students remember the order of operations is PEMDAS.
Another way to remember this is the phrase “Please Excuse My Dear Aunt Sally.”
Critical Thinking Challenge
Can you think of another phrase that could help you remember the order of operations?
Video Source (04:50 mins) | Transcript
Video Source (01:07 mins) | Transcript
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Remember to take it one step at a time and rewrite your equation after completing an operation. Doing this will help you keep track of what you’ve already done and make sure you don’t skip any steps.
Remember to continue to work on memorizing your single digit multiplication if they aren’t memorized yet. As you are beginning to see, we are using multiplication a lot in these lessons and they will be easier if you know your multiplication.
Additional Resources
- Khan Academy: Intro to order of operations (09:39 mins, Transcript)
- Khan Academy: Order of operations example (04:26 mins, Transcript)
- Khan Academy: Order of operations (PEMDAS) (05:19 mins, Transcript)
Practice Problems
Evaluate the following expression:
- \(0\div4\times7+5^{2}\div5\times5 = ?\)
- \(6 - 4 ^{2} \div 2 - 2^{3} + 3 = ?\)
- \(6 \div 1 - \lgroup 7 - 5 \rgroup \times 3^{2} \times 7 = ?\)
- \(7 + 2 \times 3^{3} + 12 \div 2 = ?\)
- \(2^{3} \times 5 \div \lgroup 5 - 1 \rgroup \div \lgroup 2 - 1 \rgroup \times 6 = ?\)
- \(5^{3} - \lgroup 5 \times 2 \rgroup^{2} - 2^{4} - 2^{3} = ?\)