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.