Order of Operations

The order of operations is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.

Definition

The order of operations is typically remembered using the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e. Powers and Square Roots, etc.)
• MD: Multiplication and Division (left-to-right)
• AS: Addition and Subtraction (left-to-right)

The order of operations is important when more than one operation is present in a mathematical expression. It helps to eliminate ambiguity while performing different operations.

Example

Consider the expression: 3 + 4 * 2.

Without a convention on the order of operations, this could be interpreted as either (3 + 4) * 2 = 14 or as 3 + (4 * 2) = 11. The order of operations tells us to perform the multiplication before the addition, so the second interpretation is the correct one.

Note

When expressions have more than one operation with the same level of priority (e.g., addition and subtraction), we operate from left to right. However, in some mathematical and programming languages, operations with equal priority are performed from right to left, so it's always best to clarify and use parentheses when there could be ambiguity.

Applications

The order of operations rule is used universally in mathematics and computer science. It is vital for the correct interpretation of mathematical expressions and equations in fields ranging from basic arithmetic to advanced calculus, algebra, and beyond.