Order of Operations

Definition

The order of operations is a mathematical rule that dictates the sequence in which calculations are performed to ensure consistent and accurate results. It helps eliminate ambiguity in mathematical expressions by establishing a hierarchy for operations, which is critical for interpreting complex expressions correctly.

5 Must Know Facts For Your Next Test

1. The order of operations helps avoid confusion when evaluating expressions with multiple operations, ensuring everyone interprets the expressions the same way.
2. Mathematicians use the phrase 'Please Excuse My Dear Aunt Sally' as a mnemonic device to remember the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
3. If an expression includes parentheses, all calculations within them must be completed first before applying other operations.
4. Exponents must be calculated immediately after parentheses but before multiplication, division, addition, or subtraction.
5. When only multiplication and division or addition and subtraction are present in an expression, they should be evaluated from left to right.

Review Questions

• How does the order of operations ensure consistency in solving mathematical expressions?
  • The order of operations establishes a clear set of rules for performing calculations, which prevents ambiguity in solving mathematical expressions. By following the sequence—parentheses first, then exponents, followed by multiplication and division from left to right, and finally addition and subtraction also from left to right—everyone arrives at the same result when evaluating complex expressions. This consistency is crucial for effective communication in mathematics.
• Discuss how using parentheses affects the order of operations in mathematical expressions.
  • Using parentheses significantly impacts how mathematical expressions are evaluated because they indicate that the operations enclosed should be prioritized. When parentheses are present, calculations inside them are performed first before any other operations outside. This feature allows for more complex calculations by altering the natural flow dictated by the order of operations and can lead to different results depending on their placement within an expression.
• Evaluate the expression 3 + 4 × (2 - 1)² ÷ 2 and explain each step according to the order of operations.
  • To evaluate the expression 3 + 4 × (2 - 1)² ÷ 2 using the order of operations, start by simplifying within the parentheses: (2 - 1) = 1. Then calculate the exponent: 1² = 1. Next, replace back into the expression: 3 + 4 × 1 ÷ 2. Now perform multiplication and division from left to right: 4 × 1 = 4 and then 4 ÷ 2 = 2. Finally, add: 3 + 2 = 5. Thus, following these steps ensures that we accurately arrive at the final answer.

Related terms

PEMDAS: An acronym that stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right); it is a common way to remember the order of operations.

Parentheses: Symbols used in mathematics, such as '()' or '[]', that indicate the operations within them should be performed first according to the order of operations.

Expression: A combination of numbers, variables, and operators (such as +, -, ×, ÷) that represents a value; understanding how to apply the order of operations is essential for evaluating expressions.