Math101 · Independent Ontario learning library
Math101Order of Operations
Order of operations is a convention for reading an unparenthesized expression consistently.
The convention gives expressions a shared, unambiguous meaning and underlies substitution, formulas, calculators, and programming. Understanding hierarchy is more dependable than chanting an acronym.
Intuition and core definition
Order of operations is a convention for reading an unparenthesized expression consistently. Grouping symbols come first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication does not always precede division, and addition does not always precede subtraction; each pair shares a level.
Notation, language, and conditions
Parentheses, brackets, fraction bars, and radical bars create groups. A fraction bar groups the entire numerator and denominator. Juxtaposition such as $3x$ indicates multiplication. The acronym BEDMAS or PEMDAS is a memory aid, but the left-to-right rule within equal-priority operations is part of the convention.
Why this idea matters
Operation conventions make a written expression unambiguous, with grouping and exponent structure taking priority over left-to-right operations at the same level.
A dependable method
- Mark every explicit or implied group, including fraction numerators and denominators.
- Evaluate innermost groups and exponents.
- Perform multiplication and division in the order they appear from left to right.
- Perform addition and subtraction in the order they appear from left to right.
- Rewrite one stage per line and estimate to catch implausible results.
