Order of Operations
The order of operations is a mathematical and algebraic set of rules. It is used to evaluate (solve) and simplify expressions and equations. The order of operations is the order that different mathematical operations are done. The standard mathematical operations are addition (+), subtraction (−), multiplication (*, ×, · or n), division (/, ÷, : or ‘), grouping/parenthesizing/bracketing ((), [] or {}) and exponentiation (^ or ⁿ), which includes root extraction (√’).
Mathematicians have agreed on a correct order to use operations, and it is very important that they know these rules. When people are solving a problem with more than one operation, they will need to know the correct order to solve the problem correctly. Otherwise, the answer will be wrong.
Rules
Follow all the rules in this order from left to right in the equation.
Brackets and parentheses
Use operations inside brackets and solve any indices. You should always solve brackets first when solving an equation.
Example:
Exponents
When seeing an exponent, solve it first after solving the brackets. (5 3 = 5 * 5 * 5 = 125)
Example:
Multiplication and division
Solve any multiplication and division in the problem. Note that multiplication does not precede division; this is a common mistake. Both are solved from left to right as they occur.
Example:
Addition and subtraction
Lastly, solve any addition or subtraction. It always goes from left to right.
Example:
Acronyms
The acronym for the order of standard operations is GEMDAS/PEMDAS/BOMDAS/BIMDAS, which stands for Grouping/Parenthesization/Bracketing, Exponentation, Multiplication, Division, Addition and Subtraction.
When solving 8 - 7 + 5, some people say that 7 + 5 must take precedence, but this is incorrect. Instead, one should look from left to right to find the correct answer. This rule also applies to multiplication and division.
Written for younger readers
Order of Operations, in simpler words
This version comes from Wikijunior, a set of books written for children aged 8 to 11. It is shorter and uses plainer language than the article above.
From Wikijunior: Introduction to Mathematics
- Order of Operations
- Precedence
- Brackets
Evaluate the expression 3 + 4 × 5.
If you add first, it is 7 × 5 and evaluates to 35.
If you multiply first, it is 3 + 20 and evaluates to 23.
Is the first or second answer true?
With no order of operations all of the two answers are true. If an expression evaluates to more than one answer, math does not work. The order of operations says the order you do operations, so there is only one way to evaluate an expression.
You use precedence to make the order of operations. All operations have a precedence that is more, less, or the same as an other operation. You do operations with more precedence before you do operations with less precedence. You do operations with the same precedence from left to right.
This list says the order of precedence for operations. Operations at the top of the list have more precedence than operations at the bottom of the list. Operations on the same line have the same precedence.
- Brackets ( ) or [ ]
- Raise ^
- Multiply ×, Divide ÷
- Add +, Subtract -
Brackets is a special operation that has the most precedence. You use the ( and ) signs to make a separate expression from a group of numbers. You evaluate an expression in parenthesis first. You use parenthesis if you need to do an operation with less precedence first.
Where this page comes from
The article above is adapted from “Order of Operations” on Simple English Wikipedia, by its contributors. We removed reference markers, navigation boxes and tables, expanded measurement templates into readable numbers, and kept the prose otherwise intact. The simpler version is adapted from Wikijunior on Wikibooks.
Both sources are published under CC BY-SA 4.0, so this page is published under the same licence. You may share and adapt it, including commercially, as long as you credit the original and keep the same licence.
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