Order of Operations in Math (PEMDAS or BEDMAS)

This entry was posted on by (updated on )
Order of Operations - PEMDAS or BEDMAS

The order of operations is a set of rules in mathematics for the correct sequence to solve mathematical expressions. Without a standardized order, expressions could yield different results. To ensure consistency and accuracy, mathematicians follow a specific hierarchy when performing calculations. This order is commonly summarized by the acronym PEMDAS and PEDMAS in the United States and France or BEDMAS and BEMDAS in other English-speaking regions like Canada and the UK.

Order of Operations – Key Points

  • The order of operations is the set of rules for performing operations in a math problem.
  • There are several different acronyms, but they all outline the same order.
  • PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
  • BEDMAS/BEMDAS: Brackets are equivalent to parentheses.
  • GEMDAS: Groupings include parentheses, brackets, and absolute values.
  • Multiplication/Division and Addition/Subtraction are performed left to right.
  • Basic calculators evaluate expressions sequentially, not by order of operations.
  • Always solve nested parentheses from the innermost grouping outward.

Why Is the Order of Operations Important?

Consider the expression:

8 + 2 × 3

  • If you add first, you get (8 + 2) × 3 = 10 × 3 = 30.
  • If you multiply first, you get 8 + (2 × 3) = 8 + 6 = 14.

The correct answer is 14 because multiplication takes precedence over addition. The order of operations prevents ambiguity and ensures that everyone solves mathematical expressions the same way.

What Does PEMDAS Stand For?

The acronym PEMDAS helps remember the sequence of operations:

  • P: Parentheses (solve operations inside parentheses first)
  • E: Exponents (calculate powers and roots)
  • M: Multiplication (from left to right)
  • D: Division (from left to right)
  • A: Addition (from left to right)
  • S: Subtraction (from left to right)

Note: Multiplication and Division are performed in the order they appear from left to right, as are Addition and Subtraction. They are not prioritized over one another.

Order of Operations: Examples Using PEMDAS

Here are three examples illustrating the use of PEMDAS:

Example 1:

Solve: 5 + 3 × 2

Solution:

  1. Perform multiplication first: 3 × 2 = 6.
  2. Add: 5 + 6 = 11.

Answer: 11

Example 2:

Solve: 4 + (6 – 2)² ÷ 2

Solution:

  1. Parentheses: 6 – 2 = 4.
  2. Exponent: 4² = 16.
  3. Division: 16 ÷ 2 = 8.
  4. Addition: 4 + 8 = 12.

Answer: 12

Example 3:

Solve: 10 – 2 × 3 + 4

Solution:

  1. Multiplication: 2 × 3 = 6.
  2. Subtract and add from left to right:
    • 10 – 6 = 4
    • 4 + 4 = 8.

Answer: 8

Example 4 (Nested Parentheses):

Solve: (2 + (3 × 4)) – 5

Solution:

  1. Solve the innermost parentheses first: 3 × 4 = 12.
  2. Replace and solve the outer parentheses: 2 + 12 = 14.
  3. Subtract: 14 – 5 = 9.

Answer: 9

PEMDAS vs. PEDMAS

While PEMDAS is widely taught, some people use PEDMAS instead. The only difference lies in the acronym’s arrangement:

  • P: Parentheses
  • E: Exponents
  • D: Division
  • M: Multiplication
  • A: Addition
  • S: Subtraction

The rules remain identical. Multiplication and Division, as well as Addition and Subtraction, are performed left to right.

BEDMAS and BEMDAS

In some regions, particularly in countries that use British English, the acronyms BEDMAS and BEMDAS are common:

  • B: Brackets (the British term for parentheses)
  • E: Exponents
  • D: Division
  • M: Multiplication
  • A: Addition
  • S: Subtraction

These acronyms follow the same rules as PEMDAS and PEDMAS, with “brackets” being synonymous with parentheses.

GEMDAS: Another Variant

Some schools teach the acronym GEMDAS, where G stands for Groupings. Groupings include parentheses, brackets, and other forms such as absolute value bars (| |).

  • G: Groupings (e.g., parentheses, brackets, absolute values)
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

The rules remain the same: work left to right for operations at the same level.

How Basic Calculators Operate

Basic calculators do not follow the order of operations. Instead, most evaluate expressions sequentially as you type them. For example:

Input: 2 + 3 × 4

  • A basic calculator performs 2 + 3 = 5, then 5 × 4 = 20.
  • The correct answer following PEMDAS is 14.

Be aware, some calculators perform operations differently from others. To avoid errors, manually include parentheses or perform each operation in the correct sequence.

How to Use the Order of Operations When There Are No Parentheses

If an expression does not include parentheses (or brackets), proceed with the remaining operations in the correct sequence:

  1. Solve exponents (if any).
  2. Perform multiplication and division from left to right.
  3. Perform addition and subtraction from left to right.

Example: Solve 6 ÷ 2 × 3 – 4.

  1. Division and multiplication (left to right): 6 ÷ 2 = 3, then 3 × 3 = 9.
  2. Subtract: 9 – 4 = 5.

Answer: 5

FAQs About the Order of Operations

1. Do calculators follow the order of operations?

  • Most modern scientific and graphing calculators follow the order of operations automatically. However, basic calculators usually do not, so it is important to enter calculations step by step or include parentheses.

2. Are multiplication and division equally important in PEMDAS?

  • Yes, multiplication and division are performed in the order they appear from left to right. Neither takes precedence over the other.

3. Does subtraction always come last?

  • No, subtraction and addition are equally important. Perform them left to right as they appear.

4. What happens if I forget to follow PEMDAS?

  • Failing to follow the order of operations can result in incorrect answers or confusion in math problems.

5. Why are there different acronyms (PEMDAS, BEDMAS, etc.)?

  • Different acronyms are used in various regions for teaching purposes. Despite the variations, the rules remain the same.

6. What if I see nested parentheses?

  • Solve the innermost parentheses first and work outward.

7. Can I solve from right to left instead of left to right?

  • No. The rules specifically require left-to-right evaluation for multiplication/division and addition/subtraction.

Not Everyone Agrees

The important point about order of operations is that it is a set of rules that should be understood by both the person writing the mathematical expression and the ones solving it. Not everyone follows PEMDAS, BEDMAS, or whatever acronym you’re using.

For example, solve the following problem:

8 ÷ 2(2 + 2) = ?

My answer is 1. Strictly using PEMDAS, the answer is 16. While pretty much everyone agrees on doing the operation inside the parentheses first, a lot of mathematicians and scientists (but not all) perform operations on the expression inside parentheses over reading the expression left to right.

Who is right? No one. It’s an ambiguous expression that needs more brackets!

References

  • Angel, Allen R.; Runde, Dennis C.; Gilligan, Lawrence; Semmler, Richard (2010). Elementary Algebra for College Students (8th ed.). Prentice Hall. ISBN 978-0-321-62093-4.
  • Lennes, N. J. (1917). “Discussions: Relating to the Order of Operations in Algebra”. The American Mathematical Monthly. 24 (2): 93–95. doi:10.2307/2972726
  • Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010). NIST Handbook of Mathematical Functions. National Institute of Standards and Technology. ISBN 978-0-521-19225-5.
  • Strogatz, Steven (2019). “The Math Equation That Tried to Stump the Internet“. The New York Times.
  • Swokowski, Earl William (1978). Fundamentals of Algebra and Trigonometry (4th ed.). Boston: Prindle, Weber & Schmidt. ISBN 0-87150-252-6.