PEMDAS & BODMAS

The mnemonic device PEMDAS and BODMAS are used to remember the order of operations for elementary functions in mathematics.

PEMDAS stands for:

• Parenthesis
• Exponentiation
• Multiplication
• Division
• Addition
• Subtraction

BODMAS stands for:

• Brackets
• Orders (Exponents)
• Division
• Multiplication
• Addition
• Subtraction

Because [multiplication and division] and [addition and subtraction] are commutative sets (meaning they can be done in any order), some teachers encourage the acronym to be written as such:

P
E
MD
AS

Phrases¶

There exist some common phrases used to help remember order of operations:

• Please Excuse My Dear Aunt Sally
• Purple Elephants Might Destroy A School

Examples¶

Lets look at some examples of how an order of operations acronym is used with the following example:

$$10-5*2 + (2 +4) +2^3 +1 \div 3$$

If we were to blindly execute these operations from left to right (ignoring parenthesis), we would get the incorrect answer of 24 (or possibly $1944 \tfrac{1}{3}$). Let us again calculate this expression by following the steps of PEMDAS.

1. Parenthesis, execute all steps inside parenthesis

   $10-5*2 + \mathbf{6} +2^3 +1 \div 3$

2. Exponentiation, evaluate all exponents

   $10-5*2 + 6 + \mathbf{8} +1 \div 3$

3. Multiply & Divide

   $10- \mathbf{10} + 6 + 8 + \mathbf{\tfrac{1}{3}}$

4. Add & Subtract

   $14\tfrac{1}{3}$

When solving an expression with PEMDAS it is important to recognise that the first P (or B in BODMAS) is recursive, meaning that when you solve an expression inside of brackets that you must continue to apply the PEMDAS order of operations.

Incorrect, does not follow recursive order of operations:

$2*1+(2+3*4) \neq 22$

Correct, follows recursive order of operations:

$2*1+(2+3*4)=16$