# Order of operations

There are quite a few mnemonics for the order of operations. A common one in the UK is BODMAS:

• Brackets
• Orders
• Division
• Multiplication
• Subtraction

Elsewhere, PEMDAS is popular.

But division and multiplication have equal precedence, and so do addition and subtraction. A common convention is that operations with equal precedence are evaluated from left to right.

This leads to all sorts of misunderstandings.

$$8 \div 2(1+3) = 16$$ or $$1$$

Some people think that the presence or omission of a multiplication symbol in the above expression is important: implicit multiplication might bind more tightly than the division symbol.

Several ways of resolving the ambiguity have been suggested, but all the ones I've seen introduce other problems.

## Suggested resolutions

Reverse Polish Notation: The expression $$(x-2)(x-1)$$ would be written instead $$\times \, - \, x \, 2 \, - \, x \, 1$$, or something like that. There's no need for brackets or operator precedence, but it is hard to see at a glance what each operator applies to.

Make up a new rule: At MathsJam Gathering 2020, Christian Lawson-Perfect suggested adding a rule "M before D except after 3". So $$6 \div 2 \times 3 = 1$$, but $$3 \div 2 \times 4 = 6$$.