The Order of Operations

The idea in one sentence

An expression with more than one operation would have more than one answer, so mathematics agreed on a reading order — and the agreement is what makes writing expressions down possible at all.

The order

1. Brackets first — anything inside them.
2. Exponents next.
3. Multiplication and division, left to right.
4. Addition and subtraction, left to right.

Read strictly left to right you’d get 20, and that is exactly the ambiguity the rule exists to kill. The acronym for it is PEMDAS in some countries and BODMAS in others; they say the same thing, and both are memory aids for a convention rather than a law about numbers.

The part the acronym gets wrong

PEMDAS looks like six steps. It is four, because multiplication and division share a step, and addition and subtraction share a step. Within a shared step you work left to right, not letters-first.

Doing the multiplication first because “M comes before D” gives 1, which is wrong. The same trap sits in \(10 - 3 + 2\): left to right gives 9, addition-first gives 5.

This is the error that survives longest, because the acronym actively encourages it.

Why this particular order

It isn’t arbitrary. Multiplication is repeated addition, and exponents are repeated multiplication — each level is a shorthand for many of the level below it. Resolving the compressed operations first unpacks the expression in the order it was built.

\(3 + 2 \times 5\) means three plus five twos. Doing the multiplication first is just counting those five twos before adding the three.

Brackets sit above all of it because they are the override: they let the writer say “this part is one quantity, handle it first,” which is how you’d write three plus two, all times five, when that’s what you meant.

When in doubt, bracket it

Nobody will thank you for a technically correct expression that takes three readings. If you mean \((2 + 3) \times 4\), write the brackets even when the order rules would have got you there anyway. Brackets cost nothing and remove all doubt — which is also what a fraction bar does, since everything above it is grouped and everything below it is grouped without any brackets being written.

Common mix-ups

• Multiplication before division, always. They rank equally; go left to right.
• Addition before subtraction, same error one level down.
• Treating a minus sign as part of the next number when it’s a subtraction.
• Forgetting an exponent applies only to what it touches. \(-3^2\) is \(-9\), because the exponent attaches to the 3. \((-3)^2\) is 9.

Check yourself

Evaluate \(18 \div 3 \times 2 + 4\) and \((5 + 3)^2 \div 4\).

First: \(18 \div 3 = 6\), then \(6 \times 2 = 12\), then \(+4 = 16\). Second: the bracket gives 8, squaring gives 64, dividing gives 16.