Order of Operations (BIDMAS and PEMDAS) Explained

Why $2 + 3 \times 4$ is $14$ and not $20$ , what BIDMAS and PEMDAS really mean, and the left-to-right rule that catches almost everyone out.

3 min read Supports 2 lessons

Work out $2 + 3 \times 4$ . If you go left to right you get $5 \times 4 = 20$ . If you multiply first you get $2 + 12 = 14$ . Both feel reasonable, but a calculation can only have one answer, so mathematicians agree on an order.

That agreement is the order of operations. It is a convention, like driving on one side of the road. It works because everyone follows it.

Order of operations

(\;)\;\to\; x^n \;\to\; \times\,\div \;\to\; +\,-

Brackets, then powers, then multiplication and division, then addition and subtraction.

BIDMAS, BODMAS and PEMDAS

Different countries teach different memory aids, but they all describe the same order:

BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction.

BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction.

PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Indices, orders and exponents all mean powers, like the $2$ in $3^2$ .

The rule people miss: left to right

The acronyms make it look as if division comes before multiplication (BIDMAS) or multiplication before division (PEMDAS). Neither is true. Multiplication and division share a rank, and so do addition and subtraction. Within a rank you work from left to right.

$12 \div 3 \times 2$
Division and multiplication have equal rank.

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Watch out

The same applies to subtraction: $10 - 4 + 3 = 9$ , not $3$ . Go left to right.

A full example

Worked example

Evaluate $(8 - 2)^2 \div 4 \times 3 + 1$ .

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Brackets first.
$6^2 \div 4 \times 3 + 1$

Hidden brackets

Some notation carries brackets you cannot see. A fraction bar groups everything above it and everything below it:

\frac{6 + 4}{2} = \frac{10}{2} = 5 \qquad\text{not}\qquad 6 + \frac{4}{2} = 8

A square root sign does the same to everything under it, and an exponent applies to everything written in its superscript.

Watch out

$-3^2 = -9$ , because the power is applied before the minus sign. To square negative three you need brackets: $(-3)^2 = 9$ .

Tip

When you write your own calculations, add brackets whenever there is any doubt. Extra brackets never change a correct answer, and they make your intention clear.

Common questions

Do you always do division before multiplication?: No. Multiplication and division have equal priority, so you work through them from left to right in the order they appear. The same goes for addition and subtraction.
Is BIDMAS the same as PEMDAS?: Yes. They are two memory aids for the same rule. Indices and exponents both mean powers, and brackets and parentheses are the same thing.

Practise order of operations

The Order of Operations lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Do you always do division before multiplication?

No. Multiplication and division have equal priority, so you work through them from left to right in the order they appear. The same goes for addition and subtraction.

Is BIDMAS the same as PEMDAS?

Yes. They are two memory aids for the same rule. Indices and exponents both mean powers, and brackets and parentheses are the same thing.