Order of Operations: Why 2 + 3 × 4 Is 14

One expression, one meaning

What is $2 + 3 \times 4$?

Working left to right gives $5 \times 4 = 20$. Multiplying first gives $2 + 12 = 14$.

Both are reasonable readings, and that is the problem. An expression has to mean one thing, so mathematicians agreed on an order. The agreed answer is $14$.

The order

Step	Operation
1	parentheses
2	exponents
3	multiplication and division, left to right
4	addition and subtraction, left to right

$$2 + 3 \times 4 = 2 + 12 = 14$$

Steps 3 and 4 each cover two operations of equal rank. That is the part people get wrong, and it has its own section below.

Equal rank means left to right

Multiplication does not outrank division. They are the same step, done in the order they appear.

$$12 \div 3 \times 2$$

Left to right: the division comes first.

$$12 \div 3 = 4 \quad\Rightarrow\quad 4 \times 2 = 8$$

Doing the multiplication first would give $12 \div 6 = 2$, which is wrong.

The same holds for addition and subtraction:

$$10 - 4 + 3 = 6 + 3 = 9$$

Adding first would give $10 - 7 = 3$. Left to right settles it.

Parentheses override the order

Parentheses say do this first, whatever the usual order would be.

$$2 + 3 \times 4 = 14 \qquad (2 + 3) \times 4 = 20$$

Same numbers, same operations, different answers — because the parentheses changed which came first.

That is what parentheses are for. Without them there would be no way to write “add first, then multiply”, and the fixed order would be a restriction rather than a convention.

When parentheses are nested, work from the innermost outward:

$$2 \times \big(3 + (4 - 1)\big) = 2 \times (3 + 3) = 2 \times 6 = 12$$

Some writers use square brackets or braces for the outer layers — $\{$, $[$, $($ — purely so the pairs are easier to match by eye. They all mean the same thing.

Working through a long expression

Take one step at a time and rewrite the whole expression each time. The temptation is to do two things at once, and that is where errors come from.

$$5 + 2 \times (8 - 3)^2 \div 5$$

Parentheses:

$$5 + 2 \times 5^2 \div 5$$

Exponents:

$$5 + 2 \times 25 \div 5$$

Multiplication and division, left to right:

$$5 + 50 \div 5 \quad\Rightarrow\quad 5 + 10$$

Addition:

$$15$$

Why the order is what it is

The convention is not arbitrary, even though it is an agreement.

Multiplication is repeated addition, so it is a shorthand for a group of additions. In $2 + 3 \times 4$, the $3 \times 4$ stands for $4 + 4 + 4$ — a single bundled quantity. Evaluating that bundle before adding it to the $2$ keeps the shorthand meaning what it stands for.

The same logic puts exponents above multiplication. $3 \times 2^4$ has the $2^4$ standing for $2 \times 2 \times 2 \times 2$, one bundled quantity, so it resolves first.

Each level of the order is a shorthand for the level below it, and the shorthand is unpacked before the level below is applied.

Where this is used

Every calculator, spreadsheet and programming language follows this order. A spreadsheet formula =2+3*4 returns $14$, not $20$, and so does every programming language you are likely to meet.

It also underlies algebra. In $3x + 5$, the $3x$ is a single term precisely because multiplication binds tighter than addition. That reading is what makes $3x + 5$ and $3(x + 5)$ different expressions.

Worked examples

Common mistakes

Practice problems