Order of Operations

1. The ambiguity problem

Look at the expression $2 + 3 \times 4$ and ask a room full of people for its value. Half will say $20$ (they added $2 + 3 = 5$ first, then multiplied by $4$). The other half will say $14$ (they multiplied $3 \times 4 = 12$ first, then added $2$). They're both performing valid arithmetic — they just disagree on what the expression means.

There are only two ways out of this. Either you write parentheses everywhere — $(2 + (3 \times 4))$ — so that the order is forced explicitly, or you agree on a default order that everyone uses when parentheses are absent. Mathematics has overwhelmingly chosen the second route. The default order is what this topic is about.

Two things to keep in mind throughout:

• The rule is a convention. There's no theorem that says multiplication "should" come before addition; the rule could in principle have gone the other way. But it didn't, and the worldwide consensus is real: every textbook, every calculator, every programming language obeys it.
• Parentheses always win. Whatever the rule says about the default order, $(\,\ldots\,)$ overrides it. When in doubt, add parentheses — they cost nothing and they kill ambiguity dead.

2. The convention: PEMDAS

In the English-speaking world, the rule is usually memorized with the mnemonic PEMDAS — "Please Excuse My Dear Aunt Sally." In Britain and the Commonwealth it's BODMAS or BIDMAS. The letters are different but the rule is the same:

Two things make this list less innocent than it looks:

• Multiplication and division occupy the same tier; they're not two separate steps. When both appear, work left to right. Same for addition and subtraction.
• The mnemonic PEMDAS lists "M" before "D" and "A" before "S" purely because PEMDAS is easier to say than PEDMAS or PEDMSA. It does not mean multiplication beats division. This is the single most common misreading of the rule.

Applied to the opening puzzle: $2 + 3 \times 4$. There are no parentheses or exponents. Multiplication (tier 3) is more tightly bound than addition (tier 4), so we do the multiplication first: $3 \times 4 = 12$, and the expression becomes $2 + 12 = 14$. That is the universally agreed answer.

4. Why this order — the principle

You can memorize PEMDAS as raw rule, or you can see the pattern that makes it the natural order. The pattern is short:

Each operation is a compressed version of the one below it. Stronger compression binds tighter.

Walk up the ladder:

• Multiplication is repeated addition. $3 \times 4$ is shorthand for $4 + 4 + 4$. Multiplication is the "compressed" form of addition.
• Exponentiation is repeated multiplication. $3^4$ is shorthand for $3 \times 3 \times 3 \times 3$. Exponentiation is the compressed form of multiplication.

The convention says: do the most-compressed operation first, then unfold one level at a time. When you read $2 + 3 \times 4$, the $\times 4$ is hugging the $3$ — it's a single compact unit that means "three fours." That unit should resolve before the looser $+$ tries to combine it with the $2$.

The same logic explains why exponents come before multiplication. In $2 \times 3^2$, the $3^2$ is the tightest unit; it resolves to $9$ first, and then the $2 \times 9$ happens. Subtraction inherits its tier from addition (it's just adding a negative); division inherits its tier from multiplication (it's just multiplying by a reciprocal) — which is also why each pair lives on the same tier rather than two adjacent ones.

5. Subtleties that catch people out

Left-to-right within a tier

When the same tier appears more than once, work left to right. The classic gotcha is $8 \div 2 \times 4$:

• Reading left to right: $8 \div 2 = 4$, then $4 \times 4 = 16$.
• Reading "M before D": $2 \times 4 = 8$, then $8 \div 8 = 1$. Wrong.

The convention is left-to-right, so the answer is $16$. Anyone who tells you "$1$" has been bitten by the PEMDAS-mnemonic-as-strict-ranking misreading.

The unary minus and exponents

This one is genuinely confusing because it looks like an exception even though it isn't. Compare:

• $-3^2 = -(3^2) = -9$. The exponent binds tighter than the negation, so it acts on the $3$ first, and the negation takes the result.
• $(-3)^2 = 9$. The parentheses force the negation to apply before the squaring, so we're squaring a negative number, which gives a positive.

The takeaway: in $-3^2$, the minus sign is not part of the base. If you want it to be, you have to say so with parentheses. Many calculators get this wrong, and most spreadsheet programs disagree on the convention — yet another reason to lean on explicit parentheses when the stakes are real.

Implicit multiplication

When you write $2(3+1)$ or $2x$, the multiplication is implicit — there's no $\times$ symbol. Strictly by PEMDAS, implicit multiplication should sit at the multiplication tier, no different from $2 \times (3+1)$. In practice, some mathematicians and some calculators treat implicit multiplication as binding tighter than ordinary multiplication or division. That's the source of the periodic viral debate over expressions like $6 \div 2(1+2)$, which gives $9$ under the strict convention and $1$ under the "implicit multiplication binds tighter" reading.

The honest answer is that this case is genuinely ambiguous in the wild — the standard rules don't all agree. Working mathematicians avoid the problem by writing the expression as $\tfrac{6}{2(1+2)}$ or $\tfrac{6}{2} \cdot (1+2)$ — whichever they actually mean — and never relying on the reader to guess.

6. Common pitfalls

7. Worked examples

Try each before opening the solution.