Order of operations: why 6÷2(1+2) splits the internet

Every so often a short problem goes viral: 6÷2(1+2). Half the comments swear it’s 9, the other half swear it’s 1, and someone always posts a photo of two calculators disagreeing. “Calculator” is one of the most searched words in the US, and part of that curiosity comes from exactly this: if machines don’t make mistakes, how can two of them give different answers?

The answer teaches something important: math notation is a language, and languages run on conventions. Knowing them is what separates getting expressions right from getting them right most of the time.

You read an expression in layers:

1. Parentheses and other grouping symbols, innermost first.
2. Exponents and roots.
3. Multiplication and division, same priority, left to right.
4. Addition and subtraction, same priority, left to right.

Step 3 trips people up. PEMDAS makes it look like M comes before D, but it doesn’t: they share one level, and position decides. The same goes for A and S. That’s why some teachers write it as P-E-(MD)-(AS).

The split comes down to implied multiplication, where the times sign is left out, as in $2(1+2)$ or $2x$. Two conventions are in use:

• Strict convention: implied multiplication is just multiplication. So 6÷2(1+2) = 6÷2×3 = 3×3 = 9. Most phone calculators work this way, and programming languages won’t even accept a missing sign.
• Juxtaposition convention: things written side by side form a block with higher priority. So 2(1+2) = 6 becomes one number, and 6÷6 = 1. Several scientific calculator models and many physics and engineering texts read it this way. Someone who writes $1/2x$ almost always means $1/(2x)$, not $x/2$.

Neither is wrong. They’re different reading rules applied to notation that doesn’t say what the author meant. That’s why working mathematicians never write 6÷2(1+2): they use a stacked fraction.

Let’s solve an unambiguous expression carefully: 18 − 12 ÷ 3 × 2 + 1.

Look at step 2: anyone doing 3 × 2 first would get 12 ÷ 6 = 2 and end at 18 − 2 + 1 = 17, which is wrong. Now compare (18 − 12) ÷ (3 × 2) + 1: the parentheses change everything, giving 6 ÷ 6 + 1 = 2. Same numbers, different grouping, 11 versus 2.

The stakes can be real. Say a hypothetical order costs $40 plus 3 items at $5 each. Typing 40 + 3 × 5 gives $55, the right total; typing (40 + 3) × 5 gives $215.

• Always multiplying before dividing. They share a level; go left to right.
• Always adding before subtracting. 10 − 4 + 2 is 8, not 4.
• Mixing up $-3^2$ and $(-3)^2$. The exponent comes before the sign: $-3^2 = -9$, while $(-3)^2 = 9$.
• Stacked exponents. $2^{3^2}$ is read top-down: $2^9 = 512$, not $8^2 = 64$.
• Trusting the keypad blindly. Before an exam, type 6÷2(1+2) into your calculator. If it says 1, it uses the juxtaposition convention.

If it can be read two ways, add parentheses. Extra parentheses never change the value of a correct expression; they only remove doubt. On paper, prefer a fraction bar to ÷. On a calculator or spreadsheet, type (6/2)(1+2) or 6/(2(1+2)) with an explicit times sign, and every device will agree.