6÷2(1+2) = 1 or 9? Order of operations on a calculator
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Why 6÷2(1+2) gives 1 on some calculators and 9 on others, and the exact order of operations a scientific calculator uses: brackets, functions, powers, implied multiplication, then × ÷ and + −.
On this calculator it is 1, because the implied multiplication 2(1+2) is grouped first, making it 6÷6. If you apply PEMDAS strictly left to right for × and ÷, you get 9. When an expression can be read two ways, add brackets so it can only mean one thing.
Why there are two answers
PEMDAS (or BODMAS) says multiplication and division have equal rank and go left to right. Read that way, 6÷2(1+2) is 6÷2 = 3 first, then 3×3 = 9.
But many people, textbooks and calculators treat a number written right against a bracket, like 2(1+2), as one unit. It is the same reason 1/2x is usually read as 1/(2x). Read that way, 6÷(2×3) = 1.
Neither reading is a math error; they are two writing conventions. That is why calculators differ by brand and model: some give 1 and some give 9. For example, the calculation priority table in the Casio fx-570ES PLUS / fx-991ES PLUS manual puts “multiplication where the multiplication sign is omitted” above multiplication and division. This calculator follows the same rule.
Try 6÷2(1+2) ▸ Try 6÷2×(1+2) ▸
The order this calculator uses
Rows higher in the table are worked out first; within a row, left to right.
| Step | What | Example |
|---|---|---|
| 1 | Brackets, fraction boxes, root boxes, functions | (1+2), □/□, √□, sin(30) |
| 2 | Signs written after a number (x², x! keys) | x², x⁻¹, x!, %, °′″ |
| 3 | Powers in an exponent box (x□ key) | 2³ |
| 4 | Leading minus sign | −2² = −(2²) = −4 |
| 5 | Implied multiplication | 2π, 2(1+2), 3sin(30) |
| 6 | Permutations and combinations | 5P2, 5C2 |
| 7 | Multiply and divide | 6÷2×3 = 9 |
| 8 | Add and subtract | 1−2+3 = 2 |
Examples that trip people up
| Expression | This calculator | Why |
|---|---|---|
| 6÷2(1+2) | 1 | Implied product 2(1+2) first |
| 6÷2×(1+2) | 9 | With an explicit ×, left to right |
| −2² | −4 | Square first, then the sign |
| (−2)² | 4 | The bracket squares −2 |
| 1÷2π | ≈ 0.159 | Same as 1÷(2π). With π in the denominator the answer is shown as a decimal |
| 3sin(30), DEG | 3/2 | sin 30° = 1/2, times 3 |
Spreadsheets differ again: Excel evaluates =-2^2 as 4 because it applies the sign first. The same expression can mean different things in different programs.
Powers of powers
With natural display, an exponent sits in its own small box. Press x□ again inside the exponent box and you get a tower: 232 = 2⁹ = 512. Leave the exponent box with ▶ and press x² instead, and you get (2³)² = 64. Because you can see the boxes, there is little room for confusion.
How to avoid ambiguity
- Use the fraction key □/□ for division, so you can see exactly what is in the denominator. 6 over 2(1+2) gives 1; the fraction 6/2 times (1+2) gives 9.
- When you send an expression to someone else, write 6÷(2(1+2)) with the extra bracket.
- Spreadsheets and most programming languages have no implied multiplication, so you must type
6/2*(1+2), which evaluates left to right to 9.
Found a mistake, or a rule that has changed? Email contact@lumenlab.page. We check it, fix it and put the new date on this guide.