Implicit multiplication: what 2(3 + 4) really means
Implicit multiplication is multiplication written without a × sign. It is harmless on its own, but when a division comes before it, two sensible conventions can give different answers. This guide shows exactly when that happens.
What it is
Writing a number right next to a bracket, as in 2(3 + 4) or (1 + 2)(3 + 4), means multiply. In algebra the same idea appears as 3x. Nobody disputes its value in isolation: 2(3 + 4) is 14 either way.
When it causes trouble
The trouble starts when a division sits directly before it, as in 8 ÷ 2(2 + 2). Two conventions exist.
- Same rank as ×. Treat 2(2 + 2) as 2 × (2 + 2) and go left to right. That gives 8 ÷ 2 × 4 = 16.
- Tighter than ×. Treat 2(2 + 2) as one unit, worked out before the division. That gives 8 ÷ 8 = 1.
Both readings, side by side
These were computed by the same solver you can use on the solver page.
| Expression | Left to right | Implicit first | Same answer? |
|---|---|---|---|
| 2(3 + 4) | 14 | 14 | Yes |
| 3(4 + 5) − 2 | 25 | 25 | Yes |
| 6 ÷ 2(3) | 9 | 1 | No, they differ |
| 10 ÷ 5(2) | 4 | 1 | No, they differ |
| 12 ÷ 3(2 + 2) | 16 | 1 | No, they differ |
| 8 ÷ 2(2 + 2) | 16 | 1 | No, they differ |
| 48 ÷ 2(9 + 3) | 288 | 2 | No, they differ |
Notice the pattern: the two conventions agree whenever no division comes before the bracket, and split whenever one does.
The gap is always a square
When the expression has the form a ÷ b(c), the left-to-right reading is (a ÷ b) × c and the other reading is a ÷ (b × c). Their ratio is c². In the table, 6 ÷ 2(3) gives 9 and 1, a ratio of 3². The 48 ÷ 2(9 + 3) row differs by 12² = 144. The bigger the bracket, the bigger the disagreement.
Where each convention is used
Many school courses teach PEMDAS with all multiplication and division at one rank, so implicit multiplication gets no special treatment. Some scientific and mathematical writing, and some calculators, give it higher priority, because in handwritten algebra a term like 2x behaves as one unit. Neither is a mistake. They are different house styles.
What to do about it
- When you write an expression, add brackets around anything that must stay together, or use a stacked fraction.
- When you read one, look for a ÷ followed by a number touching a bracket. That is the danger sign.
- In an exam, follow your course's convention and show the bracket you assumed.
See it in famous puzzles
Questions
Is implicit multiplication the same as regular multiplication?
In value, yes: 2(3 + 4) and 2 × (3 + 4) are both 14. The only question is how tightly it binds when a division comes before it.
Which convention is correct?
Neither is wrong. School courses that teach PEMDAS usually treat it as ordinary multiplication and go left to right. Some scientific writing and some calculators give it higher priority. When it matters, add brackets.
Does this affect algebra?
Sometimes. An expression like 1/2x can be read as (1/2)x or 1/(2x). Writing the fraction with brackets or a stacked fraction removes the doubt.