Why fixed rules exist
A calculation such as 8 + 2 × 5 can be read in two ways: as (8 + 2) × 5 = 50 or as 8 + (2 × 5) = 18. So that everyone arrives at the same result, there is a binding order of operations. Multiplication and division are carried out before addition and subtraction. The correct result is therefore 18.
The complete order
The full order is: first, brackets are calculated, from the inside out. Then come powers and roots. After that, multiplication and division, and finally addition and subtraction. Common memory aids are PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) and BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction). In German schools the short form is "Punkt vor Strich", literally "dot before dash".
An example with all levels: 2 + 3 × (4 + 1)² − 6 ÷ 2. First the bracket: 4 + 1 = 5. Then the power: 5² = 25. Then multiplication and division: 3 × 25 = 75 and 6 ÷ 2 = 3. Finally addition and subtraction: 2 + 75 − 3 = 74. If you type the calculation exactly like this into the calculator, you will also get 74.
Same level: left to right
If several operations of the same level follow one another, calculation runs from left to right. 20 − 5 − 3 gives 12, because 20 − 5 = 15 is calculated first and then 15 − 3 = 12. Starting from the right would give 20 − 2 = 18, which is wrong. The same applies to division: 100 ÷ 10 ÷ 2 is 5, not 20.
Mixed multiplication and division work the same way: 12 ÷ 3 × 2 gives 8. First 12 ÷ 3 = 4, then 4 × 2 = 8. Many people intuitively calculate 3 × 2 = 6 and then 12 ÷ 6 = 2. According to the rules that is not correct. If that is what you mean, use brackets: 12 ÷ (3 × 2).
The exception: power towers
Powers are the only operation grouped from right to left. 2^3^2 means 2^(3^2) = 2^9 = 512, not (2^3)^2 = 64. The reason: (a^b)^c can be written as a^(b × c) anyway, so the tower would otherwise be redundant. The online calculator follows this convention.
The minus sign as a sign
Take particular care with the sign of a number. −3² means −(3²) = −9, because the power binds more strongly than the minus. If the negative number is to be squared, it belongs in brackets: (−3)² = 9. Many students stumble at exactly this point. So on a calculator, too, you should set brackets deliberately.
Implicit multiplication
In formulas the multiplication sign is often omitted: 2π, 3a or 4(x + 1). The calculator understands this notation and treats it like a normal multiplication. 3(4 + 1) gives 15. A well-known viral example is 6 ÷ 2(1 + 2). If implicit multiplication is calculated from left to right like any other multiplication, the result is 6 ÷ 2 × 3 = 9. Some textbooks, however, treat it as binding more strongly and arrive at 1. The solution is simple: write unambiguously, i.e. (6 ÷ 2) × 3 or 6 ÷ (2 × 3).
Typical everyday pitfalls
The most common mistake concerns averages and fractions. If you want the mean of 12, 15 and 18, you have to enter (12 + 15 + 18) ÷ 3. Without brackets, only the 18 is divided by 3 and the result is 33 instead of 15. It is similar with discounted prices: (49.90 + 29.90) × 0.8 gives the total after a 20% discount. Without the bracket, only the second item is reduced.
A second classic arises with simple desk calculators that use immediate execution. They ignore the order of operations and return 20 for 2 + 3 × 4. A formula-based calculator such as this one evaluates the whole expression and correctly arrives at 14.
Tips for error-free calculating
When in doubt, use one bracket too many rather than too few; superfluous brackets do not change the result. Check the preview in the display before pressing Enter. Estimate the result roughly in your head. And when fractions are involved, the fraction calculator helps: it works with numerator and denominator and reduces the result automatically.