The same expression can give different results on different calculators. Almost always, the reason is how each one defines operator precedence. This guide lays out SciKey's full precedence table and looks closely at three expressions that regularly cause arguments: -2^2, 2^3^2 and 6÷2(1+2).
1. SciKey's precedence (highest first)
| Level | Operation | Associativity | Example |
|---|---|---|---|
| 1 | Brackets, function calls | (1+2), sin(30) | |
| 2 | Postfix ! % ° | Left to right | 5! = 120, 50% = 0.5 |
| 3 | Power ^ | Right to left | 2^3^2 = 512 |
| 4 | Unary - + | -2^2 = −4 | |
| 5 | ×, ÷, mod, implied multiplication | Left to right | 6÷2(1+2) = 9 |
| 6 | +, − | Left to right | 10−4−3 = 3 |
Operators on the same level follow their associativity: 8÷2÷2 is (8÷2)÷2 = 2 and 10−4−3 is (10−4)−3 = 3.
2. Why -2^2 is −4
In mathematics, −2² is read as −(2²) = −4: the power comes first and the minus sign applies to the result. It is the same rule that makes −x² = −4 when x = 2. SciKey ranks unary minus below exponentiation, so -2^2 returns −4. For (−2)², use brackets: (-2)^2 = 4.
A sign in the exponent is allowed: 2^-3 is 2⁻³ = 0.125.
Excel is different
In Microsoft Excel, =-2^2 returns 4. Microsoft's documentation, "Calculation operators and precedence in Excel", lists negation (the minus in −1) above percent (%) and exponentiation (^) in its precedence table, so Excel computes (−2)². If a spreadsheet and a calculator disagree, suspect this. In Excel, =-(2^2) gives −4.
Programming languages vary too. Python evaluates -2**2 as −4 (matching maths convention), while JavaScript considers -2**2 ambiguous and rejects it as a syntax error unless you add brackets.
3. Powers go right to left: 2^3^2 = 512
In mathematics, a tower like 2^3^2 means 2^(3²) = 2⁹ = 512, because the exponent of the exponent is evaluated first. SciKey treats ^ as right-associative, so 2^3^2 = 512. For (2³)² = 64, write (2^3)^2.
Excel's documentation says operators of equal precedence are evaluated left to right, and exponentiation is no exception, so =2^3^2 gives 64 there. Again, brackets make the intent unambiguous everywhere.
4. The 6÷2(1+2) debate
This one circulates online constantly, with camps insisting on 9 or 1.
- Reading as 9: an implied multiplication is just a multiplication. 6÷2×3 = (6÷2)×3 = 9.
- Reading as 1: 2(1+2) is a single unit, like 2a, so it goes first. 6÷(2×3) = 1.
The honest conclusion is not that one side is "wrong" but that the notation is ambiguous. Professional writing avoids it and uses a fraction bar or brackets instead.
SciKey's choice
SciKey gives implied multiplication the same precedence as ordinary multiplication and division and evaluates left to right. So:
6÷2(1+2)= 91/2π= (1/2)×π = 1.57079632679 (also shown as π/2)- For 1/(2π), type
1/(2π)= 0.159154943092
We chose this because a single rule is easier to predict: multiplication and division go in the order written, whether or not the sign is shown.
Other calculators
Some manufacturers and models are designed to evaluate a multiplication with an omitted sign before an ordinary × or ÷, so the same expression can give 1. Behaviour varies by model, so check your exam calculator's manual. The safest approach is never to type an ambiguous expression in the first place.
5. Postfix operators and %
!, % and ° apply only to the value immediately before them.
3!2= 3! × 2 = 12 (implied multiplication)2×3!= 12, while(2×3)!= 720200×15%= 200 × 0.15 = 30100+10%= 100 + 0.1 = 100.1
Watch the last one. Some desk calculators treat 100 + 10 % as "add 10% to 100" (= 110), but SciKey's % simply divides by 100. For a 10% markup, type 100×(1+10%) = 110.
6. mod precedence and sign
mod sits at the multiplication level: 2+7 mod 3 is 2 + (7 mod 3) = 3. The result takes the sign of the divisor: -7 mod 3 = 2 and 7 mod -3 = −2. In -7 mod 3, the unary minus binds to 7 first, because unary signs rank above the multiplication level.
7. Functions, constants and powers
Function calls have the highest precedence, so sin(30)^2 is (sin 30°)² = 0.25, the same as sin²30° in a textbook (DEG). For the sine of 30², put it inside: sin(30^2). Because SciKey requires brackets on functions, an ambiguous input like sin 30^2 is never accepted.
A power after a constant binds tighter than implied multiplication. 2π^2 is 2×(π²) = 19.7392088022, not (2π)² = 39.4784176044. That is also why the surface area of a sphere, 4πr², typed as 4πx^2 works as intended.
8. Remove ambiguity with brackets
More reliable than memorising rules is using brackets generously.
| Intent | Ambiguous input | Clear input |
|---|---|---|
| a/(bc) | a/bc | a/(b×c) |
| (−x)² | -x^2 | (-x)^2 |
| e^(−t/RC) | e^-t/RC | e^(-t/(R×C)) |
| (a+b)/2 | a+b/2 | (a+b)/2 |
The letters a, b, t, R and C in the table are placeholders; in SciKey, substitute numbers or the x, y, z variables (note that R is already taken by the gas constant). The third row's e^-t/RC is evaluated as (e^−t)/RC. Whenever the exponent is an expression, bracket it. Extra brackets never make an answer wrong; missing ones often do.
9. Summary
- Powers before unary minus:
-2^2= −4 (Excel gives 4) - Powers right to left:
2^3^2= 512 - ×, ÷ and implied multiplication share a level, left to right:
6÷2(1+2)= 9 %divides by 100:100+10%= 100.1- When in doubt, add brackets