Why Your Calculator Gives a Different Answer: −3², 6÷2(1+2), and Other Order-of-Operations Traps

Two calculators, two answers, same keystrokes. A clear explanation of PEMDAS, implied multiplication, unary minus, and how to type expressions so every calculator agrees.

Type 6÷2(1+2) into two different calculators and you may get 9 on one and 1 on the other. Neither is "broken". They implement different — and both defensible — conventions. If you write firmware, spreadsheets, or exam answers, you need to know exactly where the ambiguity lives.

The baseline: PEMDAS / BODMAS

1. Parentheses (Brackets)
2. Exponents (Orders)
3. Multiplication and Division — equal precedence, left to right
4. Addition and Subtraction — equal precedence, left to right

The most common misreading is treating M before D. Under the standard rule, 8 ÷ 4 × 2 = 4 (left to right), not 1.

Trap 1: −3² is −9, not 9

In standard mathematical notation and in almost every programming language, exponentiation binds tighter than unary minus: −3² = −(3²) = −9. To square a negative number you must write (−3)² = 9.

Excel is the famous exception: =-3^2 returns 9 because Excel gives unary minus higher precedence. If you port a formula from Excel to Python, MATLAB, or a scientific calculator, this silently flips signs.

Trap 2: implied multiplication — 6÷2(1+2)

There are two conventions for a number placed next to a parenthesis:

• Strict PEMDAS: 2(1+2) is just 2 × (1+2), same precedence as ÷, evaluated left to right → 6 ÷ 2 × 3 = 9.
• Implied multiplication binds tighter (PEJMDAS): 2(1+2) is a single term, so 6 ÷ [2(3)] = 1. This is how physicists read 1/2π — nobody means (1/2)·π.

Casio calculators historically returned 1; many TI models return 9; some Casio firmware switched to 9 after complaints, then some switched back. Wolfram Alpha gives 9. There is no authority to appeal to — the expression is badly written. Fix it with parentheses: (6÷2)(1+2) or 6÷(2(1+2)).

Trap 3: stacked exponents are right-associative

2^3^2 means 2^(3^2) = 2^9 = 512, not (2^3)^2 = 64. Python, MATLAB and standard notation agree. Some calculators and Excel evaluate left to right and return 64. Again: parentheses.

Trap 4: the fraction bar is a parenthesis

On paper, a horizontal fraction bar groups everything above and below it. When you flatten it to one line, you must add the parentheses the bar implied:

   a + b
  ───────   →   (a + b) / (c × d)     NOT   a + b / c × d
   c · d

The same applies to square roots and exponents: √(x²+y²) becomes sqrt(x^2 + y^2), and e^(−t/τ) needs exp(-t/tau), not exp(-t)/tau.

Trap 5: percent keys are not arithmetic

On many basic calculators, 200 + 10 % gives 220 (adds 10% of 200), while 200 × 10 % gives 20. Scientific calculators usually treat % as ÷100 instead. When writing a formula, write × 1.10 or × 0.10 and leave the % key alone.

How to write expressions everyone agrees on

1. Parenthesize every negative base before an exponent: (-x)^2.
2. Parenthesize every numerator and denominator that has more than one term.
3. Never rely on implied multiplication next to a division sign; write * explicitly.
4. Parenthesize stacked exponents.
5. Replace % with a decimal factor.

Extra parentheses cost nothing. A wrong sign in a control-loop gain, or a factor of 2π in a resonance frequency, costs a lot more.

Quick self-test