Why Calculators Disagree: Order of Operations Explained

Two calculators can show different answers for the same keystrokes because they apply different rules about which operation comes first. A scientific calculator that reads a whole expression follows standard precedence: parentheses, then powers, then multiplication and division, then addition and subtraction. Once you know the few places where those rules surprise people, you can type any expression and trust the result.

What order does a scientific calculator use?

The Scientific Calculator evaluates expressions with these levels, from strongest to weakest:

Level	Operations	Groups
1	Parentheses and functions such as sin( ), log( ), sqrt( )	inside out
2	Powers ^	right to left
3	A leading minus sign (negation)
4	×, ÷ and mod	left to right
5	+ and −	left to right

The postfix operators ! (factorial) and % apply directly to the number in front of them before anything else touches it.

Five expressions that trip people up

1. Stacked powers: 2^3^2 = 512, not 64. Powers group from the right, so this means 2^(3^2) = 2^9 = 512. If you want 64, write (2^3)^2. This matches the convention in mathematics, where a tower of exponents is read from the top down.

2. Negative squares: -2^2 = −4. The power is applied first, then the minus sign: −(2²). To square negative two, type (-2)^2, which gives 4. This one matters in physics and statistics formulas where a squared term is supposed to stay positive.

3. Implicit multiplication: 6/2(1+2) = 9. The calculator inserts a multiplication sign between a number and a parenthesis, so 2π and 3(4+1) work as you would write them on paper. That makes 6/2(1+2) the same as 6 ÷ 2 × 3, which is evaluated left to right: 3 × 3 = 9. Some textbooks treat implied multiplication as binding tighter and would get 1. The safe habit is to add parentheses so nobody has to guess: 6/(2(1+2)).

4. Percent: 200*15% = 30, but 200+15% = 200.15. Here % simply divides the number before it by 100, so 15% is 0.15. Multiplying works as expected. Adding does not do the "plus 15 percent" shortcut that some pocket calculators have. For a price increase, type 200*(1+15%) or 200*1.15, both of which give 230. For everyday percentage questions, the Percentage Calculator labels each case for you.

5. Chained division: 8/2/2 = 2. Division groups from the left: (8 ÷ 2) ÷ 2. It is not 8 ÷ (2 ÷ 2) = 8.

Degrees or radians?

Trigonometry depends on the angle unit, and this is the single most common source of "wrong" answers. In DEG mode, sin(30) is 0.5. Switch to RAD and the same keystrokes return −0.98803162409286, because 30 radians is a very different angle. The toggle sits in the top-left corner of the display.

Use degrees for geometry, construction and navigation. Use radians for calculus, physics formulas and anything involving π directly, such as sin(pi/6), which is 0.5 in RAD mode. If you need to convert an angle on its own, the Angle Converter does it.

The inverse functions sin⁻¹, cos⁻¹ and tan⁻¹ answer in the current unit too: asin(0.5) returns 30 in DEG mode.

Why some inputs show an error

The calculator reports an error instead of printing a misleading number:

• tan(90) in degrees: tangent is undefined there, because cos(90°) is zero.
• sqrt(-4): there is no real square root of a negative number. cbrt(-8) works and gives −2.
• log(0) or ln(-1): logarithms need a positive input.
• 1/0: division by zero.
• sin 30: functions need parentheses, so type sin(30).

Very large results switch to scientific notation. 170! is shown as 7.2574156153e+306, and 171! exceeds what the number format can hold, so it shows ∞.

Small details that save time

• log is base 10 and ln is base e: log(1000) = 3 and ln(e) = 1. There is also log2( ) if you type it: log2(1024) = 10.
• Results are rounded to 14 significant digits, so 0.1+0.2 shows a clean 0.3 instead of floating-point noise.
• mod gives a non-negative remainder for a positive divisor: 17 mod 5 = 2 and -7 mod 3 = 2.
• Extra functions you can type include abs, exp, floor, ceil, round, min, max, nroot(x, n), and the constants tau and phi.

How to use the calculator step by step

1. Type an expression in the display, such as sqrt(3^2+4^2), or tap the buttons. A live preview of the result appears as you type.
2. Check DEG or RAD before any trigonometry.
3. Press = or Enter. The result moves into Ans, and the expression and result are added to the History panel.
4. Reuse a result by tapping Ans, or click any line in the history to insert that value. Escape clears the current line; ⌫ deletes one character; AC clears everything.

For example, compound growth of 1,000 at 5% for 10 years is 1000*(1+0.05)^10, which returns 1628.8946267774. Expressions are parsed by the page itself rather than executed as code, and nothing you type is sent anywhere.

FAQ

Why does my phone calculator give a different answer?

Basic calculators often apply each operation the moment you press it, without precedence. Typing 2 + 3 × 4 on one of those can give 20 instead of 14. A scientific calculator that sees the whole expression gives 14.

How do I enter a root other than square root?

Use nroot(x, n), for example nroot(27, 3) = 3, or raise to a fractional power: 27^(1/3).

Does the history survive a page reload?

No. The history and Ans value live in the open page only and reset when you reload it.

Try the examples above in the Scientific Calculator.