Scientific calculator

Order of operations in this calculator

The calculator reads a whole line before evaluating it, so the order of operations decides the answer. It follows the convention taught as PEMDAS or BODMAS, with a precise rule for each level. From the tightest binding to the loosest:

Take 8/2(2 + 2), a common test of these rules. The bracket is worked out first, giving 8/2 × 4. Division and multiplication share a level, so they run left to right: 8/2 = 4, then 4 × 4 = 16. If the intended meaning was 8 divided by 2(2 + 2), write 8/(2(2 + 2)), which gives 1.

The same left-to-right rule means 1/2pi is (1/2) × π = 1.5708, not 1/(2π). Type 1/(2pi) for 0.159155. Powers, on the other hand, group from the right: 2^3^2 is 2^9 = 512, not 8² = 64.

A leading minus sign is applied after the power, so -3^2 = −9, matching the algebra convention that −x² means −(x²). To square a negative number, bracket it: (-3)^2 = 9. A minus sign directly after ^ belongs to the exponent, so 2^-1 = 0.5.

Implicit multiplication rules

A multiplication sign can be left out when a number or closing bracket is followed by a name, a function, an opening bracket or a root sign: 2pi, 3(4 + 5), 2 sin(30), (1 + 2)(3 + 4) and 4√9 all multiply. Implicit multiplication has exactly the same priority as ×, which is why 8/2(2 + 2) and 1/2pi come out as above. If you copy an expression written for a reader, add brackets around anything meant as a denominator.

A digit is multiplied in only after a closing bracket or a factorial: (1 + 2)3 = 9 and 4!/(2!2!) = 6. Two bare numbers side by side, as in 2 3, stop with "Expected an operator", because a missing operator is more likely than a product. Letters that are not a constant, function or suffix, such as the x in 2x, give "Unknown name".

A function name without brackets takes only the next number and any power attached to it. In radians, sin 2pi means sin(2) × π = 2.8566, while sin(2pi) is 0; in degrees, sin 30^2 is sin(900°) = 0, not (sin 30°)² = 0.25. The keypad inserts brackets after every function name, which avoids the question.

Three typing details change a result without any error:

• E-notation. 2e3 is 2 × 10³ = 2000 and 2e+1 is 20, but 2e + 1, with spaces, is 2 × e + 1 = 6.4366.
• Commas. Outside brackets, 1,000 is one thousand. Inside a function's brackets a comma separates arguments, so max(1,000, 2) reads as max(1, 0, 2) and returns 2. Type 1000 there.
• Suffixes. k, M, B and T multiply by 10³, 10⁶, 10⁹ and 10¹², and lakh and cr by 10⁵ and 10⁷: 2.5M is 2,500,000 and 3 cr is 30,000,000.

Degrees, radians and gradians

Angles start in degrees. A full turn is 360°, 2π radians or 400 gradians, so each unit converts to the others by a fixed ratio.

The mistake to watch for is a textbook angle in the wrong mode: sin(30) is 0.5 in degrees but −0.988032 in radians, where it means 30 radians. If a trig answer looks wrong, check the mode shown at the top of the display first. Inverse functions answer in the current unit, so atan(1) returns 45 in degrees and 0.785398 in radians.

The functions deg() and rad() convert a number between radians and degrees whatever the mode, but they do not switch the mode. In degree mode, sin(rad(30)) takes the sine of 0.5236 degrees and returns 0.0091384. The graphing calculator always works in radians.

In degree mode, multiples of 90° and the angles whose sine or cosine is ±1/2, or whose tangent is ±1, come out exact: sin(150) is exactly 0.5 and cos(90) is exactly 0. Other angles, such as sin(45) = 0.70710678…, are irrational and carry 50 significant digits. tan(90) has no value and returns an error rather than a huge number.

Exact fractions and 50-digit decimals

Addition, subtraction, multiplication, division, whole-number powers, percentages and repeating decimals keep an exact fraction, as do abs, floor, ceil, round, trunc, sign, min, max, sum, avg and mod. When the answer is not a whole number, the exact fraction appears in front of the decimal, provided its numerator and denominator have 18 digits or fewer between them: 100/7 shows as 100/7 = 14.2857142857.

A root, logarithm, trig function, π, e or a non-whole power switches the calculation to 50 significant digits, and the fraction line disappears even when the true value is rational. sqrt(4)/3 shows 0.666666666667 without 2/3; type 2/3 to keep it exact. The last of the 50 digits can be off by one: sqrt(2)^2 is held as 1.999… with 49 nines, and acos(0.5) in degrees as 59.999…998. The display rounds to 12 significant figures by default, so both read as 2 and 60.

Working in decimal also settles rounding ties the way they look on paper. round(2.675, 2) returns 2.68 here. In binary floating point 2.675 is stored as 2.67499999…, which is why Python's round() documentation notes that the same call gives 2.67 there. Halves round away from zero: round(2.5) is 3 and round(-2.5) is −3.

Function reference

Every function below can be typed; the keypad covers the common ones, with a 2nd key for inverse and hyperbolic trig. Results are for degree mode.

Two rows behave differently from some other tools. The remainder takes the sign of the divisor, so -7 mod 3 is 2, where JavaScript's % operator gives −1. A negative base with a fractional power returns a real result only for odd roots: (-8)^(1/3) is −2, while (-4)^0.5 stops with an error. Square roots of negative numbers belong in the complex number calculator, and the permutations and combinations calculator lists the arrangements behind nCr and nPr.

Ans, memory and the tape

After =, pressing an operator key continues from the answer: the line becomes ans followed by the operator. ans holds 50 significant digits rather than the 12 on the display, but it is a decimal, not a fraction, so exactness ends at =: 1/7, then × 7, displays 1 while holding 0.999…98. Typing 1/7*7 on one line gives exactly 1. Pressing a digit instead starts a new calculation. M+ and M− add the current result to, or subtract it from, a memory value M, and MR inserts M into the line. The tape keeps the last 50 results on this device; tap one to reuse it.