Scientific Calculator

What you can work out here

Type or tap a whole expression, such as a formula from homework or a lab sheet, and the answer appears as you type, in textbook order of operations. It covers trigonometry in degrees or radians, logarithms, powers and roots, factorials, π and e, parentheses and implicit multiplication (2π). Beside the answer it shows how the expression was read, with parentheses around every operation, so you can check that it did what you meant before you copy the number.

How to use the scientific calculator (keyboard and keypad)

• Expression: type it or tap the keys. Function names need parentheses: sin(30), log(1000), sqrt(2). A missing ) at the end is added for you. On the example the page opens with, a number key starts your own expression.
• Angle unit: Degrees or Radians, shown as DEG or RAD on the display. Alt + A switches it from the keyboard.
• = or Enter finishes a calculation: the result becomes Ans and joins the History list (the last 10). Right after =, an operator such as × continues from the answer (Ans×), while a number or function starts a new expression.
• 2nd switches the next key to its partner: sin to sin⁻¹, ln to eˣ, log to 10ˣ, √ to ∛, xʸ to ʸ√x, x² to x³, n! to nCr.
• Keyboard: ^ for powers, ! for factorials, * and / for × and ÷, pi, e, Ans, asin, ln, log2, cbrt. Esc clears the expression. You can paste a whole expression.

The result updates as you type. The Try buttons load six cases explained on this page: sin(30) in each angle unit, −2^2, 2^3^2, 20! and a base-2 log.

Worked example: how far a ball launched at 35° travels

A ball leaves level ground at 12 m/s, 35° above the horizontal. Ignoring air resistance, its range is R = v² sin(2θ) ÷ g, with g = 9.81 m/s². Type 12^2 sin(2×35) ÷ 9.81 with the angle unit on Degrees:

1. Read as ((12^2) × sin(2 × 35)) ÷ 9.81: the power first, then the multiplications and the division from left to right.
2. 12^2 = 144 and 2 × 35 = 70.
3. sin(70°) = 0.939692620786.
4. 144 × 0.939692620786 = 135.315737393.
5. 135.315737393 ÷ 9.81 = 13.7936531492, so the ball lands about 13.79 m away.

With the angle unit left on Radians, the same keys give 11.3598632155, because sin(70) then means the sine of 70 radians. Both look plausible, which is why the unit is shown on the display and next to the result.

Degrees or radians: which mode do I need?

Use degrees when the angle is written with a degree sign (35°, a 12° roof pitch). Use radians when it is written with π (π/6) or comes from calculus and physics formulas that measure angles as arc length. The two are linked by degrees ÷ 180 = radians ÷ π, so 30° is π/6 radians.

Why sin(30) sometimes shows −0.988

Because the calculator is in radians: sin(30) is then the sine of 30 radians, −0.988031624093, not of 30°, which is 0.5. Here a warning appears when an expression looks written for the other unit: sin(30) in radians, or π inside sin, cos or tan in degrees. Its Switch to degrees button changes the unit only when you press it. Whenever an expression uses angles, a tile also shows its value in the other unit.

Order of operations: why −2² = −4 and 2^3^2 = 512

Parentheses come first, then powers, then × and ÷ from left to right, then + and − from left to right. Three rules decide the cases calculators disagree on:

• The sign comes after the power. −2^2 means −(2^2) = −4. To square −2, type (−2)^2, which is 4. Spreadsheets such as Excel apply the sign first and return 4 for −2^2.
• Stacked powers work from the top down. 2^3^2 means 2^(3^2) = 2^9 = 512, not (2^3)^2 = 64.
• Implicit multiplication is ordinary ×. 2π is 2 × π, and 1/2π is (1 ÷ 2) × π, not 1 ÷ (2π). Add parentheses when you mean the second.

The “Read as” line shows each of these with its parentheses, and the steps under the result list the operations in the order they were done.

log vs. ln: what’s the difference?

Only the base. log is the common logarithm, base 10: log(1000) = 3, because 10³ = 1000. ln is the natural logarithm, base e ≈ 2.71828: ln(1000) = 6.90775527898. log₂ is base 2, and log(x, b) takes any base b: log(8, 2) = 3, because 2³ = 8. For logarithms in more detail, including solving for an exponent, use the Exponent, Root and Logarithm Calculator listed under Related calculators.

Powers, roots and factorials

• Powers: x² and x³ square and cube, xʸ raises to any power (2^10 = 1024), and 1/x is the power −1 (4^(−1) = 0.25).
• Roots: √ is the square root, ∛ the cube root, which also works for negative numbers (∛(−27) = −3). For other roots, ʸ√x raises to the power 1 ÷ y: 32^(1÷5) = 2.
• Factorials: n! multiplies 1 × 2 × … × n, and 0! = 1. It needs a whole number from 0 to 170. 20! = 2.43290200818E+18, and the calculator also writes out all its digits: 2,432,902,008,176,640,000.

Reading scientific notation (what “E” means)

E stands for “times ten to the power”. 2.43290200818E+18 is 2.43290200818 × 10¹⁸, and 1E−7 is 1 × 10⁻⁷ = 0.0000001. The result switches to this form below 0.000001 and from 1E+12 up, where plain digits would run off the display. You can type E yourself: 6.02E23 is 6.02 × 10²³. The EXP key writes the same E after a number, so the power of ten stays part of it: 1 ÷ 2 EXP 3 is 1÷2E3 = 1 ÷ 2000 = 0.0005. The result tiles show the same number in scientific notation (one digit before the point) and in engineering notation, where the power of ten is a multiple of 3.

Why you might see an error (domain, division by zero, overflow)

An error means the expression has no real-number value, or can’t be read as typed. The message names the problem and, when it is one part of a longer expression, the character where it starts; that part is highlighted under the expression.

• Outside a function’s domain: sin⁻¹ and cos⁻¹ need a value from −1 to 1, so asin(2) has no answer. Logarithms need a number above 0, and √ a number of at least 0.
• Division by zero: tan = sin ÷ cos, and cos 90° = 0, so tan(90°) has no value, just like 1 ÷ 0. In radians, tan(π/2) gives a huge number instead of an error, because π/2 can’t be stored exactly.
• Overflow: the largest number the calculator can hold is about 1.8 × 10³⁰⁸. 170! ≈ 7.26 × 10³⁰⁶ fits; 171! ≈ 1.24 × 10³⁰⁹ does not, so it is reported rather than shown as infinity.
• Typing slips: a missing number after an operator, a ) with no (, or sin 30 without parentheses.

How the result is shown

The result shows 12 significant digits. Behind it are two kinds of arithmetic:

• Exact fractions for typed numbers joined by + − × ÷, whole-number powers, ! and %. 0.1 + 0.2 − 0.3 gives 0, where plain binary arithmetic gives 5.55111512313E−17 because 0.1 can’t be stored exactly in binary. Rational results also show as a fraction: 1/3 + 1/6 = 1/2, and 7 ÷ 3 = 7/3 = 2 1/3, the decimal 2.(3) with the 3 repeating.
• Double precision (about 16 significant digits) for π, e, Ans and every function. When a result is only rounding left over from these, it is shown as 0 and the note says so: sin(π) in radians computes to 1.22464679915E−16 because the stored π is not exactly π.

The steps under the result list every operation with its value, and the last line gives the full stored value.

Assumptions and limitations

• Real numbers only. There are no complex results, so √(−4) is an error rather than 2i.
• Degree mode is exact at multiples of 30° and 45°; other angles, and every radian value, carry up to about 16 significant digits.
• Close to a zero of the answer, fewer of the 12 digits shown are reliable. The sine of an angle typed just short of π, or exp(1E−15) − 1, is tiny because almost all its digits cancel, and what is left carries the rounding of the 16-digit values it came from, sometimes from the first digit on. Treat such a result as approximate.
• Numbers beyond about 1.8 × 10³⁰⁸, or closer to zero than about 2.2 × 10⁻³⁰⁸, are reported instead of shown.
• There are no statistics, unit conversions, graphs or equation solving here. For quadratic equations, use the Quadratic Equation Solver listed under Related calculators.

Common mistakes

• Wrong angle unit. An answer of −0.988 for sin(30) means radians; check DEG or RAD before copying a trig result.
• Squaring a negative without parentheses. −3^2 is −9; (−3)^2 is 9.
• Dividing by a product without parentheses. 1/2π is π/2. For 1 ÷ (2π), type 1/(2π).
• Using log for ln. Formulas with e, such as continuous growth, need ln.
• Copying all 12 digits into a final answer. Round to the precision of your measurements: the ball’s 13.7936531492 m is better given as 13.79 m.