🔢 Scientific calculator — brackets, powers and angles
Everything the basic calculator does, plus brackets, powers and roots, logarithms, trigonometry, factorials and a memory. Angles can be in degrees or radians, which is the one setting on this pad that changes what an answer means rather than how big it is — so it says which one is in use on the display, where you can see it.
The calculator
0
0
DEG
The keyboard works as well: the digits, + − * / ^ % and brackets, Enter for =, Backspace to take a key back, Esc to clear. 2nd swaps the eight function keys that have an inverse and says so on the display. The % key divides by 100 — for percentage questions the percentage calculator is the better tool.
The answer
The sum is read as written — brackets, then powers, then × ÷, then + − — and worked out in double precision, about fifteen to seventeen significant digits, with the display rounded to fifteen of them: 0.1 + 0.2 reads as 0.3 rather than as 0.30000000000000004, because 0.1 has no exact form in binary. A number too big or too small for digits comes out in the e form, where 1.234e+21 means 1.234 × 10²¹. In degrees the quarter turns are answered exactly, so sin(180) is 0; in radians the same sum shows 1.22464679914735e-16, because π itself is not exactly representable — the questions above are about that. The % key divides by 100. Nothing is sent anywhere: the sum, the memory and the last answer live in this tab, and closing it takes them with it.
Why use this calculator?
For the sums a basic pad cannot hold: a bracket, a power, a root that is not a square, a sine, a logarithm, a factorial. Those are the keys that make a scientific calculator worth the extra space, and they all behave the way the notation does — so what you would write on paper is what you type here. The one setting worth knowing about is the angle: degrees or radians, because sin(30) is 0.5 in one and −0.988 in the other, and a calculator that does not say which it is in is a calculator you cannot trust.
Before the keypad
Three things worth knowing about the machine in your hand:
- The slide rule did this job for 350 years. William Oughtred worked out the scales in 1622, and engineers carried one until the 1970s — the Apollo missions were planned with them, and the last ones were made the same decade the pocket calculator arrived.
- The HP-35 was the first pocket scientific calculator, in 1972, and it cost $395 — which is why it is called the slide rule killer. The guidance computer that landed Apollo 11 had about 4KB of memory; the calculator app on a phone has millions of times more.
- Degrees are a human choice, radians are a mathematical one. A radian is the angle where the arc equals the radius, and 180° is exactly π radians. Calculus only works with radians — the rule that the slope of sin is cos is true in radians and false in degrees — which is why this page keeps the mode on the display.
How does this calculator work?
The sum is read as written, in the order the notation uses:
brackets, then powers, then × ÷, then + −, with powers grouping to the right
- Everything the basic calculator does applies here: 2+3×4 is 14, and −2² is −4 while (−2)² is 4. Powers group to the right, so 2^3^2 is 512, and 2^−1 is 0.5.
- A function takes what is in the bracket after it, and the keypad writes that bracket for you. Typing √2 works too — with no bracket, a function takes the number straight after it.
- The times can be left out before a bracket, a constant or a function: 2(3+4) is 14 and 2π is 6.28318530717959.
- A key that works on a number works on what is on the screen. A single number takes the key straight after it — 5 then x² is 5² — and a longer sum gets brackets around it, so 8+2 then x² is (8+2)² rather than the 8+2² that the digits alone would say. The same goes for sin, ln, √ and 1/x pressed after a number: 30 then sin is sin(30), and 8+2 then √ is √(8+2).
- Every function key also works the other way round, which is how a calculator is usually used: press sin first and it writes its bracket open and waits for the number.
- The postfix keys work on what is already there: x² and x³ square and cube it, n! takes the factorial, and % divides by 100.
- n! of anything that is not a whole number comes from the gamma function, which is the way mathematics extends the factorial: 0.5! is √π ÷ 2, or 0.886226925452758. 170! is the last one a double can hold; 171! says it is too big rather than pretending.
- 2nd swaps the eight keys that have an inverse: sin, cos and tan become sin⁻¹, cos⁻¹ and tan⁻¹, ln becomes eˣ, log becomes 10ˣ, x² becomes x³, √ becomes ∛, and xʸ becomes the yth root — which writes x^(1/y), so 8 with the yth-root key and 3 gives 8^(1/3) = 2.
- In degree mode the quarter turns are answered exactly, so sin(180) is 0 and cos(90) is 0. In radian mode nothing is exact, because π cannot be written down exactly in binary: sin(π) comes back as 1.22464679914735e-16 rather than 0.
- MS stores the figure on the display, MR puts it back into the sum, M+ and M− add it to or take it from the memory, and MC clears it — with the M flag on the display while something is in there. ANS puts the last answer into the sum, and EXP writes the e of 2.5e6, the way a calculator does.
- Nothing is stored anywhere but this tab: the memory and the last answer go when the page does.
Frequently asked questions
When should I use radians instead of degrees?
Degrees for anything you would say out loud — a roof pitch, a ramp, a bearing. Radians for mathematics and physics, because every rule of calculus involving angles is written for them: the slope of sin is cos in radians and something with a π in it in degrees. The flag on the display says which one is in use, and it is always visible while you type, because the same sum means two different things in the two modes.
How do I take a root that is not square or cube?
The yth-root key, which is xʸ after 2nd: the yth root of x is x^(1/y), and that is exactly what the key writes. The fifth root of 32 is 32^(1/5) = 2, and the cube root of 27 is 27^(1/3) = 3. The ordinary √ and ∛ keys are shortcuts for the two cases worth their own key — and ∛ knows that the cube root of −8 is −2, which the generic power does not.
What is 4.5!?
It is what the factorial means when the number is not whole: 52.3427777845535. The factorial of a whole number is a product — 4! is 4×3×2×1 — and there is no such product for 4.5, so mathematics extends it with the gamma function, which agrees with the factorial at every whole number and keeps going in between. 0.5! comes out as √π ÷ 2, which is the neatest thing on this page. Above 170! there is no room left in a double, so the page says the number is too big rather than showing infinity.
Why is sin(180) exactly 0 but sin(π) not?
Because 180 in degree mode is a whole number the page can check for, and π is not a number anyone can write down exactly. In degrees the quarter turns are answered exactly — sin(180), cos(90), sin(90) — while in radians π is rounded to the nearest double before the sine is taken, so sin(π) comes out as 1.22464679914735e-16 rather than 0. That is not an error in the arithmetic; it is what the arithmetic is being asked to do.
What do the memory keys keep, and where?
One number, in the page: MS stores the figure on the display, MR puts it back into the sum, M+ and M− add it to or take it from what is stored, and MC clears it. The M flag on the display lights while something is in there so you are never left wondering what a key did. Nothing is saved to the device and nothing is sent anywhere — closing the tab empties the memory.
What is the difference between e and EXP?
e is Euler's number, 2.71828182845905 — the constant. EXP writes the e of scientific notation into the number being typed, so 2.5 EXP 6 is 2.5e6, which is 2,500,000. They are the same character on the display and the difference is where they sit: digits straight after the e of an exponent belong to the exponent, while an e standing on its own is the constant and is multiplied in.
What does 2nd do?
It swaps the keys that have an inverse, and puts a 2nd flag on the display so the swap is never a secret: sin, cos and tan become their inverses, ln becomes eˣ, log becomes 10ˣ, x² becomes x³, √ becomes ∛ and xʸ becomes the yth root. The keys without an inverse — the digits, the operators, π, n!, the brackets, ANS and EXP — stay where they are.