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🏗️ Engineering & constructionFree
Power from force and velocity
A 50 N force pulling a load along at 4 m/s delivers 200 W, because power is force times velocity when both point the same way.
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🏗️ Engineering & constructionFree
Projectile from height calculator
Throw at 20 m/s and 45 degrees from 10 m up and it lands 49.0821614194 m away after 3.4706329175 seconds under 9.81 m/s².
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🏗️ Engineering & constructionFree
Pulley mechanical advantage calculator
Four rope strands holding a 400 N load give an ideal mechanical advantage of 4 and a pull of just 100 N.
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🏗️ Engineering & constructionFree
Rotational kinetic energy calculator
A flywheel with a moment of inertia of 2 kg·m² spinning at 4 rad/s stores 16 J. Enter the moment of inertia and the angular velocity to get the energy held in…
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🏗️ Engineering & constructionFree
Spring constant calculator
A spring that stretches 0.25 m under a 50 N force has a spring constant of 200 N/m. Enter the force and the extension or compression to get its stiffness, with…
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🏗️ Engineering & constructionFree
Terminal velocity calculator
An 80 kg body with a drag coefficient of 1 and 0.7 m² of area, falling through air of 1.225 kg/m³, settles at 42.776304713 m/s.
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🏗️ Engineering & constructionFree
Torque calculator
A force of 10 N on a 0.5 m lever arm at 90° gives a torque of 5 N·m. Enter force, arm length and angle to get the turning effect for bolts, doors or statics…
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🏗️ Engineering & constructionFree
Work done calculator
Pushing with 10 N through 5 m in the direction of motion does 50 J of work; at 90° the same push does none.
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➗ MathFree
Aliquot sum calculator
The proper divisors of 12 add up to 16, which is more than 12, so 12 is abundant; 6 gives exactly 6 and is perfect.
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➗ MathFree
Amicable numbers checker
220 and 284 are amicable: the proper divisors of 220 add up to 284, and those of 284 add up to 220.
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➗ MathFree
Armstrong number checker
153 is an Armstrong number: it has three digits, and 1 + 125 + 27, the cubes of 1, 5 and 3, add back up to 153.
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➗ MathFree
Automorphic number checker
25 is automorphic because 25 squared is 625, which ends in 25. Enter any non-negative whole number to see its exact square, its digit count and whether the…
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➗ MathFree
Digit sum calculator
The digits of 12345 add up to 15. Enter any integer, choose a base from 2 to 36 when you need one other than ten, and see every digit together with their total.
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➗ MathFree
Digital root calculator
9875 becomes 29, then 11, then 2: its digital root is 2, reached in 3 steps. Enter any non-negative whole number, even one with up to 1000 digits, and see the…
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➗ MathFree
Number of divisors calculator
12 has 6 divisors: it factors as 2² × 3, so the count is (2+1) × (1+1) = 6. Enter any positive whole number and get its number of divisors together with the…
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➗ MathFree
Sum of divisors calculator
The divisors of 12 add up to 28: 1 + 2 + 3 + 4 + 6 + 12. Enter any positive whole number up to one trillion and get this exact sum with the prime factors used…
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➗ MathFree
Euler's totient φ(n)
Euler's totient of 36 is 12: that many numbers from 1 to 36 share no prime factor with 36. Enter any positive integer up to one trillion and get the exact count…
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➗ MathFree
Extended Euclidean calculator
For 240 and 46 the greatest common divisor is 2, and 240 × (−9) + 46 × 47 = 2. Enter two integers to get their divisor, the two Bézout coefficients and the…
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➗ MathFree
Factorial prime factorization
10! factors as 2⁸ × 3⁴ × 5² × 7, and you get that list without ever computing the full product. Enter n from 0 to 100000 and see every prime up to n with its…
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➗ MathFree
Happy number checker
19 is happy: 1² + 9² gives 82, then 68, then 100, then 1, in four steps. Enter any positive integer to see whether it reaches 1 or falls into the endless…
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➗ MathFree
Jacobi symbol calculator
The Jacobi symbol (5/13) is −1, which tells you 5 is not a square modulo 13. Enter any integer and any odd positive denominator to get −1, 0 or +1 through…
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➗ MathFree
Legendre symbol (a/p)
The Legendre symbol (2/7) is 1, so 2 is a square modulo 7: 3 times 3 is 9, which leaves 2. Enter any integer and odd prime to get 1, −1 or 0 with a…
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➗ MathFree
Mersenne number checker
Enter an exponent and get the full value of 2^p − 1 with a prime verdict: 7 gives 127, a Mersenne prime, while 11 gives 2047, which is 23 × 89.
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➗ MathFree
Modular exponentiation calculator
Raise a number to a power and keep only the remainder: 2 to the 10th modulo 1000 is 24, and 3 to the 5th modulo 13 is 9.
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➗ MathFree
Modular multiplicative inverse
Find the number that undoes a multiplication modulo m: the inverse of 3 modulo 11 is 4, because 3 times 4 leaves the remainder 1.
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➗ MathFree
Next prime after n
Find the smallest prime strictly greater than any whole number: after 10 comes 11, a gap of 1, and after the prime 7 comes 11, not 7 itself.
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➗ MathFree
Pandigital number checker
Check whether a number uses every digit of its base exactly once: 1023456789 is pandigital in base 10, while a number made only of the digits 1 to 9 is not…
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➗ MathFree
Previous prime finder
Finds the largest prime strictly below your number: for 10 the primes below are 2, 3, 5 and 7, so the answer is 7.
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➗ MathFree
Prime counting function π(n)
Counts how many primes are less than or equal to n: π(10) is 4, because the primes are 2, 3, 5 and 7.
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➗ MathFree
Quadratic residue checker
Tells you whether a number has a square root modulo m, and lists every root: 2 is a square modulo 7, because 3 squared is 9, which leaves 2, and the roots are 3…
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➗ MathFree
Radix converter
Rewrites a whole number from any base between 2 and 36 into another: hexadecimal FF is 255 in decimal.
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➗ MathFree
Factorial trailing zeros
100! ends in 24 zeros: 100 holds 20 fives and 4 twenty-fives, and 20 + 4 = 24. Enter a non-negative whole number to get the count of trailing zeros of its…
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➗ MathFree
Twin prime checker
3 and 5 are both prime and differ by two, so they form a twin prime pair, just like 11 and 13. Enter a number to check it against the one two places above, or…
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🔬 ScienceFree
Brewster angle calculator
Finds the polarizing angle for any pair of media: light going from air (index 1) into glass (index 1.5) reflects fully polarized at about 56.3°.
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🔬 ScienceFree
Critical angle calculator
Tells you where total internal reflection begins: light going from glass (index 1.5) towards air (index 1) is fully reflected beyond 41.8103148958°, the arcsine…
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🔬 ScienceFree
Diffraction grating calculator
Find the angle where a grating sends each bright order, or learn that the order cannot exist: with lines 0.000001667 m apart and light of 5.5 × 10⁻⁷ m, order 3…
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🔬 ScienceFree
F-stop calculator
Turn focal length and aperture diameter into the f-number printed on a lens: a 50 mm lens with a 25 mm entrance pupil is f/2, with a relative aperture of 0.5.
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🔬 ScienceFree
Lens magnification calculator
Find out how big the image is and which way up: an object at 30 with its image at 60 gives a magnification of -2, an inverted image twice the object's size.
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🔬 ScienceFree
Lens power calculator
Switch between focal length and diopters in one step: a focal length of 0.5 m is a power of 2 diopters, a converging lens, and the same reciprocal works in…
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🔬 ScienceFree
Mirror equation calculator
Give the focal length and the object distance, and you get the image distance and magnification: an object at 30 in front of a mirror with focal length 10 forms…
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🔬 ScienceFree
Single-slit diffraction calculator
Green light at 500 nm through a 0.1 mm slit puts its first dark fringe at 0.286480091241°. Enter the slit width, wavelength and order to locate any minimum and…
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🔬 ScienceFree
Snell's law calculator
A ray going from air into glass at 30° bends to 19.4712206345°. Enter both refractive indices and the incidence angle measured from the normal to get the…
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🔬 ScienceFree
Telescope magnification calculator
A 1000 mm objective with a 25 mm eyepiece gives 40x. Enter both focal lengths in the same unit to see how much larger a distant object appears, and how each…
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🔬 ScienceFree
Beat frequency calculator
Two tones at 440 and 442 Hz beat twice per second: the beat rate is simply the gap between them. Enter both frequencies to get the beat, their average and which…
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🔬 ScienceFree
Doppler effect calculator
A 440 Hz source approaching at 20 m/s through air at 343 m/s is heard about 6% higher. Add the wave speed and how the source and the observer move to get the…
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🔬 ScienceFree
Doppler shift sound calculator
A 440 Hz siren rushing toward a listener at 20 m/s sounds roughly 6% higher, because its crests arrive packed closer together.
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🔬 ScienceFree
Period to frequency calculator
A period of 0.02 seconds is a frequency of 50 Hz, and the reverse works the same way. Enter either the period or the frequency of any repeating signal and get…
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🔬 ScienceFree
Pipe resonance calculator
A 0.5 m pipe open at both ends, with sound at 343 m/s, resonates at 343 Hz and then at 686, 1029, 1372 and 1715 Hz.
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🔬 ScienceFree
Sound intensity calculator
A 1 W source spreads its sound over a sphere, so 1 m away the intensity is 0.079577471546 W/m². Enter the acoustic power and the distance to get the intensity…
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🔬 ScienceFree
Decibel calculator
An intensity of 0.000001 W/m² is a million times the hearing threshold, which is a level of 60 dB.
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🔬 ScienceFree
Speed of sound calculator
At 20 °C sound crosses dry air at about 343 m/s, compared with 331.3 m/s at 0 °C. Enter the air temperature in Celsius, Kelvin or Fahrenheit to get the speed of…
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🔬 ScienceFree
Standing wave calculator
On a 1 m string vibrating in its second harmonic, the wavelength is 1 m and the nodes sit at 0, 0.5 and 1 m.
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🔬 ScienceFree
Wave speed on a string calculator
Enter the string tension and its linear mass density to get the speed of a transverse wave, v = √(T/μ): 100 N on a string of 0.01 kg/m gives 100 m/s.
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🔬 ScienceFree
Wave speed calculator
Give any two of frequency, wavelength and wave speed and get the missing one from v = fλ: a 50 Hz wave with a 2 m wavelength travels at 100 m/s.
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➗ MathFree
Birthday paradox calculator
In a group of just 23 people, the chance that two share a birthday is already about 50 percent, even with 365 days in the year.
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➗ MathFree
Coin flip probability
Ten fair flips land exactly five heads with probability 0.24609375, because 252 of the 1024 possible sequences have five heads.
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➗ MathFree
Conditional probability calculator
With P(A) of 0.5, P(B) of 0.4 and a joint probability of 0.2, P(A given B) is 0.5 and the events are independent; lower the joint to 0.1 and the result drops to…
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➗ MathFree
Dice sum probability
With two six-sided dice, 7 comes up 6 ways out of 36, a 1 in 6 chance, while 2 and 12 each have just 1 in 36.
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➗ MathFree
Dice roll probability
Rolling exactly 7 with two six-sided dice happens in 6 of 36 outcomes, a 1/6 chance. Pick the number of dice, the sides and your target, then choose exact or…
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➗ MathFree
Expected value calculator
A fair die with faces 1 to 6 has an expected value of 3.5: the long-run average of many rolls, though no single roll shows it.
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