Math
1,470 math calculators: percentages, fractions, equations, geometry, trigonometry, statistics and number theory. Each one shows the method in plain words, so you get the answer and understand how it was reached, free in your browser.
Compare fractions
Find out which of two fractions is larger, exactly: 3/4 beats 2/3, and 1/2 and 2/4 come out equal because they are the same value.
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Complementary angle calculator
Find the angle that completes an acute angle to a right angle, so the complement of 30° is 60° and that of 12.5° is 77.5°.
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Compatible numbers
Estimate a quotient or product the way you would in your head: 417 ÷ 6.1 becomes 420 ÷ 6 = 70, shown next to the exact 68.36, so you can tell at a glance…
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Completing the square
Rewrite ax² + bx + c in vertex form and read the parabola at a glance: 2x² − 8x + 5 becomes 2(x − 2)² − 3, with its vertex at (2, −3).
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Completing the square with worked steps
See every step that turns ax² + bx + c into vertex form: x² + 6x + 2 becomes (x + 3)² − 7, with the vertex at (−3, −7), by halving 6 to 3 and adding and…
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Complex number calculator
Do arithmetic on two complex numbers a + bi in one place: (3 + 2i) times (1 − 4i) gives 11 − 10i. Division uses the exact conjugate formula and refuses a zero…
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Complex number to polar form
Describe a complex number by distance and direction instead of coordinates: 3 + 4i sits at magnitude 5, and 2i points straight up at 90°, with the angle also…
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Complex number to rectangular form
Turn a modulus and an argument back into a + bi: a modulus of 2 at 60 degrees gives a real part of 1 and an imaginary part of about 1.732.
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Complex number to trigonometric form
Rewrite a + bi as r(cos θ + i sin θ), ready to copy: 1 + i has modulus √2 and argument 45°, also given in radians.
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Lucas theorem calculator
Get C(n, k) modulo a prime without huge factorials: C(1000, 200) modulo 7 is 1, found by writing both numbers in base 7 and multiplying small digit binomials.
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Modular nth root calculator
Find every k-th root of a number modulo a prime: the square roots of 2 modulo 17 are 6 and 11, listed in order.
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Multiplicative order calculator
Find the smallest power that brings a number back to 1 modulo n: powers of 2 modulo 9 run 2, 4, 8, 7, 5, 1, so the order is 6.
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Baby-step giant-step calculator for discrete logarithms
2 raised to the 9th power leaves a remainder of 5 when divided by 13, so the discrete logarithm of 5 to base 2 modulo 13 is 9.
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Count primitive roots modulo n
Modulo 18 there are exactly 2 primitive roots, the value of Euler's totient applied twice. Enter a modulus and see how many generators it has, or a clear notice…
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Complex root calculator
Get every nth root of a complex number, not only the principal one: the cube roots of −8 are −2 and 1 ± √3 i, all of magnitude 2, at 60°, 180° and 300°.
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Condition number calculator
See how much a matrix can magnify small errors in your data: rows (2, 1) and (1, 2) give a condition number of 3, the largest singular value divided by the…
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Cone surface area calculator
Get every area of a right circular cone from its radius and slant height: radius 3 and slant height 5 give a lateral area of 15π and a total of 24π, base…
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Idempotents modulo n calculator
Modulo 12 there are four residues that stay the same when squared: 0, 1, 4 and 9. Enter any modulus up to one trillion to get the complete sorted list, plus the…
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Deterministic Miller–Rabin test
97 is prime: it passes all seven fixed bases, 2, 3, 5, 7, 11, 13 and 23, proven to settle every whole number up to 9007199254740991.
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Nth power residue checker
Tells you whether a number is a k-th power modulo a prime: 2 is a square modulo 7, because 2 raised to the power 3 leaves remainder 1.
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Quadratic non-residues list mod p
Lists every remainder that is never a square modulo an odd prime: modulo 7 the squares are 1, 2 and 4, so the non-residues are 3, 5 and 6.
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Solovay-Strassen primality test with a chosen base
For 13 with base 2, both residues come out as 12, so 13 passes and is a probable prime to base 2. A mismatch would prove the number composite, with the base as…
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Conic section classifier
Enter the six coefficients of a general second-degree equation and see which curve it draws: x² + y² − 25 = 0 is a circle centered at (0, 0), with discriminant…
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Fermat pseudoprime checker for any base
Find out if a composite number passes Fermat’s test anyway: 341 with base 2 leaves residue 1, so it is a Fermat pseudoprime to base 2.
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Lucas pseudoprime checker
Spot the odd composites that fool the Lucas test: with P equal to 1 and Q equal to −1, the composite 323 still passes, making it the smallest Fibonacci…
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Prime signature calculator
Shows the shape of a number's prime factorization without the primes: 360 = 2³ × 3² × 5 has signature [3, 2, 1].
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Proth primality test
Proves a Proth number k × 2^n + 1 prime with a single witness; taking k as 3 and n as 2 gives 13, and witness 2 raised to 6 leaves 12, that is −1 modulo 13, so…
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Smallest prime factor calculator
Finds the least prime that divides your number, plus what is left after dividing: for 84 the smallest prime factor is 2 and the cofactor is 42.
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Complex conjugate calculator
Enter the real and imaginary parts and get the conjugate: 3 + 2i becomes 3 − 2i. It is the step behind dividing complex numbers and finding a modulus squared…
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Consecutive integers calculator
Find the run of consecutive integers that adds up to the total you choose: three numbers summing to 15 are 4, 5 and 6.
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B-smooth numbers list
Up to 30 there are 18 numbers whose prime factors are all 5 or less, the 5-smooth or Hamming numbers.
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Integer square root calculator
The integer square root of 82 is 9, because 9 squared is 81 and 10 squared is already 100. Paste any non-negative whole number, even one with dozens of digits…
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Mertens function calculator
Add up the Möbius function from 1 to n in one step: M(100) is 1, the exact signed balance between numbers with no repeated prime factor and an even or an odd…
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Perfect cube checker
Find out whether a whole number is a perfect cube and get its integer root: 27 is 3 cubed, and −8 is −2 cubed, since cubes also exist below zero.
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Pollard's p−1 factorization
Finds a factor of a composite number when one of its primes p has a smooth p − 1: 1003 with a bound of 16 splits into 17 and 59, since 17 − 1 is 16.
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Squarefree Number Checker
30 is squarefree because it factors as 2 × 3 × 5, each prime once, while 12 is not, since 2 squared divides it.
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Unitary divisor sum calculator
The unitary divisors of 12 are 1, 3, 4 and 12, which add up to 20, while all its divisors add up to 28.
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Convolution calculator
Slide one sequence across another and add up the products at each step: [1, 2, 3] convolved with [0.5, 0.5] gives [0.5, 1.5, 2.5, 1.5].
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Coordinate distance calculator
Enter two points and get the straight-line distance between them: from (1, 2) to (4, 6) it is exactly 5, because the sides of the right triangle are 3 and 4.
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Cototient calculator
The cototient of 36 is 24: that many numbers from 1 to 36 share a factor with 36. Enter any positive integer up to one trillion and get the exact count, n minus…
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Highly Composite Number Checker
12 is highly composite: it has 6 divisors, while no smaller number has more than 4. Enter any whole number up to 1,000,000 to see whether it sets a new…
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Jordan totient calculator
With order 1, the Jordan totient of 36 is 12, the same as Euler’s totient. Choose a higher order k and the calculator applies n^k times the product of (1 −…
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Perfect number checker
Check whether a number equals the sum of its proper divisors: 28 is perfect because 1, 2, 4, 7 and 14 add up to 28.
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Practical number checker
Tells you whether every whole number below n can be built from distinct divisors of n: 12 is practical, since 1, 2, 3, 4, 6 and 12 reach every value from 1 to…
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Sociable numbers checker
Follows the chain of proper-divisor sums from your number and tells you whether it comes back: 12496 returns to itself after 5 steps, visiting four other…
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Totatives list calculator
The numbers up to 12 that share no factor with 12 are 1, 5, 7 and 11, four in all. Enter a positive whole number to see the full ordered list and its count, not…
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Corner point calculator
Enter the objective coefficients and the corners of the feasible region and see which corner scores highest: for 3x + 5y over (0, 0), (0, 4), (3, 3) and (5, 0)…
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Calkin-Wilf fraction index
3/2 is the 5th fraction in the Calkin-Wilf sequence, two levels below the root 1/1. Enter a reduced fraction with numerator and denominator from 1 to one…
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Egyptian fraction term count
With the greedy method, 3/4 becomes 1/2 + 1/4, so it needs 2 unit fractions. Enter a proper fraction and get the number of terms plus each unit denominator in…
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Quadrant finder
Enter x and y and see where the point sits on the plane: (−3.5, 2) lies in quadrant II. A point with a zero coordinate is reported on its axis or at the origin…
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Cosine calculator
Type an angle in radians and get its cosine, rounded to twelve decimals: 1.0471975511965976, which is π/3, gives exactly 0.5.
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cos⁻¹(x) calculator
Enter a cosine between -1 and 1 and get the angle back: 0.5 gives 60 degrees, or 1.047197551197 radians.
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Bézout coefficients calculator
For 240 and 46 the greatest common divisor is 2, and 240 × (−9) + 46 × 47 gives 2, so x is −9 and y is 47.
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Two-coin representable checker
Find out if an exact total can be made from two coin values: with coins of 4 and 7, the amount 23 works as four 4s plus one 7.
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Mediant of Two Fractions Calculator
The mediant of 1/3 and 2/5 is 3/8: the numerators and the denominators are added separately, with no reduction, so you see exactly how the new fraction was…
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Negative Pell equation solver
Find out whether x² − Dy² = −1 has whole-number solutions and get the smallest one: when D is 13, the answer is x 18 and y 5, found from a continued fraction…
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Pierce expansion calculator
Enter a fraction between 0 and 1 and get its exact Pierce digits: 2/3 gives 1 and 3, because 1/1 − 1/(1·3) rebuilds two thirds.
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Pythagorean triples from a leg
Enter one leg of a right triangle and get every whole-number triangle that shares it: a leg of 20 has four partners, from 20-15-25 to 20-99-101.
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cos(2θ) calculator
Enter an angle θ in radians and get the cosine of its double, worked out as cos²θ minus sin²θ: a θ of 0.7853981634, a quarter of π, gives exactly 0.
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Hyperbolic cosine calculator
Get the hyperbolic cosine of any number: cosh(1.5) is 2.352409615243247. It is the average of e^x and e^-x, the curve traced by a hanging chain or power line…
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