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.
Distinct parts partition count
The number 5 splits into different parts in 3 ways: 5, 4+1 and 3+2. Enter any whole number from 0 to 5000 and get the exact count of its distinct-part…
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Gaussian Integer Division Calculator
Dividing 17 + 9i by 4 − 3i gives quotient 2 + 3i and remainder 3i, whose norm 9 is below the divisor’s 25.
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Gaussian integer factorization
Factor any nonzero Gaussian integer into Gaussian primes: 5 becomes −i · (1+2i) · (2+i), with the unit, exponents and norms listed.
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Goldbach partition count
10 splits into two prime pairs, 3 + 7 and 5 + 5, so its Goldbach partition count is 2. Enter any even number from 4 up to ten million to get that count, with…
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Sum of three squares representation calculator
310 can be written as 2² + 9² + 15², and the result shows those three numbers with the full equation checked.
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Sum of Two Squares Count
65 can be written as a sum of two squares in 16 ordered, signed ways, so its count is 16. Enter a non-negative whole number and get the exact total from its…
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Waring problem representation
With squares, 30 becomes 25 + 4 + 1, that is 5² + 2² + 1², by always taking the largest square that still fits.
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Cosine similarity calculator
Score how closely two vectors point the same way, from -1 for opposite to 1 for identical direction.
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Cullen number generator
At index 5 the Cullen number is 161, because 5 × 32 + 1 = 161. Enter any whole index from 0 upwards and get the full term with every digit exact, however long…
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Digit factorial chain length calculator
Starting from 69, the digit factorial chain visits 5 distinct values before one repeats; its first step is 6! + 9! = 363600.
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Factorion checker
145 is a factorion: 1! + 4! + 5! is 1 + 24 + 120, which adds back up to 145. Enter a positive whole number and see each digit's factorial, the total and a plain…
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Fermat number generator
Type an index and see the Fermat number 2^(2^n)+1 written out in full: index 0 gives 3 and index 4 gives 65537.
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Repunit checker
Checks whether a positive whole number is written only with the digit 1: 111 is the three-digit repunit, equal to (10^3 − 1)/9.
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Vampire Number Checker
1260 is a vampire number because 21 × 60 = 1260 and the digits of 21 and 60 are exactly 1, 2, 6 and 0.
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Additive prime checker
23 is an additive prime: it is prime, and its digits add up to 5, which is prime too. Enter a whole number from 2 and see both checks side by side, so you know…
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Balanced prime checker
53 is a balanced prime: the nearest primes are 47 below and 59 above, both 6 away, and their average is 53.
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Permutable Prime Checker
Check whether a prime stays prime however you rearrange its digits: 113 passes, since all 3 distinct orderings of its digits are prime.
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Safe prime checker
Checks whether a prime p is safe, meaning (p − 1)/2 is prime too: 23 is safe because its half, 11, is prime, which also makes 11 a Sophie Germain prime.
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Wieferich prime checker
1093 is a Wieferich prime: 2 raised to 1092 leaves remainder 1 when divided by 1093², which is 1194649.
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Dudeney number checker
512 is a Dudeney number: its digits add up to 8, and 8 cubed is 512. Enter any positive whole number to see its digit sum, whether it is a perfect cube, its…
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Evil number checker
3 is an evil number: in binary it is 11, with 2 ones, and 2 is even. Enter any non-negative whole number to see its binary form, how many ones it has and…
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Home Prime Calculator
12 factors as 2 × 2 × 3, joining those digits gives 223, and 223 is prime, so the home prime of 12 is 223.
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Lucas prime checker
Enter an index and see whether the Lucas number at that position is prime: L(5) is 11, a two-digit prime, while L(6) is 18 and composite.
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Mersenne prime checker using the Lucas-Lehmer test
Give a prime exponent and the Lucas–Lehmer test tells you whether 2^p − 1 is prime: for 31 it runs 29 steps, ends with a residue of 0 and confirms the prime…
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Prime triplet finder
Lists every prime triplet up to your limit in the two shapes p, p + 2, p + 6 and p, p + 4, p + 6: up to 50 there are 7, from (5, 7, 11) to (41, 43, 47).
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Alternating digit sum calculator
For 918082 the alternating digit sum is 22, a multiple of 11, so the number divides by 11 exactly.
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Balanced ternary converter
20 in balanced ternary is 1 −1 1 −1: three cubed, minus three squared, plus three, minus one. Enter any integer, negative ones included, to get its signed-digit…
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Digital root in base
In base 16, 255 is written FF, its digits add up to 30 and then to 15: the digital root is 15, reached in 2 steps.
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Equidigital Number Checker
27 is equidigital: it has 2 digits, and its factorization 3³ also uses 2, the base 3 and the exponent 3.
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Multiplicative persistence calculator
Multiply a number's digits again and again until one digit is left: 39 becomes 27, then 14, then 4, so its multiplicative persistence is 3.
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Depressed cubic solver
t³ − 3t + 2 = 0 has the roots 1, 1 and −2, since it factors as (t − 1)²(t + 2): enter p and q and get all three roots with the discriminant and Cardano's…
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Inverse variation calculator
If x is 4 when y is 6, the constant of variation is 24, so an x of 8 gives a y of 3, with each substitution step shown.
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Linear equation from slope and point
Slope 2 through the point (1, 3) gives y − 3 = 2(x − 1), which simplifies to y equals 2x + 1 because the y-intercept is 3 − 2 × 1, that is 1.
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Literal equation solver
Solve y = mx + b for x and get x = (y − b) / m, with the steps that undo each operation. No numbers are needed: the result stays a formula you can reuse.
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Solve quadratic by factoring
Factor a quadratic the way your textbook does: x² − 5x + 6 becomes (x − 2)(x − 3), so the roots are 2 and 3.
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Quadratic with complex roots calculator
Get exact complex roots, not rounded decimals: x² + x + 1 has discriminant −3, so its roots are −1/2 + (√3/2)i and −1/2 − (√3/2)i, a conjugate pair.
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Vertex form to standard form calculator
Turn a vertex-form parabola into standard form without expanding by hand: 2(x − 3)² − 5 becomes 2x² − 12x + 13, using b = −2ah and c = ah² + k.
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Quadratic in form solver
Crack a biquadratic equation in steps: with u in place of x², x⁴ − 5x² + 4 = 0 becomes u² − 5u + 4 = 0, giving 1 and 4 for u and four real solutions, −2, −1, 1…
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Parabola x-intercepts from vertex form
Get the x-intercepts straight from vertex form, without expanding: y = (x − 2)² − 4 crosses the x-axis at 0 and 4, each 2 units from the vertex at (2, −4).
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Cube equation solver
x³ = 8 has three solutions, not one: the real root 2 and the complex pair -1 ± i√3. Enter any real k and get all three roots of x³ = k as real and imaginary…
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Sum and product of roots calculator
For x^2 - 5x + 6 = 0 the roots add up to 5 and multiply to 6, read straight from the coefficients with Vieta's formulas: the sum is −b/a and the product is c/a.
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Two-step linear equation solver
6x + 5 = 20 becomes 6x = 15 and then gives 5/2 for x, kept as an exact fraction. The solver removes the constant, divides by the coefficient and checks the…
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Y-intercept of a line calculator
The line 2x − y + 6 = 0 crosses the y-axis at (0, 6): enter the three coefficients of the general form and get the intercept, or a clear notice when the line is…
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Casting out nines check
Casting out nines checks a sum or product in seconds: a claimed total of 579 leaves remainder 3 mod 9, the same as its addends’ remainders 6 and 6 combined, so…
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Imaginary quadratic class number calculator
Enter a squarefree d and get the exact class number of Q(√−d): for 23 the discriminant is −23 and the class number is 3, while 1 gives the Gaussian rationals…
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Fibonacci coding converter
Turn any positive integer into its Fibonacci code: 6 becomes 10011, built from the summands 1 and 5.
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Hofstadter Q sequence
In Hofstadter’s Q sequence the 10th term is 6 and the 20th is 12. Each new term looks back by distances that earlier terms decide, which makes the growth…
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Narayana's cows sequence calculator
Get any term of Narayana's cows sequence from index 0 to 97: N(10) is 28, because each term adds the one just before it to the one three places back, 19 plus 9.
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p-adic valuation calculator
Count how many times a prime divides a whole number: 72 is 8 times 9, so it contains the prime 2 three times, its 2-adic valuation is 3 and the leftover part is…
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Perrin sequence term
Get any term of the Perrin sequence, which starts 3, 0, 2 and adds the terms two and three places back: P(10) is 17, that is 10 plus 7.
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Thue-Morse sequence bit
Position 7 is 111 in binary, three ones, so its Thue-Morse bit is 1. Enter any non-negative position to see its binary form, how many ones it has and the…
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Lower Wythoff sequence calculator
The 10th term of the lower Wythoff sequence is 16, and the sequence opens 1, 3, 4, 6, 8. Enter a position counted from one to get that exact term without…
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Absolute value inequality solver
|2x − 3| < 5 holds for −1 < x < 4, the interval (−1, 4), while a greater-than version gives two outer rays instead.
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Cubic inequality solver
x³ − 2x² − 5x + 6 ≥ 0 holds on [−2, 1] ∪ [3, ∞): enter the four coefficients and the comparison, and see the real roots, the sign on every interval and the…
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2x2 system elimination method calculator
Solve two linear equations exactly by elimination: for 2x + 3y = 7 paired with x minus y equal to 1, x cancels out, 5y = 5 remains, and the point is (2, 1).
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Line and parabola intersection
The line y = x + 2 meets the parabola y = x^2 at (-1, 1) and (2, 4): the calculator sets both equal, solves the quadratic and reads the discriminant, here 9, to…
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Two-variable linear inequality calculator
For 2x + y < 6 you get a dashed boundary where 2x + y equals 6, the half-plane below y = −2x + 6, and the origin as a test point that works.
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Factor difference of cubes calculator
Check that two whole numbers are perfect cubes and factor their difference: 125 − 8 becomes (5 − 2)(25 + 10 + 4), because 125 is 5³ and 8 is 2³.
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Homogeneous linear system 2x2 calculator
Enter the four coefficients of two equations with zero on the right and get the full solution set: with 1, 2, 3 and 4 the determinant is −2, so only x and y…
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Polynomial composition calculator
Substitute one polynomial into another and get the expanded result: with f(x) = x² and g(x) = 1 + x, f(g(x)) is 1 + 2x + x², of degree 2. Each step is shown.
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