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.
Egyptian fraction calculator
5/6 becomes 1/2 + 1/3: enter a fraction and get it as a sum of different unit fractions, the way Egyptian scribes wrote it.
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Eigenvalue and eigenvector calculator
The matrix with rows (2, 1) and (1, 2) has eigenvalues 3 and 1, each paired with its unit eigenvector.
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Elimination method
2x + 3y = 7 and 4x − 5y = 1 meet at x ≈ 1.7272727273 and y ≈ 1.1818181818. Enter the six coefficients of two linear equations and get the solution with every…
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Ellipse circumference calculator
An ellipse with semi-axes 5 and 3 has a circumference of 25.526986, from Ramanujan's 1914 formula.
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Ellipse perimeter
Semi-axes of 5 and 3 give an ellipse perimeter of 25.526986, found with Ramanujan's formula because no exact elementary formula exists.
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Ellipse standard form
4x² + 9y² − 8x − 18y − 23 = 0 is an ellipse centered at (1, 1) with semi-axes 3 and 2 and no rotation.
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Endpoint calculator
If one end of a segment is at (0, 0) and its midpoint is at (2, 3), the other end is at (4, 6). Enter the known end and the midpoint to get the missing point…
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Engineering notation
47,000 becomes 47×10^3 and 0.000025 becomes 25×10^-6: the exponent always moves in steps of three, so it tells you the prefix, kilo or micro, at a glance.
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Equation of a circle from diameter endpoints
The ends (1, 2) and (5, 6) give a circle centred at (3, 4) with radius squared 8, so its equation is (x − 3)² + (y − 4)² = 8.
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Equation of a plane
The plane through (1, 0, 0), (0, 1, 0) and (0, 0, 1) has a, b and c all equal to 0.57735026919 and d equal to −0.57735026919.
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Equilateral triangle area calculator
A side of 10 gives an equilateral triangle an area of about 43.301270, and a side of 2 gives √3, about 1.732051.
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Equivalent fractions calculator
Scaling 3/4 by 2 gives 6/8, and scaling it by 5 gives 15/20: the same value written with a bigger denominator.
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Equivalent ratio calculator
Scaling 3:4 by 5 gives 15:20, and both still equal 0.75, so the proportion has not changed. Enter a ratio and any non-zero factor, including 0.5 to scale down…
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Error function calculator
erf(1) is 0.842700689748 and erfc(1) is 0.157299310252: the two numbers you need for normal probabilities, heat spread and measurement uncertainty.
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Euclidean algorithm calculator
For 1071 and 462 the greatest common divisor is 21, reached in three divisions: 1071 = 462 × 2 + 147, then 462 = 147 × 3 + 21, then a remainder of zero.
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Euclidean distance calculator
From [0, 0] to [3, 4] the distance is exactly 5, and from [0, 0, 0] to [1, 2, 2] it is 3. Enter two vectors of the same length, in any number of dimensions, and…
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Euler's polyhedron formula calculator
A cube with 8 vertices and 12 edges must have 6 faces, so give any two counts and the third follows from Euler's rule that vertices minus edges plus faces…
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Exact value of trig functions
A calculator shows sin 45° as 0.70710678; this gives you the exact √2/2 that textbooks and exams expect.
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Expanded form calculator
Type 204.75 and get 200 + 4 + 0.7 + 0.05: every non-zero digit becomes a term worth its place value, and the terms always add back to your number.
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Expanding logarithms
Log base 2 of x·y/z becomes log₂(x) + log₂(y) − log₂(z). List each factor and say whether it multiplies, divides or carries a power, and you get one term per…
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Exponent calculator
2 to the power of 10 is 1024. Enter any base and exponent, negative or fractional ones included, and get the exact power.
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Exponential form
1234.5 becomes 1.2345 × 10^3 and 0.00072 becomes 7.2 × 10^-4. Type any finite number, plain or already in scientific notation, and get its mantissa, its power…
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Exponential function calculator
Start at 2, triple for 4 periods and you reach 162. Enter the coefficient a, the base b and the exponent x to get a·b^x for growth, decay or any scaled power…
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Exterior angles of a triangle calculator
Interior angles of 60, 70 and 50 degrees give exterior angles of 120, 110 and 130, which always total 360.
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Most frequent value calculator
In red, blue, red, green, red, blue the most frequent value is red, which appears 3 times; blue follows with 2 and green with 1.
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Percent change between values calculator
Going from 80 to 100 is a 25% increase, yet going from 100 back to 80 is only a 20% decrease. Enter the old and the new value to get the signed percentage, the…
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Fourth root calculator
The fourth root of 16 is exactly 2, and the fourth root of 10 is 1.778279410039. Enter any non-negative number and get the value that, used four times as a…
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Fractional exponent calculator
8 raised to the power 2/3 is 4: the cube root of 8 is 2, and 2 squared is 4. Type the base and the two whole numbers of the fraction, which is reduced first, so…
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Frustum cone area
With radii of 3 and 4 and a slant height of 5, a truncated cone has a side area of 109.9557428756 and a total surface of 188.4955592154.
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Fundamental counting principle calculator
Three shirts, four pairs of trousers and two pairs of shoes make 24 outfits. Enter how many options each step offers and get the exact number of combinations…
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Galileo's paradox of infinity
Galileo paired every counting number with its square, 1 with 1, 2 with 4, 3 with 9, and found that neither list runs out first.
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Gamma function calculator
Γ(2.5) is 1.3293403882: the gamma function extends the factorial to every real number, so Γ(5) equals 4!, which is 24.
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Gauss-Jordan elimination calculator
The classic system that opens with 2x + y − z equal to 8 solves to (x, y, z) = (2, 3, −1). Type each equation as a row of numbers and see whether there is one…
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GCD calculator
The greatest common divisor of 1071 and 462 is 21, found in three steps of Euclid's method without factoring either number.
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GCF calculator
The greatest common factor of 48 and 18 is 6: it is the largest number on both lists of divisors. Enter two integers, signed or with a zero, and get the exact…
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General form equation of a circle
A circle with center (3, −2) and radius 5 becomes x² + y² − 6x + 4y − 12 = 0 in general form. Enter the center and radius and get the three coefficients, with…
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General to standard form of a circle
x² + y² − 6x + 4y − 12 = 0 is the circle with center (3, −2) and radius 5. Enter the three coefficients and the square is completed for you, with a warning when…
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Rectangle calculator
A rectangle 5 long and 3 wide has an area of 15 and a perimeter of 16. Enter the two sides once and read both figures in your own unit, squared for the area, so…
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Golden ratio calculator
Split a length in the ratio phi and get both pieces at once: 100 becomes 61.803399 and 38.196601, and the larger divided by the smaller gives phi, about…
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Gradient calculator
Give two points and get the slope of the line that joins them: from (1, 2) to (4, 8) the gradient is 2, so the line climbs two units for every unit across.
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Gram-Schmidt calculator
Paste independent vectors and get an orthonormal basis of the same space: (1, 1, 0), (1, 0, 1) and (0, 0, 3) become three unit vectors at right angles, the…
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Graph linear inequalities on a number line
Enter an inequality such as 2x + 3 < 11 and see how to draw it: the answer x < 4, an open circle at 4, shading to the left and the interval (-∞, 4).
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Graphing quadratic inequalities
Type the three coefficients and the sign, and see which part of the line to shade: x² − 5x + 6 ≤ 0 has roots 2 and 3, the parabola opens upward, and the answer…
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All primitive roots modulo n
Modulo 14 there are 6 invertible residues and exactly two primitive roots, 3 and 5. Enter a modulus from 2 to 10,000 to get the complete sorted list of…
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Cipolla modular square root
Cipolla's algorithm gives the actual square root modulo an odd prime, not just a yes or no: for 2 modulo 7 it returns 3 and 4, since 3 squared is 9, which is 2…
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Fermat's little theorem checker
Enter a base and a modulus and see whether a^(n−1) mod n equals 1: for 2 and 7 the remainder of 2^6 is 1, for 2 and 15 it is 4.
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Hensel lifting calculator
Starting from 3, a root of x² − 2 modulo 7, Hensel lifting gives 10 modulo 49, 108 modulo 343 and 2166 modulo 2401.
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Modular division calculator
Divide in modular arithmetic: 8 divided by 3 modulo 11 is 10, because the inverse of 3 is 4 and 3 times 10 leaves the remainder 8.
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Pisano period calculator
Find the Pisano period for any modulus up to 1,000,000: modulo 10 the Fibonacci remainders repeat every 60 steps, and modulo 2 every 3.
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Pocklington primality test
Proves a number prime from a partial factorization of n − 1: for 31, the primes 2, 3 and 5 of 30 give a factored part of 30, and since 900 is at least 31, one…
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Difference of squares factorization
5959 equals 80² − 21², and that split reveals its factors 59 and 101. Enter an odd composite number and get the two squares, the closest factor pair and an…
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Euler-Jacobi pseudoprime checker
561 is an Euler–Jacobi pseudoprime to base 2: the power with exponent 280 leaves 1, matching its Jacobi symbol, although 561 is composite.
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Integer nth root floor calculator
The cube root of 1000 is exactly 10, while the cube root of 999 rounds down to 9. This calculator gives that whole-number floor for any root degree from 1 to…
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Liouville function calculator
λ(12) is −1, because 12 is 2 × 2 × 3, three prime factors counted with repetition. Enter any positive integer to get 1 for an even count or −1 for an odd one…
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Lucas primality test
29 is proven prime with base 2: 2 raised to 28 leaves 1 modulo 29, and the checks for 2 and 7, the prime factors of 28, both pass.
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Powerful part calculator
Splits any positive whole number into its largest powerful divisor and a squarefree cofactor: 12 becomes 4 × 3, because 2 appears squared and 3 only once.
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Prime factor count with multiplicity
Counts every prime factor of n, repeats included: 12 is 2 × 2 × 3, so Ω(12) is 3, while its distinct primes ω(12) are only 2.
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Abundancy index calculator
The divisors of 12 add up to 28, so its abundancy index is 28/12, reduced to 7/3: above 2, which makes 12 abundant.
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Aliquot sequence calculator
Starting at 12, the aliquot sequence runs 12, 16, 15, 9, 4, 3, 1, 0 and ends at zero after 7 steps.
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Calkin–Wilf sequence term
Term 5 of the Calkin-Wilf sequence is 3/2, reached from the root 1/1 by one step left and one step right.
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