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➗ MathFree
Square of a Complex Number Calculator
(3 + 4i)² is −7 + 24i: enter the real and imaginary parts and get the square in a + bi form, with real part a² − b² and imaginary part 2ab.
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➗ MathFree
Common ratio calculator
Enter two consecutive terms and get the constant multiplier of the sequence: 6 followed by 18 gives a common ratio of 3, because 18 divided by 6 is 3.
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➗ MathFree
Nth roots of unity calculator
The fourth roots of unity are 1, i, -1 and -i, evenly spaced at 0°, 90°, 180° and 270° on the unit circle.
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➗ MathFree
Arithmetic sequence missing term calculator
In 4, 7, ?, 13, 16 every step is 3, so the missing third term is 10. Mark the gap with a question mark and get the missing value, its position, the step and the…
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➗ MathFree
Geometric sequence first term calculator
Work back to the start of a geometric sequence: if the fifth term is 162 and the ratio is 3, the first term is 2, because 162 divided by 81 is 2.
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➗ MathFree
Geometric mean of two numbers
Get the multiplicative middle of two positive numbers: the geometric mean of 4 and 9 is 6, the square root of their product 36.
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➗ MathFree
Sigma Notation Sum Calculator
The sum of k^2 + 2*k for k from 1 to 5 is 85, over 5 terms: enter the term, the index letter and both bounds, and every whole value in the range is added for…
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➗ MathFree
Sum of First n Natural Numbers Calculator
1 + 2 + … + 100 is 5,050: the formula n(n + 1)/2 gives the total of the first n natural numbers at once, without adding each one.
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➗ MathFree
Sum of first n odd numbers calculator
1 + 3 + 5 + 7 + 9 adds up to 25: the first n odd numbers always total exactly n squared. Enter n and get the sum along with the last odd number in the list…
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➗ MathFree
Binomial coefficient calculator
A committee of 3 chosen from 10 candidates can be formed in 120 ways: enter n and k and get the exact binomial coefficient, every digit correct and with no…
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➗ MathFree
Characteristic polynomial 2x2 calculator
For the matrix with rows 4, 1 and 2, 3, the trace is 7 and the determinant 10, so the characteristic polynomial is λ² − 7λ + 10, with coefficients 1, −7 and 10.
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➗ MathFree
2x2 determinant calculator
For rows 3, 8 and 4, 6, the diagonal products are 18 and 32, so the determinant is −14 and the matrix is invertible.
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➗ MathFree
Hadamard product calculator
Multiply matching cells of two equal-size matrices: entries 1, 2, 3, 4 times 5, 6, 7, 8 give a top row of 5 and 12.
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➗ MathFree
Matrix minor calculator
For the matrix with rows 1, 2, 3; 0, 4, 5; 1, 0, 6, crossing out row 1 and column 1 leaves rows 4, 5 and 0, 6, whose determinant 24 is the minor.
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➗ MathFree
Matrix of cofactors calculator
For rows 1, 2, 3; 0, 4, 5; 1, 0, 6, you get nine minors and nine cofactors: the minor at row 1, column 2 is −5, and after the alternating sign its cofactor is…
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➗ MathFree
Symmetric matrix checker
Find out at once whether a square matrix is symmetric: the 3×3 matrix with rows 1 2 3, 2 4 5 and 3 5 6 passes after 3 pairs are compared, and a failing one…
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➗ MathFree
Absolute Value Expression Evaluator at a Given Input
With x set to 4, |2*x − 5| + 3 gives 6, because the bars turn 3 into its distance from zero before adding 3.
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➗ MathFree
Multinomial Coefficient Expansion Calculator
Count how many ways distinct items can be shared among labeled groups: 8 people split into groups of 3, 3 and 2 give 560 arrangements, computed exactly and…
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➗ MathFree
Multiply rational expressions calculator
Multiply two polynomial fractions and get the reduced answer with its domain: (x + 1)/(x − 2) times (x − 2)/(x + 3) simplifies to (x + 1)/(x + 3), excluding the…
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➗ MathFree
Direct rule of three calculator
4 is to 10 as 7 is to 17.5: the direct rule of three scales the known value by the same factor, here 1.75, so both pairs keep the same ratio.
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➗ MathFree
Set-builder to interval notation converter
{ x | -1 < x ≤ 4 } becomes (-1, 4]: a parenthesis where the bound is left out, a square bracket where it is kept.
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➗ MathFree
Distributive property calculator
Expand one factor over a sum in parentheses and see each product separately: −2a(3a − 4b + 5) becomes −6a² + 8ab − 10a.
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➗ MathFree
Interval notation from inequality calculator
Write x < 4 as the interval (−∞, 4) without guessing which bracket goes where: an included limit gets a square bracket, an excluded one a parenthesis, and…
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➗ MathFree
What percent of calculator
Answer "what percent of the whole is this part?" in one step: 18 students out of 24 is 75%, and 30 against 24 is 125%.
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➗ MathFree
Midpoint of a segment calculator
Enter both end points and get the exact center of the segment: from (0, 0) to (4, 6) the midpoint is (2, 3).
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➗ MathFree
Solve for x in a ratio calculator
Solve the missing number in a proportion: 3/4 = 9/x gives 12 for x, and both cross-products equal 36, so you can see the answer holds.
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➗ MathFree
Slope from linear equation calculator
For 3x + 2y = 12 the slope is -1.5 and the y-intercept is 6: the calculator collects the terms from both sides and rewrites the line in slope-intercept form.
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➗ MathFree
Cross multiplication calculator
If 2/3 equals 4/d, then d is 6, because 2 × 6 and 3 × 4 both give 12: enter any three terms of a proportion and get the fourth, with the cross products and each…
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➗ MathFree
GCF of monomials calculator
Get the greatest common factor of algebraic terms, variables included: 12x²y, 18xy³ and 24x³y share 6xy.
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🔬 ScienceFree
Aphelion distance from eccentricity calculator
An orbit with a semi-major axis of 10 and an eccentricity of 0.2 reaches an aphelion of 12, in the same unit.
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🔬 ScienceFree
Orbital speed at altitude calculator
Find the speed needed to stay in a circular orbit at a set height: 400,000 m above Earth's 6,371,000 m radius takes 7,672.598648 m/s.
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🔬 ScienceFree
Perihelion velocity calculator
Around the Sun, an orbit with a semi-major axis of 149597870700 m and eccentricity 0.0167086 reaches its closest point at 147098299718 m, moving at…
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🔬 ScienceFree
Specific orbital energy from semi-major axis calculator
Get the energy per kilogram of a two-body orbit from its size alone: a semi-major axis of 7000000 m around Earth, with μ of 398600441800000, gives…
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🔬 ScienceFree
Apparent magnitude from absolute magnitude and distance calculator
An object of absolute magnitude 4.83 at 10 parsecs appears at magnitude 4.83, because the distance modulus there is zero.
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🔬 ScienceFree
Apsis speed ratio calculator
An orbit with eccentricity 0.5 is three times faster at periapsis than at apoapsis, while a circular orbit gives 1.
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🔬 ScienceFree
Combined maneuver Δv calculator
Going from 7.8 to 7.5 km/s while turning the orbit 30° takes 3.970519835 km/s in a single combined burn.
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🔬 ScienceFree
Escape burn delta-v calculator from a circular orbit
From a circular Earth orbit with a radius of 6,771 km, escaping takes a prograde burn of 3178.094418806551 m/s, not the full escape speed.
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🔬 ScienceFree
Escape velocity at altitude calculator
At 400,000 metres above Earth's surface, escape velocity is 10850.5399996432 m/s. Enter the body's mass, its radius and the altitude to get the ideal escape…
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🔬 ScienceFree
Orbital flight path angle calculator
On an orbit with eccentricity 0.5, the flight path angle at a true anomaly of 60° is 19.106605350869°, so the object is moving outward after periapsis.
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🔬 ScienceFree
Hohmann Total Delta-V Calculator for Circular Orbits
A Hohmann transfer from a 7000000 m orbit to 42164000 m around Earth needs a total delta-v of 3770.7272333041296 m/s in two burns.
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🔬 ScienceFree
Hohmann transfer orbit semi-major axis calculator
A Hohmann transfer from a radius of 7,000 km to 42,164 km follows an ellipse with a semi-major axis of 24,582 km, the average of the two.
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🔬 ScienceFree
Hyperbolic semi-major axis calculator
For Earth, a flyby with an excess speed of 3 km/s follows a hyperbola with a semi-major axis of −44288.9379778 km.
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🔬 ScienceFree
Gravity balance point distance calculator between two masses
Find the spot on the line between two masses where their pulls cancel out: for Earth and the Moon, 384,400,000 m apart, it sits about nine tenths of the way…
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🔬 ScienceFree
L4 and L5 Lagrange Point Geometry Calculator
Place the two triangular Lagrange points from a single distance: with the bodies 2 units apart, L4 and L5 sit halfway along the line and form an equilateral…
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🔬 ScienceFree
Brightness ratio to magnitude difference calculator
A source 100 times brighter than another sits 5 magnitudes lower: a ratio of 100 gives a difference of -5.
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🔬 ScienceFree
Rocket final mass calculator
A 100000 kg rocket that gains 9400 of velocity with an exhaust velocity of 3500 ends with 6817.248197125795 kg and burns 93182.7518028742 kg of propellant.
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🔬 ScienceFree
Orbital semi-latus rectum calculator
An orbit with a semi-major axis of 7000 and eccentricity 0.05 has a semi-latus rectum of 6982.5, in the same unit as the axis.
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🔬 ScienceFree
Sphere of influence calculator
Find how far a moon or planet's own gravity dominates: the Moon, 384400 km from Earth, has a sphere of influence of 66170.0117012 km under the classic…
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🔬 ScienceFree
Rocket structural coefficient calculator
See how much of a stage's mass is dry hardware: 1200 kg of structure and 8800 kg of propellant make 10000 kg, a structural coefficient of 0.12, with the cargo…
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🔬 ScienceFree
Rocket thrust from mass flow
Multiply how much propellant leaves each second by how fast it leaves to get thrust: 100 kg/s at 3000 m/s gives 300000 N, the ideal figure for engine sizing and…
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🔬 ScienceFree
Rocket thrust-to-weight ratio calculator
Compare engine thrust with the vehicle's weight at one moment: 7600000 N against 549054 kg under 9.80665 m/s² gives a thrust-to-weight ratio of…
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🔬 ScienceFree
Differential tidal force calculator
Measure how strongly gravity stretches an object: the Moon, 384400000 m away, pulls the near and far sides of Earth apart by 0.00000219855615754 m/s², and…
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🔬 ScienceFree
True anomaly from eccentric anomaly calculator
Turn the eccentric anomaly into the real angle of a body around its focus: with an eccentricity of 0.5, an eccentric anomaly of 90° becomes a true anomaly of…
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🔬 ScienceFree
Blackbody radiant exitance calculator
Double a surface's absolute temperature and it radiates sixteen times more power per square metre; triple it and the figure rises eighty-one times.
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🔬 ScienceFree
Bolometric magnitude calculator
A visual magnitude of 4.83 with a bolometric correction of -0.07 gives a bolometric magnitude of 4.76.
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🔬 ScienceFree
Atmospheric extinction magnitude calculator
With an extinction coefficient of 0.2 magnitudes per airmass, a star seen through 1.5 airmasses loses 0.3 magnitudes to Earth's atmosphere.
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🔬 ScienceFree
Mass ratio from barycenter calculator
Read the mass ratio of two orbiting bodies straight from their distances to the shared centre of mass: distances of 1,500,000,000 and 450,000,000,000 give a…
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🔬 ScienceFree
Proper motion calculator
A star moving sideways at 20 km/s at a distance of 10 parsecs drifts 0.42194092827 arcseconds per year across the sky.
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🔬 ScienceFree
Surface brightness calculator
Spread a galaxy's total light over its patch of sky: a magnitude of 10.5 across 120 square arcseconds gives a mean surface brightness of 15.697953 mag/arcsec²…
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🔬 ScienceFree
Combined magnitude of two stars
A magnitude 1 star and a magnitude 2 star seen as one point shine at magnitude 0.6361488422, brighter than either alone.
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