Math Calculators

Solve every everyday percent problem with the working shown: X% of Y, X is what percent of Y, percent change, percent difference, increase or decrease by a percent, and reverse percentages.

Find the percent increase or decrease between a starting value and an ending value, with the direction and absolute change.

Add, subtract, multiply or divide fractions and mixed numbers with every step shown — common denominator, cross-cancelling, simplifying — and get the answer as a fraction, mixed number, decimal and percent. Also simplifies and compares fractions.

Convert a fraction or mixed number to a decimal with the repeating digits marked — 1/3 = 0.(3), 5/6 = 0.8(3) — rounded to any number of places, as a percent, with the long division worked out step by step.

Turn any decimal into a simplified fraction (and mixed number), with denominators up to 10,000.

Solve a proportion A : B = C : D for the missing term in any position, with cross-multiplication shown. Also simplifies ratios (including decimals and 3-part ratios), converts to 1 : n form, compares two ratios, scales a ratio and splits a total in a given ratio.

Find the greatest common factor and least common multiple of two whole numbers.

Solve ax² + bx + c = 0 with the quadratic formula — real or complex roots, plus the discriminant.

Find the missing side of a right triangle — hypotenuse or either leg — using a² + b² = c².

Find a triangle’s area from its three side lengths with Heron’s formula — no height needed.

Get a circle’s diameter, circumference and area from its radius in one step.

Calculate the area of a rectangle, triangle, circle, trapezoid or ellipse from its dimensions.

Find the volume (and surface area) of a box, cube, sphere, cylinder or cone.

Find the slope, y-intercept and line equation y = mx + b through two points.

Get the distance between two points and the midpoint halfway between them.

Raise any base to any power, including negative and fractional exponents, with the answer in plain and scientific notation.

Take the square root, cube root or any nth root of a number, with a built-in check of the answer.

Compute the logarithm of any number in any base, with natural log and a power check included.

Convert any number to scientific, E and engineering notation (with the SI prefix), turn scientific notation back into a plain number, and add, subtract, multiply, divide, raise to a power or take square roots in scientific notation — with significant-figure rounding and every step shown.

Round any number to a chosen number of decimal places or to the nearest 10, 100 or 1,000.

Paste a list of numbers and get the sample or population standard deviation, variance, mean, standard error, confidence interval, quartiles, z-scores, a frequency table and a histogram, with every step worked out.

Paste a list of numbers and get the mean, median, mode, range, sum and count at once.

Compute nPr and nCr — how many ways to arrange or choose r items out of n.

Work out a probability with the steps shown: favorable ÷ total outcomes with odds, two events (and, or, neither, conditional), repeated trials with the binomial distribution, and the area under a normal curve.

Compute n! for any whole number up to 170, with digit count and trailing zeros.

Break any whole number up to a billion into its prime factors, with divisor count and a primality check.

Average a list of numbers with the working shown: mean, sum, count, median, mode, range, geometric, harmonic and trimmed means, plus a weighted-average mode for grades and prices.

Find the percent error between a measured value and the accepted true value with the working shown, plus absolute, signed and relative error. Average several lab trials, check a tolerance, or work back from a percent error.

Solve a proportion a/b = c/x for any missing term by cross-multiplication.

Convert numbers to Roman numerals and Roman numerals back to numbers, from 1 to 3999.

Convert a binary number to decimal, with the hexadecimal and octal forms.

Convert a whole number to binary, plus its hexadecimal and octal forms.

Convert a hexadecimal number to decimal, with the binary and octal forms.

Find the remainder and quotient of a division, with both truncated and floored modulo.

Round a number to a chosen number of significant figures and count how many it already has.

Convert scientific notation (mantissa and exponent) to a plain decimal and engineering notation.

Enter an angle in degrees or radians and get all six trig ratios — sine, cosine, tangent and their reciprocals cosecant, secant and cotangent.

Find the angle from a ratio with arcsin, arccos or arctan — answered in degrees, radians and as a multiple of π.

Solve a triangle from two angles and a side using the law of sines — remaining sides, the third angle and the area.

Find the third side and remaining angles of a triangle from two sides and the included angle using the law of cosines.

Convert degrees to radians or radians to degrees, with the answer also expressed as a clean multiple of π.

From the two legs of a right triangle, get the hypotenuse, both acute angles, the area and the perimeter.

Find the surface area of a sphere, cube, cylinder, cone or rectangular box, with its volume shown as a bonus.

Find the arc length of a circle from the radius or diameter and the central angle (degrees, radians or gradians) — or from the chord, the arc height, the sector area or the arc itself. Also gives the exact π form, sector and segment area, chord, sagitta, perimeter and worked steps.

Add, subtract or multiply two matrices — enter each as rows separated by semicolons and columns by commas, and get the result matrix instantly.

Compute the determinant of a 2×2, 3×3 or 4×4 square matrix by cofactor expansion, and see whether the matrix is invertible.

Find the inverse of a 2×2 or 3×3 matrix using the adjugate over the determinant, with a check for singular matrices that have no inverse.

Solve a system of 2 or 3 linear equations with Cramer’s rule — enter each equation’s coefficients and constant to get x, y and z.

Enter two 2D or 3D vectors to get the dot product, cross product, magnitudes and the angle between them in degrees.

Solve ax³ + bx² + cx + d = 0 for its real roots using the trigonometric and Cardano methods, with near-integer roots snapped cleanly.

Check whether a number is prime, see its prime factorization, and find the next and previous primes — all by trial division.

Add, subtract or multiply two polynomials entered as coefficient lists from highest to lowest degree, and read the result as a clean expression.

Reduce any fraction or mixed number to lowest terms with the steps shown three ways — Euclid’s GCF, prime factorization or dividing by common factors one at a time — plus the mixed number, decimal (repeating digits marked), percent and equivalent fractions.

Convert an improper fraction to a mixed number, or a mixed number back to an improper fraction, with the decimal value.

Divide one whole number by another to get the quotient, the remainder and the exact decimal result.

Find the cube root of any number, positive or negative, and see whether it is a perfect cube.

Find the absolute value of a number — its distance from zero — for any positive, negative or decimal input.

Find the least common denominator (LCD) of two or more fractions, mixed numbers, whole numbers or decimals, rewrite each one over the LCD, and see the sum and the order from smallest to largest, with steps by prime factors, listing multiples or the GCF formula.

Compute the antilogarithm — the inverse of a logarithm — for base 10, base e or any custom base.

Find the nth Fibonacci number and the full sequence up to that term, plus the running sum.

Find the nth term and the sum of an arithmetic sequence from its first term and common difference.

Find the nth term and the sum of a geometric sequence from its first term and common ratio.

Add a list of numbers and get the sum, count, mean and running details — the sigma (Σ) of your data.

Split a length by the golden ratio φ ≈ 1.618 — find the matching longer, shorter and total segments from any one of them.

Find the area of a trapezoid from its two parallel sides and the perpendicular height, plus the length of its median.

Find the area of a parallelogram from its base and perpendicular height, and its perimeter when you also give a side length.

Find the area of an ellipse from its semi-major and semi-minor axes, with an accurate Ramanujan estimate of the circumference.

For a regular polygon with any number of equal sides, get its area, perimeter, interior angle and apothem.

Find the volume of a right circular cone from its radius and height, plus its slant height and surface area.

Find the volume of a cylinder from its radius or diameter and height, with the capacity in litres and gallons, the lateral and total surface area and the answer in terms of π. It also solves for the height or radius from the volume or surface area, and handles hollow cylinders such as pipes and tubes.

Find the volume of a sphere from its radius, together with its surface area, diameter and great-circle circumference.

Find the volume of a square or rectangular pyramid from its base dimensions and height, with base area and surface area.

Find the volume, surface area and space diagonal of a rectangular prism (box) from its length, width and height.

Find the area of a triangle from its three side lengths using Heron’s formula, with the perimeter and semi-perimeter.

Find the equation of a line in slope-intercept form y = mx + b from two points, a point and a slope, or standard form Ax + By = C — with exact fractions, point-slope and standard forms, both intercepts, the angle, a graph and every step worked out.

Find the perimeter of a rectangle, square, triangle or circle from its dimensions — the total distance around the shape.

Enter the three sides of a triangle (SSS) and get all three interior angles by the law of cosines, plus whether the triangle is acute, right or obtuse.

From any one measurement of a 30-60-90 triangle — short leg, long leg, hypotenuse, height, area or perimeter — get all three sides in the 1 : √3 : 2 ratio as exact radicals and decimals, plus the area, perimeter, height, inradius, circumradius, angles and step-by-step working.

From one side of a 45-45-90 right isosceles triangle, get both legs and the hypotenuse in the 1 : 1 : √2 ratio, with area and perimeter.

Find the straight-line distance between two points in 3D space from their (x, y, z) coordinates, plus the midpoint between them.

Find the vertex, axis of symmetry, minimum or maximum and vertex form of a parabola y = ax² + bx + c.

Compute the discriminant b² − 4ac of a quadratic ax² + bx + c and see how many real or complex roots it has.

Add, subtract, multiply or divide two complex numbers a + bi and c + di, with the result in a + bi form and its modulus.

Convert polar coordinates (r, θ) to Cartesian (x, y) using x = r·cos θ and y = r·sin θ, with the angle in degrees.

Convert Cartesian coordinates (x, y) to polar (r, θ) using r = √(x² + y²) and θ = atan2(y, x), with the angle in degrees.

From a circle’s centre (h, k) and radius, get its standard equation, general form, area and circumference.

Find the surface area of a sphere from its radius with SA = 4πr², plus its volume, diameter and great-circle circumference.

Convert a percentage to a simplified fraction and a decimal — for example 62.5% becomes 5/8 and 0.625.

Find the volume of a pipe, tube or washer from its outer radius, inner radius and length — the material in the wall, not the empty bore.

Find the volume of a truncated cone (frustum) — a bucket, lampshade or drinking cup — from its two radii and height, with slant height and surface area.

Find the volume of a truncated square pyramid (frustum) from its two square bases and height, with slant height and surface area.

Find the perimeter (circumference) of an ellipse with Ramanujan’s accurate approximation, plus its area, eccentricity and focal distance.

Find the area, height and perimeter of a trapezoid from its two parallel sides and its two legs — no height needed.

Find the area of a circular sector, its arc length, chord and perimeter from the radius and central angle in degrees or radians.

Add up the terms of an arithmetic sequence — enter the first term, common difference and number of terms to get the series sum and last term.

Sum a geometric sequence: enter the first term, common ratio and number of terms for the finite sum, nth term and infinite sum when it converges.

Compute the double factorial n!! — the product of every other integer down from n — with its ordinary factorial and digit count.

Compute combinations (nCr) and permutations (nPr), plus their with-repetition counterparts, for choosing r items from n.

Compute log base b of any positive number using the change-of-base formula, alongside natural log, base-10 and base-2.

Raise any base to any exponent — whole, negative or fractional — and get the power, its reciprocal and scientific-notation form.

Reduce a ratio like 18:24 to its simplest whole-number form, and see the equivalent single-number ratio and decimal.

Solve the proportion A/B = C/x for the unknown x by cross-multiplication, with the full working shown.

Find the percent increase or decrease from an old value to a new one with the working shown, or apply a percent change, undo one to find the original, chain several changes and get the average rate per period.

Work out what a given percentage of a number is — for example 15% of 200 — with the plain-decimal method shown.

Convert any fraction to its decimal and percentage value by dividing the numerator by the denominator.

Turn a terminating decimal into an exact fraction in lowest terms, with the reduction shown.

Convert a mixed number like 2 3/4 into an improper fraction, and see it reduced to lowest terms.

Find the greatest common factor of two or three whole numbers using the Euclidean algorithm.

Find the least common multiple of two or three whole numbers using the GCF relationship.

Round a number to a chosen number of decimal places using round-half-up, round up, round down or round to nearest.

Convert any number into scientific notation (a × 10ⁿ) and see its normalized coefficient, exponent and E-notation.

Compute a mod n — the remainder after division — with both the truncated and floored (always non-negative) result.

Find the slope of the line through two points, plus its rise, run and angle of inclination — with a vertical-line guard.

Find the midpoint of the segment between two points (x₁, y₁) and (x₂, y₂) — the exact center of the line segment.

Compute the straight-line distance between two points using the distance formula, plus the horizontal and vertical gaps.

Turn two points into the slope-intercept equation y = mx + b, with the slope, y-intercept and x-intercept — vertical lines handled.

Find the hypotenuse of a right triangle from two legs, one leg and an angle, one leg and the area, or a 45-45-90 or 30-60-90 triangle — or find a missing leg from the hypotenuse. Shows the exact radical form, both angles, area, perimeter and the worked steps.

Enter the two legs of a right triangle and get the hypotenuse, both acute angles, area and perimeter in one step.

Given two sides and the included angle, find the third side of any triangle with the law of cosines, plus the remaining angles.

Use the law of sines to find an unknown side from an angle-side pair and a second angle, with the full triangle solved.

From a circle’s radius get its circumference, area and diameter — the three measurements that depend only on the radius and π.

Reduce a square root to simplest radical form, e.g. √72 = 6√2, and show its exact decimal value.

Find the distance between two numbers on the number line as |a − b|, plus each value’s absolute value and their midpoint.

Enter a list of numbers to get their average (mean), along with the sum, count, minimum, maximum and range.

Find the percentage increase or decrease between two values, plus the raw difference and which direction it moved.

Answer “X is what percent of Y?” — turn a part and a whole into the percentage they represent.

Compare two percentages the right way — the gap in percentage points versus the relative percent change between them.

Split a restaurant bill fairly across a group with sales tax and tip added — and choose whether the tip is figured before or after tax.

Apply up to three discounts one after another and see why “20% + 10% + 5%” is never a 35% discount.

Find the nth term and the running sum of an arithmetic or geometric sequence from its first term and common difference or ratio.

Break any whole number into its prime factors in exponent form, with the count of prime factors and distinct primes.

Check whether a whole number is prime, and if it isn’t, see its smallest divisor as proof.

Convert numbers to Roman numerals and back, from 1 to 3999, with full validation of the numeral you enter.

Convert a value between binary, decimal, hexadecimal and octal in one step, with validation for the base you enter.

Round any number to a chosen number of significant figures, the way scientific and engineering work reports precision.

Combine up to four scores with different weights into a single weighted average — the math behind grades, ratings and indexes.

Solve a 2×2 system a₁x + b₁y = c₁ and a₂x + b₂y = c₂ for x and y using Cramer's rule, with the determinant shown.

Solve ax² + bx + c = 0 with the quadratic formula — real or complex roots, plus the discriminant that tells you which.

Find a real root of ax³ + bx² + cx + d = 0 numerically, then factor out the remaining quadratic for any other real roots.

Compute the determinant and inverse of a 2×2 matrix [[a, b], [c, d]], with a clear warning when the matrix is singular.

Compute the dot product and 3D cross product of two vectors, plus the angle between them, from their components.

Find the magnitude (length) of a 2D or 3D vector and its direction angle from its components.

Evaluate a polynomial at a given x using Horner's method — enter the coefficients from highest to lowest degree.

Paste a list of numbers and get the arithmetic mean, median, mode and range instantly — with the sorted values shown.

Combine two or three ratios into their compound ratio by multiplying antecedents and consequents, then simplify to lowest terms.

Turn any quantity-and-count pair into a unit rate — the amount per one unit — and see the reciprocal rate too.

Find the constant of variation from a known pair, then predict y for a new x under direct (y = kx) or inverse (y = k/x) variation.

Convert a number in any base from 2 to 36 into decimal (base 10), including fractional digits, negative numbers and very long values, with each digit’s place value, Horner’s-method steps, the exact fraction, two’s-complement reading and the same number in binary, octal, hex or any other base.

Find the exact area of any regular polygon from its number of sides and side length, using the apothem-and-perimeter formula.

Compute the interior and exterior angles and the total angle sum of any regular polygon from its number of sides.

Find the apothem — the distance from the centre of a regular polygon to the midpoint of a side — from the side length and number of sides.

Find a circle arc’s length, chord, sector area, segment area and height (sagitta) from any two of radius, central angle, chord, segment height or arc length, in degrees or radians, with steps.

Calculate the total, lateral and base surface area of a right circular cylinder from its radius and height.

Find the total and lateral surface area of a right circular cone, plus its slant height, from the radius and vertical height.

Compute the volume and surface area of a torus (a doughnut shape) from its centre-to-tube radius and tube radius.

Enter the three side lengths of any triangle and get its area by Heron’s formula, plus the semi-perimeter and perimeter.

Convert an angle between degrees, radians, gradians and full turns instantly, in either direction.

Enter an angle in degrees or radians and get its sine, cosine, tangent, and the reciprocal ratios cosecant, secant and cotangent.

Find an angle from its sine, cosine or tangent value using arcsin, arccos or arctan — with domain checking for asin and acos.

Evaluate sinh, cosh, tanh and their reciprocals coth, sech and csch for any real number, from the exponential definitions.

Add or subtract two fractions using a common denominator, then see the answer reduced to lowest terms and as a mixed number.

Multiply or divide two, three or four fractions, mixed numbers or whole numbers, with every step shown — mixed to improper, keep-change-flip, cross-cancelling, multiplying across and simplifying — and the answer as a fraction, mixed number, decimal and percent.

Reduce any fraction to its simplest form by dividing the numerator and denominator by their greatest common divisor.

Add, subtract, multiply or divide two mixed numbers — converted to improper fractions and returned as a reduced mixed number.

Convert any percentage into its decimal value and its equivalent fraction in lowest terms, all in one step.

Divide whole numbers the long way and see every bring-down, multiply and subtract step laid out in a table.

Divide one number by another and get the whole-number quotient and the remainder, the answer as “q R r”, a mixed number, a fraction and a repeating decimal, with long-division steps and the truncated, floored and Euclidean conventions for negative numbers.

Apply the product, quotient and power-of-a-power rules to two exponents on the same base and see the simplified expression.

Find the square and the cube of any number at once, along with its square root and cube root.

Compute the nth root of any number — square root, cube root, fourth root and beyond — with negative-base cases handled correctly.

Spell out any number in English or Spanish words — plain words, cheque/check writing with the currency, money amounts in words, or ordinals (first, twenty-third) — with decimals, UK “and”, Indian lakh and crore, and letter-case options.

Paste a list of values — numbers or words — and get a frequency table with counts, percentages and the most common value.

Find what share one part is of a total — turn a part and a whole into a clean percentage, plus the leftover share.

Work backwards from a value to the original amount before a percentage increase, decrease, or "is X% of what number".

Compare a measured value to the true value: get absolute error, relative error and percent error in one step.

Convert a ratio like 3:2 into a simplified fraction, a decimal and the percentage share of each part of the whole.

Convert between map (or model) distance and real-world distance using a scale ratio like 1:50,000.

Split a length into the golden ratio — get the longer and shorter segments so the whole is to the longer as the longer is to the shorter.

List the first n Fibonacci numbers, with the nth term, the running sum, and the ratio that approaches the golden ratio.

Find the nth triangular number — the count of dots in a triangle with n rows — with the formula and the running sum shown.

Add up a run of consecutive whole numbers from a start to an end value using Gauss’s pairing formula.

Add the squares of the first n whole numbers, 1² + 2² + … + n², with the closed-form formula and mean square.

Compute the arithmetic-geometric mean (AGM) of two positive numbers by iterating the arithmetic and geometric means to a shared limit.

Expand a decimal into its continued fraction and read off the best rational approximations (convergents) to the value.

Mix two solutions of different strengths and get the volume and concentration of the blend — the classic mixture word problem, solved.

If one worker finishes a job in A hours and another in B hours, how long together? Solve the classic work-rate problem instantly.

Solve any of distance, speed or time from the other two using d = s × t — pick the unknown and get the answer with the working.

In how many years will one person be a given multiple of another’s age? Solve the classic "times as old" age puzzle with the algebra shown.

Solve coin word problems step by step: how many of each coin from the total count and value, from a relation such as “4 more nickels than quarters” or “twice as many dimes”, or every combination of three coin types.

Solve for interest, principal, rate or time in a simple-interest problem using I = P·R·T — pick the unknown and see the rearranged formula.

Divide an amount in a given ratio like 2:3:5 and get each share exactly — the standard "share in the ratio" problem, done in one step.

Drive out at one speed and back at another over the same distance — your true average speed is the harmonic mean, not the simple average.

Convert a value with any conversion factor you supply — enter how many target units make one source unit and get the result plus the inverse factor.

Chain up to three conversion factors to convert a quantity step by step — the factor-label method that makes units cancel cleanly.

Multiply or divide two measurements and round the answer to the correct number of significant figures — the fewest that either input carries.

Round any number to the nearest 5, 10, 25, 50, 100 or 1000, and see the round-down and round-up values alongside it.

See markup and margin side by side for one cost — and read a full conversion table so you never confuse the two percentages again.

Convert a price between tax-inclusive and tax-exclusive, and see the exact tax amount either way — for VAT, GST or sales tax.

Work out the exact discount percentage needed to bring an original price down to the price you want to hit.

Find how many units you must sell to cover fixed costs, using the contribution-margin break-even formula step by step.

Compute the probability of a single event from favorable and total equally-likely outcomes — as a fraction, decimal, percentage and odds.

Convert betting odds — “A to B” for or against — into a clean probability percentage, decimal and implied fraction.

Combine two independent event probabilities: the chance both happen (AND), either happens (OR), and neither happens.

Find the long-run average outcome of a bet or game — enter the win amount, loss amount and win probability to see if it pays.

Score two options against weighted criteria and see which wins — a simple weighted decision matrix for any choice.

Split a restaurant bill and tip evenly across a group, with optional rounding so each person pays a clean amount.

Split a shared bill in proportion to what each of three people ordered, then add tip and tax fairly on top.

Combine weighted course components into your overall grade, and see the letter grade the weighted average earns.

Find the volume, total surface area, lateral area and space diagonal of a rectangular prism (box) from its length, width and height.

Enter a sphere’s radius to get its volume, surface area, diameter and great-circle circumference in one step.

Compute the volume, curved surface area, total surface area and base area of a hemisphere from its radius.

Find the volume and surface area of a triangular prism from its three base-triangle sides and its length.

Get the area, perimeter and diagonals of a parallelogram from its base, height and slant side.

Find the area, side length and perimeter of a rhombus from its two diagonals.

Compute the area and perimeter of a kite from its two diagonals and its two distinct side lengths.

Find the area, apothem and perimeter of a regular pentagon or hexagon from its side length.

Find the area of an annulus — the ring between two concentric circles — from the outer and inner radii.

Find the area of a circular segment, plus its chord and arc length, from the radius and central angle.

Find the volume, curved surface area and base radius of a spherical cap from the sphere radius and cap height.

Find the volume and approximate surface area of an ellipsoid from its three semi-axes.

Find the area enclosed between two quadratic curves over an interval by numerically integrating the absolute gap between them.

Approximate the definite integral of a quadratic with the composite trapezoidal rule and compare it against the exact value.

Estimate the first and second derivative of a cubic at any point using the central-difference method, plus the tangent-line equation.

Estimate the limit of a rational function as x approaches a point by evaluating it from both sides — great for removable 0/0 forms.

Find a root of a cubic with Newton's method — fast quadratic convergence from a starting guess, with iteration count and residual.

Locate a root of a cubic inside a bracketing interval with the bisection method — guaranteed convergence when the endpoints differ in sign.

Interpolate (or extrapolate) a y-value at any x from two known points using the straight-line formula, with the fitted slope shown.

Fit the exact parabola y = ax² + bx + c through three points, then predict y at any x — the unique second-degree curve on your data.

Fit y = A·e^(kx) through two points, recovering the amplitude A and growth-rate constant k, then predict y at any x.

Solve for the constant per-period growth rate that takes a starting value to an ending value over a number of periods.

Convert a constant per-period growth rate into the exact doubling time, and compare it against the Rule of 72 estimate.

Turn a continuous decay constant λ into half-life and mean lifetime, and read off how much remains after a chosen time.

Turn a starting value and an ending value into a compound annual growth rate (CAGR) — the single yearly percentage that connects them.

Average a list of numbers where each value carries its own weight — enter values and matching weights to get the true weighted mean.

Compute the harmonic mean of a list of positive numbers — the right average for rates, ratios and speeds.

Find the root mean square of a list of numbers — the quadratic mean used for AC signals, deviations and magnitudes.

Build the running total and running average of a list step by step — see how the sum and mean evolve as each value is added.

Standardize a list of numbers to z-scores — subtract the mean and divide by the standard deviation so the data has mean 0 and SD 1.

Rescale a list of numbers to 0–1, 0–100, −1 to 1 or any range with min-max normalization, using the data’s own min and max or fixed bounds. Normalize or reverse a single value, with steps, a table, outlier check and the linear scaling equation.

Flag outliers in a list with the interquartile-range rule — values beyond Q1 − 1.5·IQR or Q3 + 1.5·IQR are marked as outliers.

Smooth a series with a simple moving average — pick a window size and see the averaged values that reveal the underlying trend.

Rank a list of numbers from highest to lowest (or lowest to highest), with average ranks for ties — the standard statistical ranking.

Turn a list of numbers into a dot-plot frequency table — how often each distinct value appears, shown as dots, counts and percentages.

Get the five-number summary of a data set — minimum, first quartile, median, third quartile and maximum — plus the range and IQR.

Convert a nominal (stated) annual interest rate into the effective annual rate given how often it compounds — the true yearly cost or yield.

Grow a principal under continuous compounding with A = Pe^{rt}, and see the effective annual rate that continuous compounding implies.

Find the present value of an annuity due — equal payments made at the start of each period, like rent or lease payments paid in advance.

Compute the future value of an annuity due — equal deposits made at the start of each period — and compare it with end-of-period saving.

Compute the level payment that fully amortizes a loan using the standard annuity payment formula, plus total interest paid over the term.

Find the outstanding balance of an amortizing loan after any number of payments, using the closed-form balance formula.

Find the level deposit needed to accumulate a target amount by a future date — the classic sinking-fund payment formula.

Price a fixed-coupon bond from its yield to maturity by discounting every coupon and the face value at the required yield.

Solve numerically for a bond’s yield to maturity from its market price using bisection — the single rate that equates price to discounted cash flows.

Compute a bond’s Macaulay duration — the cash-flow-weighted average time to receive its payments — and its modified duration.

Value a stream of payments that grows at a constant rate forever with the Gordon growth formula PV = C / (r − g).

Estimate how long it takes a quantity to double at a constant growth rate using the Rule of 70, and compare it with the exact doubling time.

Find the area of a triangle from its base and perpendicular height using A = ½ × base × height, the simplest triangle-area formula.

Compute the area of a triangle from the (x, y) coordinates of its three vertices using the shoelace (surveyor’s) formula.

Find the centroid (geometric centre of mass) of a triangle from its three vertex coordinates — the average of the three points.

From the three side lengths of a triangle, find the inradius (inscribed circle) and circumradius (circumscribed circle) plus the area and perimeter.

Find the area of a trapezoid from its two parallel sides and the height between them using A = ½ (a + b) × h.

Find the area and arc length of a circular sector from its radius and central angle in degrees — the “pizza slice” of a circle.

Find the area of a circular segment — the region between a chord and its arc — from the radius and central angle in degrees.

Find the perimeter and area of a regular polygon from the number of sides and the side length, plus the apothem and interior angle.

Find the area of any simple polygon from a list of its vertex coordinates using the shoelace (surveyor’s) formula.

Find the volume of any prism from its base area and length using V = base area × length — works for any cross-section.

Find the lateral (side) surface area of a cylinder from its radius and height using LSA = 2πrh, plus the total surface area.

Find the slant height of a right circular cone from its radius and vertical height using l = √(r² + h²), plus the lateral surface area.

Update a prior probability with new evidence using Bayes’ theorem — ideal for medical-test, spam-filter and diagnostic reasoning.

Build a two-stage probability tree and get every path probability plus the marginal P(B) — great for with/without-replacement and test-reasoning problems.

Find the probability that an event happens at least once in n independent trials using the complement rule P = 1 − (1 − p)ⁿ.

Find the complement P(not A) = 1 − P(A), plus the odds for and against an event, from a single probability.

Add the probabilities of mutually exclusive events to get P(A or B) = P(A) + P(B), and check whether the events can be disjoint at all.

Multiply the probabilities of independent events for the joint P(A and B) = P(A) × P(B), and get P(A or B) and “neither” too.

Compute “n choose k” — the binomial coefficient C(n, k) — the number of ways to pick k items from n where order does not matter.

Count the arrangements of a multiset with the multinomial coefficient n! / (k₁! k₂! …) — e.g. the distinct orderings of the letters in MISSISSIPPI.

Count the ways to seat n distinct people around a circular table — (n − 1)! for rotations, or (n − 1)!/2 when reflections are the same.

Count ordered arrangements when repetition is allowed — nʳ ways to fill r positions from n choices, as for PINs, passwords and license plates.

Count derangements — permutations where nothing stays in its original place — with the subfactorial !n, plus the probability of a random derangement.

Apply the pigeonhole principle: find the guaranteed fullest container ⌈n/k⌉ items, and the items needed to force a container to a target count.

Convert between Celsius, Fahrenheit, Kelvin and Rankine with the exact affine formulas — not a rough approximation.

Convert length and distance between metric and imperial units — millimetres to miles — using exact internationally defined factors.

Convert mass between grams, kilograms, ounces, pounds, stone and tonnes with exact avoirdupois factors.

Convert volume between litres, millilitres, cubic metres and US liquid units — cups, pints, quarts and gallons.

Convert speed between km/h, mph, m/s, feet per second and knots with exact factors.

Convert pressure between pascals, kPa, bar, psi, atmospheres, mmHg and inHg with exact SI factors.

Convert energy between joules, kilojoules, calories, kilocalories, watt-hours, kWh and BTU with exact factors.

Convert area between square metres, square feet, acres, hectares and square miles with exact factors.

Convert digital storage between bits and bytes, decimal KB/MB/GB/TB and binary KiB/MiB/GiB/TiB — and see why they differ.

Convert time between seconds, minutes, hours, days, weeks, months and years using average-length definitions.

Convert angles between degrees, radians, gradians and turns — plus arcminutes and arcseconds — with exact factors.

Convert fuel economy between US mpg, UK mpg, km/L and L/100km — correctly handling the inverse relationship.

Add, subtract and multiply two complex numbers a + bi and c + di and see every intermediate step of the real and imaginary parts.

Convert a complex number a + bi to polar form: its modulus |z| and its argument (phase angle) in both radians and degrees.

Compute the determinant of any 3×3 matrix by cofactor expansion, with the three 2×2 minors shown so you can follow every step.

Multiply two 2×2 matrices A × B and get every entry of the product from the row-by-column dot products.

Solve a system of three linear equations in three unknowns with Cramer’s rule, showing the main determinant and each substituted determinant.

Find the eigenvalues of a 2×2 matrix from its trace and determinant, handling real, repeated and complex-conjugate cases.

Project vector a onto vector b in 3D: get the projection vector, its scalar component and the perpendicular (rejection) part.

Find the angle between two 3D vectors from their dot product and magnitudes, in both degrees and radians.

Compute the cross product a × b of two 3D vectors, its magnitude, and the area of the parallelogram they span.

Evaluate a point on a 3D line given in parametric form P = P₀ + t·d for any value of the parameter t.

Convert polar coordinates (r, θ) to rectangular (x, y) coordinates, with the angle entered in degrees or radians.

Convert rectangular coordinates (x, y) to polar (r, θ), with the angle reported in both degrees and radians using atan2.

Compare two pizza sizes by actual area and price per square inch to see which round pizza is really the better deal.

Find how many small pizzas (or pies) equal one large by comparing their round areas — the honest way to size a group order.

Convert a real-world measurement into its scale-model size (or back) for any modelling scale like 1:24, 1:87 or 1:144.

Turn a measurement taken off a scaled drawing into the real-world dimension (or back) for any blueprint scale.

Find the largest crop of a given width and height that fits a target aspect ratio, with the pixels trimmed from each side.

Count how many rectangular tiles fit across and down a surface as a whole-tile grid, including the grout gap between them.

Estimate how many people fit in a room or venue from its floor area and layout — standing, theatre, banquet, classroom, U-shape — after taking out the stage, bars, buffets and dance floor, or find the floor area you need for a guest count. Includes a fire-code occupant-load reference.

Estimate how many cars a parking lot holds — from its total area and the space each car needs including aisles, or by laying out rows on a rectangular lot at 90°, 60°, 45° or 30° — plus the lot size needed for a number of cars and the ADA accessible-space minimum.

Split a batch of paint into two colors by a mixing ratio, or work out the ratio you already have — for a repeatable custom shade.

Compute the gear ratio from driver and driven tooth counts, plus the output speed and torque change of the pair.

Find the driven pulley speed and belt drive ratio from two pulley diameters and the driver RPM.

Compute a lever’s mechanical advantage from its effort and load arm lengths, and the effort force needed to lift a load.

Find the percent difference between two values with the working shown — the symmetric measure that treats neither number as the “original” — plus the percent change in both directions and the gap in percentage points.

Simplify a ratio of three numbers, a : b : c, to its lowest whole-number terms and see each part as a share of the whole.

Solve everyday "if this much costs/takes that much, how much for…" problems by setting up and cross-multiplying a proportion.

Turn any pack price and size into a price per 100 g and per kilogram (or per 100 mL and per litre), including multipacks and 2-for-1 or 3-for-2 deals, and compare up to three products to see which is the better buy.

Compare two package deals by unit price and see which is the better buy and how much you save per unit.

Apply a chain of successive percentage markups to a starting cost and see the compounded final price and total markup.

Enter the first few terms of a sequence and detect whether it is arithmetic or geometric, then predict the next term.

Sum the first n terms of an arithmetic sequence from its first term and common difference — or from the first and last terms, or any two terms — sum only terms m to n, or find how many terms reach a target sum, with the nth-term and sum formulas, a term table and full worked steps.

Sum the first n terms of a geometric sequence from its first term and common ratio, with the standard closed-form formula.

Add up grade points weighted by course credit hours to compute a GPA — the credit-weighted average behind every transcript.

Count how many distinct batting orders a team can make from its roster — a permutations problem, n choose the lineup arranged.

Count the handshakes when everyone in a group shakes hands once: n(n − 1)/2, plus the reverse (people from handshakes), couples who skip their partner, two teams lining up, rounds needed and time taken, with steps and a table.

Convert clock time (hours, minutes, seconds) to decimal hours and back — the exact numbers payroll and timesheets need.

Scale any recipe up or down by servings and rescale one ingredient amount to match — with the exact scaling factor shown.

Add a tip, split the bill, and round each share up to a whole amount so nobody fumbles with coins — with the real tip you end up leaving.

Turn a single course percentage into its letter grade and 4.0-scale grade point using the standard US +/− grading table.

Convert a fraction of an hour — like 3/4 or 2/3 — into exact minutes and seconds for scheduling, billing and timing.

Convert a number of days into years, months and days — with fixed 365/30-day units, average calendar lengths, or exactly on the calendar from a start date — plus weeks, hours and the reverse conversion.

Convert a running or walking pace in minutes per mile into miles per hour — and get km/h and min/km at the same time.

Calculate heating and cooling degree days from a base temperature and average outdoor temperature — the standard measure of heating and cooling demand.

Convert a slope or grade percentage into a run-to-rise ratio (1 in N) and its angle in degrees — for ramps, roads and drainage.

Find the total fall of a drain or sewer pipe from its length and slope in inches per foot — plus the equivalent percent grade.

Turn a rise and a run into the slope angle in degrees (and degrees-minutes-seconds), percent grade, 1:n ratio, roof pitch in 12 and sloped length, or work back from an angle, grade or sloped length. Includes ADA ramp and stair checks.

Convert parts per million to percent, per mille and a plain fraction — and back — for water testing, chemistry and calibration.

Spell out a dollar amount in words the way you write it on a check — dollars in words and cents as a fraction over 100.

Find the average score you need on your remaining assignments to finish a course at or above a target grade.

Track how far along a project is from the amount done and the total, with the running total and the amount still to go.

Scale a recipe or mix written for one number of servings up or down to the servings you actually need.

Solve a proportion a/b = c/x for the missing value by cross-multiplying — enter three numbers and get the fourth.

Turn a cost and a markup percentage into the selling price, the profit per unit, and the resulting margin.

Break any dollar amount into the fewest bills and coins — perfect for cash drawers, payroll envelopes and change.

Add tax and tip to a restaurant bill and split the grand total evenly across the table, down to the per-person amount.

Type in a list of letter grades and get the unweighted GPA on the standard 4.0 scale, plus the average grade point.

Turn a list of values into a running cumulative total and the cumulative percentage of the grand total at each step.

Score one option against weighted criteria to get a single decision number — a simple weighted-average decision matrix.

Chance of winning n independent tries in a row, the chance that a streak of that length shows up somewhere in a season of games, the longest streak to expect, how many games it takes on average to see one, plus unbeaten and losing streaks with draws and a win rate taken from a team’s record.

Convert parts per million (ppm) to percent, per mille and a plain fraction.

Find the straight-line distance between two points (x1,y1) and (x2,y2) on a plane.

Find the area of a triangle from its three side lengths using Heron’s formula.

Find the volume of a truncated cone (frustum) from its two radii and height.

Compute the dot product of two 3D vectors and the angle between them.

Compute the cross product of two 3D vectors and the magnitude of the result.

Find the magnitude (length) of a 2D or 3D vector from its components.

Compute the logarithm of a number in any base, plus its natural and base-10 logs.

Find the area and arc length of a circle sector from its radius and central angle.

Convert Indian lakh and crore to thousands, millions and billions (and back) — 1 lakh = 100,000, 1 crore = 10 million — with Indian and Western comma formatting side by side.

Estimate the internal-consistency reliability of a multiple-choice or right/wrong test with the Kuder–Richardson formulas: KR-21 from the number of items, mean and standard deviation, and KR-20 from the item difficulty (p) values, with the standard error of measurement and the interpretation bands used for classroom and standardised tests.

Compute the standard error of measurement from a test's reliability and standard deviation, the 68%, 90% and 95% confidence bands around an observed score, the regressed true-score estimate, and whether two scores (or two administrations) differ by more than measurement error.

Analyse a test item from the number of high-scoring and low-scoring students who answered it correctly: the difficulty index (p-value), the discrimination index (D = upper − lower), the point-biserial approximation, the ideal difficulty for the number of options, and whether to keep, revise or drop the item.

Predict how a test's reliability changes when it is lengthened or shortened, find the number of items or raters needed for a target reliability, correct a split-half correlation to full-test reliability, and see the reliability of an average of several raters or measurements.

Convert between the score scales used on standardised tests: percentile rank, normal curve equivalent (NCE), stanine, z-score, T-score (mean 50, SD 10), deviation IQ / standard score (mean 100, SD 15) and scaled score (mean 10, SD 3), with descriptive categories and the caveats about averaging percentiles.

Evaluate any arithmetic expression step by step using PEMDAS / BODMAS: parentheses first, then exponents, multiplication and division left to right, then addition and subtraction.

Divide a polynomial by x − r with synthetic division: see the full tableau, the quotient, the remainder, the remainder theorem check and whether x − r is a factor.

Enter an angle in degrees or radians to get its point on the unit circle, exact sin, cos, tan, csc, sec and cot values, the quadrant, the reference angle and the coterminal angle.

List every possible rational root ±p/q of a polynomial with integer coefficients, test each candidate, and get the actual rational zeros with their multiplicity and the factored form.

Solve ax² + bx + c = 0 by completing the square, step by step, and get the exact solutions (radicals or complex), the vertex form y = a(x − h)² + k and the vertex.

Factor any trinomial ax² + bx + c over the integers with the ac (grouping) method, step by step: GCF, the two numbers, splitting the middle term and the final factored form.

Solve one-variable linear equations such as 5(x − 2) + 4 = 2x + 9 step by step: expand, collect terms, isolate x and check, with exact fractions.

Solve a linear inequality such as 2x − 5 < 3x + 1 step by step, flipping the sign when dividing by a negative, and get interval notation, set-builder notation and the number line.

Solve |ax + b| = c and absolute value inequalities (<, ≤, >, ≥) step by step, with both cases, interval notation and the distance form |x − h| = d.

Rationalize a / (b + d√c): multiply by √c or by the conjugate, simplify the radical and the fraction, and get the exact result with a decimal check.

Expand (ax + b)ⁿ or (ax + by)ⁿ with the binomial theorem: every term with its binomial coefficient, the Pascal’s triangle row, the middle term and the sum of coefficients.

Find the equation of the line parallel or perpendicular to Ax + By = C through a given point, in slope-intercept, point-slope and standard form, plus the distance from the point to the line.

Find the vertex, focus, directrix, axis of symmetry, focal length and latus rectum of a parabola y = ax² + bx + c or x = ay² + by + c, with its vertex and 4p forms.

From the center and semi-axes of an ellipse get its standard equation, foci, vertices, co-vertices, focal distance, eccentricity, latus rectum, directrices, area and perimeter.

From the center, a, b and orientation of a hyperbola get its standard equation, vertices, foci, asymptote equations, eccentricity, latus rectum and directrices.

Decompose a rational function N(x)/D(x) into partial fractions: factor the denominator, set up A/(x − r) terms (repeated and quadratic factors too), solve for the constants exactly.

Convert a repeating (recurring) decimal such as 2.1363636… to a fraction in lowest terms and a mixed number, with the multiply-and-subtract steps shown.

Type any boolean expression with AND, OR, NOT, XOR, NAND or NOR and variables A–E to get its full truth table, minterms, maxterms and the canonical sum-of-products and product-of-sums forms.

Simplify any boolean expression with up to five variables (A–E) to its minimal sum of products and product of sums using the Quine–McCluskey method, with prime and essential prime implicants listed.

Enter minterms and optional don't-care terms for 2 to 5 variables and get the Karnaugh map in Gray-code order, the groups, and the minimal sum-of-products and product-of-sums expressions.

Add, subtract, multiply and divide binary numbers of any length (up to 64 bits each), with the result in binary, decimal and hex, the remainder for division, and the decimal check.

X and Y coordinates of equally spaced holes on a bolt circle (PCD) for drilling and CNC — hole-to-hole distance, angle between holes, start angle, center offset and direction.

Find coterminal angles in degrees or radians: the principal angle in [0°, 360°), the largest negative coterminal angle, and a list of positive and negative coterminal angles.

Find the reference angle of any angle in degrees or radians, with the quadrant, the rule used (180° − θ, θ − 180°, 360° − θ) and the signs of sin, cos and tan.

Approximate a definite integral with Simpson's 1/3 or 3/8 rule: type f(x), the limits and n, and get the estimate, step size, weights table and a trapezoidal comparison.

Calculate left, right and midpoint Riemann sums for any f(x) on [a, b] with n rectangles — with Δx, sample points table and the trapezoidal average for comparison.

Build the Taylor (or Maclaurin) polynomial of eˣ, sin x, cos x, ln(1 + x), √(1 + x) or 1/(1 − x) about any center, evaluate it, and see the error and radius of convergence.

Solve dy/dx = f(x, y) numerically with Euler's method or the improved Euler (Heun) method: enter the initial value, step size and number of steps to get every step in a table.

Find the Lagrange interpolating polynomial through 2 to 10 points with exact fractions, evaluate it at any x, and see each basis polynomial value Lᵢ(x).

Row-reduce any matrix up to 8×8 to reduced row echelon form with exact fractions, every row operation shown, the rank and pivot columns, and the solution of an augmented system.

Normalize a 2D or 3D vector: get the unit vector û = v/|v|, the magnitude, the exact form with a simplified root, and the direction angles with each axis.

Find the shortest distance from a point (x₀, y₀) to the line Ax + By + C = 0, with the exact value, the foot of the perpendicular and whether the point lies on the line.

Find the modular multiplicative inverse of a mod m with the extended Euclidean algorithm: every division step, the Bézout identity, a check, and when no inverse exists.

Solve a system of congruences x ≡ aᵢ (mod mᵢ) with the Chinese remainder theorem: the smallest solution, the general solution, step-by-step merging, and non-coprime moduli.

Multiply two numbers the long way: every partial product by place value (ones, tens, hundreds), the column sum, and where the decimal point goes when you multiply decimals.

Divide one polynomial by another of any degree with long division: quotient, remainder, every divide-multiply-subtract step, and the check D(x)·Q(x) + R(x) = P(x).

Solve ax⁴ + bx³ + cx² + dx + e = 0 for all four roots, real and complex, with multiplicities, the real-root count and a Vieta check of the sum and product of the roots.

Solve any triangle from three known parts — SSS, SAS, ASA, AAS or the ambiguous SSA case — and get every side and angle, area, perimeter, inradius, circumradius and triangle type.

Find the rank of any matrix up to 10 × 10 by exact Gaussian elimination, with the row echelon form, pivot columns, nullity and whether the matrix has full rank or is invertible.

Factor a square matrix into a lower triangular L and an upper triangular U with exact fractions — Doolittle A = LU or partial pivoting PA = LU — plus the permutation matrix and determinant.

Get any row of Pascal’s triangle up to row 60 with exact big numbers, or print the triangle itself, plus the row sum 2ⁿ, the middle entry and the binomial coefficients C(n, k).

Solve a·b^(cx + d) + e = k for x step by step: isolate the power, take logarithms, solve the linear equation, and check the answer — with the no-solution case explained.

Find the critical points of a polynomial f(x): solve f′(x) = 0, classify each point as a local maximum, minimum or neither with the second-derivative test, and list the inflection points.

Find the volume of a solid of revolution: rotate y = f(x), or the region between f(x) and g(x), about the x-axis (disk/washer) or the y-axis (shell method) over [a, b].

Solve ax² + bx + c > 0, ≥ 0, < 0 or ≤ 0: exact roots, the solution in interval and set-builder notation, and a sign chart with test points for every interval.

Find the equation of the perpendicular bisector of the segment between two points: midpoint, slope of the segment, negative reciprocal slope, slope-intercept and standard form.

Find the circumcenter of a triangle from its three vertices, the circumradius, the equation of the circle through the three points, and whether the center lies inside, on or outside.

Find the equation ax + by + cz = d of the plane through three points in 3D: the vectors PQ and PR, the normal vector from their cross product, the intercepts and the distance from the origin.

Raise a complex number a + bi to the nth power or find all n of its nth roots with De Moivre’s theorem: modulus, argument, polar form and every root in a + bi form.

Check whether three whole numbers form a Pythagorean triple (a² + b² = c²), see if it is primitive and its Euclid parameters m and n, or list every triple up to a given hypotenuse.

List equivalent fractions for any fraction by multiplying the top and bottom by the same number, find its simplest form, or check whether two fractions are equivalent by cross-multiplying.

Compare and order fractions, mixed numbers and decimals from least to greatest or greatest to least, with each fraction rewritten over the least common denominator and as a decimal.

List all the factors (divisors) of a whole number with its factor pairs, number and sum of factors, prime factorization, prime or composite, and whether it is perfect, abundant or deficient.

Find the average rate of change of a function f(x) on an interval [a, b] — the slope of the secant line — with f(a), f(b), Δy, Δx and the secant line equation.

Free online scientific calculator: type a full expression with + − × ÷, powers, roots, sin cos tan in degrees, radians or gradians, ln, log, nCr, gcd, π, e and factorials, and see the order-of-operations steps, the fraction and other forms of the answer.

Add, subtract, multiply, divide, raise to powers and find the GCD/LCM of huge whole numbers with every digit exact — up to 5,000 digits per input, no rounding or scientific-notation loss.

Compose two polynomial functions: get f(g(x)) and g(f(x)) expanded and simplified, their degree, and the value of each composition at any x with the inner value shown.

Simplify the difference quotient [f(x + h) − f(x)] / h for any polynomial step by step: f(x + h) expanded, the simplified quotient, its limit f′(x) as h → 0 and a numeric value.

Find the inverse of a linear or rational function f(x) = (ax + b)/(cx + d): f⁻¹(x) in simplest form, the domain and range of f and f⁻¹, and a numeric check that f⁻¹(f(x)) = x.

Find every asymptote of a rational function P(x)/Q(x): vertical asymptotes, the horizontal or slant (oblique) asymptote, holes, the domain and the x- and y-intercepts.

Find every value c guaranteed by the Mean Value Theorem for a polynomial on [a, b]: the average rate of change, f′(c) = [f(b) − f(a)]/(b − a), the secant and tangent lines, and Rolle’s case.

Find the average value of f(x) on [a, b] with f_avg = (1/(b − a))∫ f(x) dx: the definite integral, the average height, the point c where f(c) = f_avg, and the minimum and maximum.

Find the arc length of a curve y = f(x) from x = a to x = b with L = ∫√(1 + f′(x)²) dx, plus the straight-line distance between the endpoints and how much longer the curve is.

For y = A sin(Bx − C) + D or y = A cos(Bx − C) + D find the amplitude, period, phase shift, vertical shift, midline, maximum, minimum, frequency and the five key points of one cycle.

Find sin 2θ, cos 2θ, tan 2θ and sin(θ/2), cos(θ/2), tan(θ/2) from an angle or from one known ratio (sin, cos or tan θ) and its quadrant, with exact fractions when they exist.

Find the orthocenter of a triangle from the coordinates of its three vertices: the point where the altitudes meet, the equation and length of each altitude, and whether it lies inside or outside.

Find the point that divides the segment from A(x₁, y₁) to B(x₂, y₂) internally or externally in the ratio m : n with the section formula, plus the midpoint and the distances AP and PB.

Identify the conic Ax² + Bxy + Cy² + Dx + Ey + F = 0 — circle, ellipse, parabola, hyperbola or degenerate — with the discriminant, center or vertex, axes, eccentricity, rotation and standard form.

Compute Euler’s totient φ(n), the count of integers from 1 to n that are coprime to n, using the prime factorization and the product formula, with the list of coprime numbers for small n.

Compute aᵇ mod m for huge exponents with fast square-and-multiply: the result, the binary form of b, every squaring step in a table and the number of multiplications used.

Solve the linear congruence ax ≡ b (mod m): check whether gcd(a, m) divides b, reduce the congruence, find the modular inverse and list every solution modulo m.

Solve ax + by = c in integers: test whether gcd(a, b) divides c, find a particular solution with the extended Euclidean algorithm, the general solution and the count of positive solutions.

Solve a linear homogeneous recurrence aₙ = p·aₙ₋₁ + q·aₙ₋₂ with initial terms a₀ and a₁: characteristic equation, roots, closed-form formula, the nth term and a table of the first terms.

Compute the scalar triple product a · (b × c) of three 3D vectors: the cross product b × c, the volume of the parallelepiped and tetrahedron, orientation and a coplanarity test.

Check whether three side lengths can form a triangle with the triangle inequality theorem, or find the possible range of the third side from two sides, plus the triangle type and area.

Apply Descartes’ rule of signs to a polynomial: count sign changes in f(x) and f(−x), list the possible numbers of positive, negative and non-real roots, and compare with the actual roots.