Mathematics RU

Mathematics

Problem solvers

Type in your own problem and get a step-by-step solution: what we do at each step and why it is allowed. Each solver comes with an article on how to solve such problems and where people usually slip.

Numbers

Fraction arithmetic Adding, subtracting, multiplying and dividing fractions, mixed numbers, brackets and powers, simplifying, comparing and converting to decimals, with every step explained. 1/2 + 1/3 Chapter 5 · Chapter 4 Solve GCD, LCM and Euclid's algorithm Type two or more numbers, and the solver finds the greatest common divisor and the least common multiple in two ways and shows why the answer is right. 84 and 126 Chapter 4 · Chapter 3 Solve Prime factorisation Type a number, and the solver breaks it into prime factors step by step, builds a factor tree, finds all the divisors and checks the answer. 360 Chapter 3 · Chapter 4 Solve Percentages, compound interest, loans A percentage of a number, a number from its percentage, the percentage change, chains of discounts and mark-ups, savings with compound interest and a loan repayment schedule, with every step explained. 15% of 240 Chapter 5 · Chapter 29 Solve Number bases Binary, octal, hexadecimal and any other base up to 36, fractions and Roman numerals, with step-by-step working and a check. 255 to 2 Chapter 1 Solve

Algebra

Quadratic equation Type the equation in any form: with brackets, fractions, x on both sides. The solver brings it to the form $ax^2 + bx + c = 0$, finds the discriminant and the roots, checks them by substitution and draws the parabola. x^2 - 5x + 6 = 0 Chapter 10 · Chapter 15 Solve Linear equation Type an equation with brackets, fractions or decimals. The solver clears the denominators, expands the brackets, moves the terms across, explains every step and at the end checks the root by substitution. 3(x - 2) = 2x + 5 Chapter 8 Solve System of linear equations Type a system, and the solver finds the answer by substitution, elimination, Cramer’s rule or Gaussian elimination, explains every step and draws the lines or planes. 2x + 3y = 7; x - y = 1 Chapter 11 · Chapter 36 Solve Higher-degree equations and factoring Cubic, biquadratic and other equations of degree above two, factorising polynomials and polynomial long division, step by step, with every move explained and a graph. x^3 - 6x^2 + 11x - 6 = 0 Chapter 14 Solve Inequalities by the interval method Linear, quadratic, rational and double inequalities: the solver marks the points on the number line, fills in the signs taking multiplicity into account, picks the intervals and checks the answer by substitution. x^2 - 5x + 6 > 0 Chapter 11 Solve Curve sketching Type a function, and the solver works through a full analysis in the standard order: domain, symmetry, zeros, asymptotes, extrema, concavity and inflection points; then it builds a sign table and draws the graph with every point it found. y = x^3 - 3x Chapter 9 · Chapter 27 Solve Exponential and logarithmic equations Type an equation or inequality with powers and logarithms, and the solver finds where it’s defined, picks a method, explains every step and weeds out extraneous roots. 4^x - 3*2^x + 2 = 0 Chapter 12 · Chapter 29 Solve Complex numbers Adding, multiplying, dividing and raising complex numbers to powers step by step, modulus and argument, polar and exponential forms, nth roots and equations, with a picture on the complex plane. (1 + 2i)(3 - i) Chapter 15 Solve Progressions Type what you know (the first term, the common difference or ratio, any terms, a sum) and the solver finds the rest: the nth term, the sum, the number of terms, the sum to infinity, with every step explained. a1 = 3, d = 4, find a20 and S20 Chapter 13 Solve

Geometry

Solving a triangle Give three elements of a triangle, and the solver finds the rest with the sine and cosine rules, sorts out the case of two solutions and draws the triangle to scale. a = 5, b = 7, C = 60° Chapter 17 · Chapter 21 Solve Trigonometric equation Type an equation, and the solver finds every family of roots, shows them on the unit circle and, if you like, picks out the roots in an interval, as school-leaving exams often require. sin x = 1/2 Chapter 21 Solve

Calculus

Derivative Type a function, and the solver finds its derivative by the same rules you’d use on paper, explains every step, simplifies the answer, checks it numerically and shows a tangent line you can slide along the graph. x^2 sin x Chapter 26 · Chapter 27 Solve Integral Indefinite and definite integrals, improper integrals and areas of regions. The solver picks a method (the table, substitution, integration by parts, partial fractions), explains every step and checks the answer by differentiating. ∫ x^2 sin x dx Chapter 28 Solve Limit Type the limit of a function or a sequence, and the solver identifies the indeterminate form, picks a technique (cancelling, the conjugate, equivalents, standard limits, L’Hôpital’s rule), explains every step and compares the answer with a table of values. lim x->0 sin(x)/x Chapter 25 Solve Differential equation First-order equations (separable, linear, Bernoulli, homogeneous) and second-order linear equations with constant coefficients. The solver recognises the type, solves it step by step, finds the constants from the initial conditions and checks the answer by substitution. y' = 2xy Chapter 31 · Chapter 29 Solve

Linear algebra

Matrices: determinant, inverse, rank, eigenvalues Enter a matrix in the table, and the solver computes the determinant three ways, finds the inverse by Gauss–Jordan elimination or through the adjugate, the rank, powers, products and eigenvectors, explaining every step. determinant [2 1 3; 0 -1 4; 1 2 0] Chapter 35 · Chapter 36 · Chapter 38 Solve

Structures

Congruences Linear congruences, systems by the Chinese remainder theorem, remainders of large powers and modular inverses, with step-by-step working and a check. 3x ≡ 4 (mod 7) Chapter 41 · Chapter 4 Solve

Chance & data

Permutations and combinations Answer three questions (all or some, does order matter, can things repeat) and the solver picks the formula, explains it, computes the answer and, for small problems, lists every option. 10 choose 3 Chapter 45 Solve Probability: Bayes and Bernoulli trials Classical probability by counting outcomes, drawing from an urn, total probability and Bayes’ theorem with a tree and “natural frequencies”, Bernoulli trials with an exact answer and the De Moivre–Laplace approximation, expectation and variance from a table. two dice, sum greater than 8 Chapter 47 · Chapter 48 Solve
Mathematics

Chapters

Introduction

  1. 0 What mathematics is

I Numbers

  1. 1 Counting
  2. 2 Zero and minus
  3. 3 Primes
  4. 4 Divisibility
  5. 5 Fractions
  6. 6 Root two

II Algebra

  1. 7 Letters
  2. 8 Equations
  3. 9 Functions and graphs
  4. 10 Quadratic equations
  5. 11 Systems and inequalities
  6. 12 Powers and logarithms
  7. 13 Sequences
  8. 14 Polynomials
  9. 15 Complex numbers

III Geometry

  1. 16 Euclid's Elements
  2. 17 The triangle
  3. 18 Pythagoras' theorem
  4. 19 The circle and π
  5. 20 Area and volume
  6. 21 Trigonometry
  7. 22 Vectors
  8. 23 Symmetry
  9. 24 Conic sections

IV Calculus

  1. 25 Limits
  2. 26 The derivative
  3. 27 Extrema
  4. 28 The integral
  5. 29 The number e
  6. 30 Series
  7. 31 Differential equations
  8. 32 Several variables
  9. 33 Fourier series
  10. 34 Complex analysis

V Linear algebra

  1. 35 Matrices
  2. 36 Gaussian elimination
  3. 37 Vector spaces
  4. 38 Eigenvectors
  5. 39 Least squares and SVD

VI Structures

  1. 40 Groups
  2. 41 Remainders and ciphers
  3. 42 Rings and fields
  4. 43 Galois theory
  5. 44 Zeta and curves

VII Chance & data

  1. 45 Combinatorics
  2. 46 Graphs
  3. 47 Probability
  4. 48 Random variables
  5. 49 Statistics
  6. 50 Markov and information

VIII Foundations

  1. 51 Logic and sets
  2. 52 Infinities
  3. 53 Rigorous analysis
  4. 54 Measure and integral
  5. 55 Functional analysis
  6. 56 Gödel and Turing

IX Horizons

  1. 57 Topology
  2. 58 Non-Euclidean geometries
  3. 59 Chaos and fractals
  4. 60 The frontier