Mathematics
made simple.

Addition combines two or more numbers into one — the sum. Learn the names of the parts, the laws of addition (commutativity, associativity, identity element) and how to add in columns with carrying.

Subtraction is the inverse of addition — take the subtrahend away from the minuend to get the difference. Learn the names, the properties, the link to addition and how to subtract in columns with borrowing.

Multiplication is repeated addition of the same number. Learn the names of the factors and the product, the laws of multiplication (commutativity, associativity, distributivity), the times tables and column multiplication.

Division is the inverse of multiplication — divide the dividend by the divisor to get the quotient. Learn the names, the link to multiplication, division with a remainder and why you must never divide by zero.

When several operations meet in one expression, we evaluate them in a fixed order: brackets first, then powers, then multiplication and division, and finally addition and subtraction. Learn the rules and practise them on examples.

Negative numbers are the numbers below zero — frost on a thermometer, an overdraft, a level underground. Learn the number line running both ways, opposite numbers, comparing negatives, and the sign rules for addition, subtraction, multiplication and division.

Rounding replaces a number with the nearest round one, and estimation tells you the size of an answer before you work it out exactly. Learn the deciding digit, the ≈ sign, rounding to tens, hundreds and thousands, and how to estimate a sum and a product.

When numbers are too big to handle in your head, you write them in columns — digit under digit. Learn the four written algorithms: addition with a carry, subtraction with a borrow, multiplication by a multi-digit number, and long division step by step.

Seven signs — I, V, X, L, C, D, M — and two rules: addition and subtraction. Learn to read and write Roman numerals in the subtractive spelling, meet the limit at 3999, and find out why this system lost to positional notation.

One formula ties three quantities together: speed, distance and time. Learn its three forms, get the units to agree, work out differences on a clock that does not count in tens, and count the days between two dates — leap years included.

A fraction writes part of a whole as a numerator and a denominator. Learn equivalent fractions, all four operations on fractions, mixed numbers, comparing fractions and reducing to lowest terms.

A decimal writes part of a whole with a decimal point instead of a fraction bar. Learn place value, converting both ways between a fraction and a decimal, all four operations, rounding and repeating expansions.

A percentage is a hundredth of a whole. Learn percentage notation, finding a percentage of a number, recovering the whole from a known part, computing percentage change, and writing an increase or a discount as a single multiplier.

A power is shorthand for multiplying the same factor by itself. Learn the base and the exponent, the laws of exponents, powers of a product and of a quotient, zero and negative exponents, the monotonicity of exponentiation, and scientific notation.

A root is the inverse of raising to a power. Learn square and cube roots, arithmetic with roots, taking a factor out of a radical and absorbing one under it, estimating and comparing roots, and rationalising a denominator.

The greatest common divisor and the lowest common multiple of two numbers. Learn the divisibility rules, the listing method, the Euclidean algorithm, and the formula tying GCD to LCM.

A prime number has exactly two divisors: one and itself. Learn the definition, the sieve of Eratosthenes, prime factorisation, and the fundamental theorem of arithmetic.

Natural, integer, rational and real numbers — four families, each one contained in the next. Learn the symbols ℕ, ℤ, ℚ, ℝ, the membership signs ∈ and ∉, the empty set ∅, the proof that √2 is irrational, and the 2k / 2k+1 notation every divisibility proof is built on.

The absolute value of a number is its distance from zero — never negative, because a distance has no direction. Learn the case-by-case definition, the distance between two numbers written as |a − b|, the properties of the modulus and equations with absolute value.

An interval is shorthand for infinitely many numbers — all the ones lying between two ends. Learn open and closed intervals, unbounded intervals with the infinity symbol, the union and intersection of two intervals, and how the condition |x − a| < r becomes a single interval.

A proportion is an equality of two ratios — one equation that rescales a recipe from four people to six and turns centimetres on a map into kilometres on the ground. Learn cross-multiplication, direct proportionality, dividing a quantity in a given ratio, and scale in both directions.

An exponent does not have to be a whole number — the notation a^(m/n) means the n-th root of a raised to the power m. Learn the formula linking powers to roots, why the base has to be positive, negative rational exponents, and the monotonicity of exponentiation.

A logarithm answers the question of which power a base has to be raised to in order to give a number. Learn the definition of log_a b, the conditions on the base and the argument, the common and natural logarithms, and the four properties that turn multiplication into addition.

Compound interest is interest charged on an amount that has already grown — which is why capital rises as a power rather than in equal steps. Learn the formula for the final balance, compounding more often than once a year, compound decline, and the difference between a percent and a percentage point.

An algebraic expression is a piece of notation where letters stand beside numbers. Learn the variable, the term and the coefficient, collect like terms, multiply brackets out — two binomials included — factor out the common term and evaluate an expression.

A linear equation is an equality with the unknown in the first power. Learn the operations that keep equations equivalent, solve one step by step, check the result, rearrange a formula to make another quantity its subject, and handle an unknown in a denominator.

A linear inequality has not one solution but a whole set of them. Learn the operations that keep the inequality sign as it is, memorise the single one that flips it, and learn to mark the solution set on a number line.

A system of equations puts two conditions on two unknowns at once. Learn substitution and elimination, see the solution as the point where two lines cross, and learn to recognise inconsistent and dependent systems.

A quadratic equation has the unknown in the second power. Learn the general form, the discriminant, the formulas for the roots, the shortcuts for incomplete equations, Vieta formulas and how to read the solutions off a parabola.

Seven identities that turn multiplying out brackets into a single line — and the same seven read right to left, which is factoring. The square of a sum and of a difference, the difference of squares, the cubes and the sum and difference of cubes, proved on a drawing and used for mental arithmetic.

A polynomial is a sum of monomials — and everything you do to one is an extension of ordinary arithmetic. Degree and coefficients, adding, subtracting and multiplying, dividing by a binomial with Horner’s scheme, the remainder and factor theorems, integer roots, factoring and biquadratic equations.

A fraction with polynomials where the numbers used to be — and the same four operations as on ordinary fractions, only preceded by factoring. The domain and the excluded numbers, cancelling, multiplying and dividing, adding and subtracting, and rational equations in product form.

Solving an inequality means finding a whole set rather than a single number — and it always takes two steps: the zeros first, then the sign between them. The parabola and the sign of a quadratic, the three discriminant cases, the sign chart, and polynomial and rational inequalities with the trap of a zero denominator.

The modulus measures a distance — and that one sentence solves the whole lesson. The equation |x − a| = r asks for the points r away from a, a strict inequality gives an interval around a, and the other direction gives a union of two. Plus reading the centre and radius back, and the cases with no solution at all.

A parameter is not a second unknown but a setting — one letter that turns a single problem into a whole family of them. The linear equation and its degenerate case, the discriminant as a condition on the number of roots, Vieta’s formulas, an inequality that holds for every x, and a line meeting a parabola.

The area of a rectangle is the product of its sides: A = a · b. See where the formula comes from, how area differs from perimeter, how to find the area of a square, how to get a side back from the area, and why the answer is written in square centimetres.

The area of a triangle is half the product of a base and its height: A = ½ · a · h. See where the half comes from, which height goes with which side, how to work out the perimeter, and why the right triangle is the easiest case of all.

The area of a disc is A = πr², its circumference is C = 2πr. See what the number π actually is, how the radius relates to the diameter, why the answer nearly always has to be rounded, and how a circle differs from a disc.

Angles are measured in degrees: acute below 90°, right exactly 90°, obtuse above it. Meet the kinds of angle, complementary and supplementary pairs, the alternate and corresponding angles at two parallel lines, and the two sum rules — 180° in a triangle and (n − 2) · 180° in a polygon.

In a right triangle a² + b² = c². See why it works, how to find the hypotenuse and a leg, what Pythagorean triples are, and how the converse lets you check whether a corner really is square.

Volume says how much fits inside a solid. See the formulas for a cuboid, a cylinder, a pyramid, a cone and a sphere, why a third appears in the pyramid and the cone, and where the cubic unit comes from.

Surface area is the area of a solid’s net — how much material it takes to cover it. See the formulas for a cuboid, a cylinder, a pyramid, a cone and a sphere, and why a cone and a pyramid are measured by the slant height rather than the height.

Litres, millilitres, cubic centimetres and cubic metres — one quantity written several ways. See why volume units convert by a thousand rather than by ten, and where the identity 1 dm³ = 1 l comes from.

Two perpendicular axes turn the plane into a map on which every point has an address made of two numbers. The order of those numbers is part of the address, their signs name the quadrant, and a segment parallel to an axis is measured by subtraction. Plus the midpoint of a segment, which is the average of the endpoints.

Three quadrilaterals whose area is not the product of two sides — and one idea that handles all of them: cut the figure so that a rectangle or two triangles fall out. Plus the angles, which always add up to 360°, and the diagonals, which are how a rhombus is recognised.

Three congruence criteria say how much data forces two triangles to be identical. Two special triangles — half a square and half an equilateral triangle — have fixed side ratios, so a single length is enough to recover the others without computing a root from scratch.

Two similar figures have the same shape and a different size, and everything that ties them together fits into a single number — the scale factor. The intercept theorem turns parallelism into a proportion, and the three powers of the scale explain why a model twice as large weighs eight times as much.

An inscribed angle is always half the central angle on the same arc — and from that one sentence follow the right angle over a diameter, the construction of a tangent and half of every exam question about circles. Plus chords, tangent segments, regular polygons, and the length of an arc and the area of a sector.

Reflection in a line, reflection in a point, translation and rotation move a figure without changing its size — and each of them comes down to a single operation on coordinates. Plus two compass-and-straightedge constructions: the perpendicular bisector of a segment and the bisector of an angle.

The volume and the surface area of a solid are already settled. This lesson measures what is inside: the diagonal of a cuboid, the angle it makes with the base, the apothem and the lateral edge of a pyramid, and the sections of a cube. Every one of these is a single right triangle found in space.

A point is a pair of numbers, a line is an equation, a circle is another equation — and geometric questions become arithmetic. The distance between points from the Pythagorean theorem, the slope from two points, the conditions for parallel and perpendicular lines, and the centre and radius read off the equation of a circle.

A vector is a segment that knows where it leads: it has a length, a direction and a sense, and in the coordinate plane it is written as a pair of numbers. Adding vectors is adding those pairs, multiplying by a number stretches the arrow, and the length comes from the Pythagorean theorem.

Sine, cosine and tangent are three ratios of the sides of a right triangle. See why they depend on the angle alone, where the exact values for 30°, 45° and 60° come from, and how to find a side you cannot measure.

The unit circle carries the sine and the cosine from a triangle onto the whole circle. See how the coordinates of a point become the values of the functions, what signs they take in the four quadrants, and what a negative angle or one past a full turn means.

sin²α + cos²α = 1 is the Pythagorean theorem written on the unit circle. See where it comes from, how to find the cosine from the sine, and why an answer is incomplete without knowing the quadrant.

The law of sines and the law of cosines carry trigonometry into any triangle — including one with no right angle. See which to pick for the data you are given, why the law of cosines generalises Pythagoras, and when there are two solutions.

A sine wave is the unit circle unrolled along an axis. See where the period of 2π comes from, why the cosine is a shifted sine, and what happens to the graph of the tangent where the function does not exist.

A radian is the angle that cuts an arc equal to the radius. See where the 2π of a full turn comes from, how to convert degrees and radians both ways, and why the arc-length and sector-area formulas collapse to a single multiplication in this measure.

Reduction formulas bring the sine, cosine and tangent of any angle back to an acute one. Instead of memorising a dozen identities, see the single rule behind them: at 180° and 360° the function stays, at 90° and 270° it swaps for its cofunction, and the sign comes from the quadrant.

The sine of a sum is not the sum of the sines. See where sin(α ± β) and cos(α ± β) come from, how the double-angle formulas fall out of them, how to get the exact value of sin 75°, and how to use them to prove trigonometric identities.

The equation sin x = 1/2 has infinitely many solutions, because the sine repeats. See how to find the base solution, how to write all the others with a parameter k, and how to pick out the ones that fall inside a given interval.

Arcsine, arccosine and arctangent answer the reverse question: which angle has this value. See why the domain had to be restricted, what the ranges of the three functions are, and when arcsin(sin x) is not x.

A function is an unambiguous assignment: every argument gets exactly one value. Meet the f(x) notation, the domain and the range, the four ways to define a function, and the vertical line test on a graph.

A linear function y = ax + b draws a straight line. Meet the meaning of the slope and the intercept, the zero, the formula of a line through two points, the general form, and the conditions for parallel and perpendicular lines.

A quadratic function f(x) = ax² + bx + c draws a parabola. Meet the vertex and the axis of symmetry, the three forms of the formula — general, canonical and factored — and how the sign of a decides between a minimum and a maximum.

One curve, four moves: up, sideways, mirrored in an axis and stretched. See how a change in the formula turns into a movement of the curve, why f(x − 2) shifts the graph to the right rather than the left, and how a plain sine becomes y = a·sin(bx + c) with any amplitude and period you like.

A graph shows the whole function at once: domain and range, zeros, sign, intervals of monotonicity and the extremum. Learn to read them off the drawing before you compute anything.

The faster you drive, the shorter the drive — and the product of speed and time stays put. See how inverse proportion differs from direct, what the hyperbola y = a/x looks like, where its two asymptotes come from, and how to read them off a quotient of two linear expressions.

Grain on a chessboard, a loan, radioactive decay and the decibel scale — one pair of functions describes them all. See what the graph of y = bˣ looks like and why y = log_b x is its mirror image, how solving an exponential equation comes down to comparing exponents, and where the domain of a logarithm comes from.

A sequence is a function that assigns a number to every natural number — which is exactly why its graph is a set of separate dots rather than a curve. See how a general formula works, how a recursive one differs from it, and how a single subtraction settles whether a sequence increases.

Two sequences describe almost everything that grows regularly: one keeps adding the same amount, the other keeps multiplying by the same amount. Meet the formulas for the nth term and for the sum of the first n terms of both, the mean property, and the reason a savings account is a geometric sequence.

Infinitely many terms, a finite total — provided the ratio is smaller than one in absolute value. See what it means precisely for a sequence to approach a limit, how limits of polynomial quotients are computed, where the number e comes from, and why 0.(3) really is one third.

Two functions set one behind the other, and one function run backwards. See why f(g(x)) is not g(f(x)), how the monotonicity of a composition follows from its parts, when a function can be inverted at all, and why the graph of the inverse is a reflection in the line y = x.

A quotient of two polynomials, a few points thrown out of the domain, and three kinds of line the curve heads for. See when a zero of the denominator makes an asymptote and when it only makes a hole, how comparing degrees gives the horizontal asymptote, and where the oblique one comes from.

A limit says where the values of a function are heading as the argument closes in on a number — even when the function has no value at that number at all. It is the first idea of analysis and the ground the derivative stands on.

A derivative measures how fast a function changes at a point — it is the slope of the tangent to the graph. Meet the definition through the difference quotient and the six rules that compute it without taking a limit every time.

The sign of the derivative says whether a function rises or falls, and its zeros point at the candidates for extrema. Learn to write the equation of a tangent, read a sign table and find a largest value.

Integration is differentiation run backwards: we look for the function whose derivative is the one we started with. An indefinite integral gives a formula up to a constant, a definite one gives a number — and that number is the area under the curve.

Three answers to one question: which single number best describes a whole data set. The mean shares the total out evenly, the median stands exactly in the middle, and the mode points at the most frequent value — and which one you pick can change the conclusion.

Two data sets can share the same mean and look nothing alike. The range, the variance and the standard deviation measure what the mean cannot see: how widely the data is spread around it.

The art of counting without listing. The rule of product, factorials, permutations, arrangements and combinations all answer one question — in how many ways can this be done — and the whole difficulty comes down to two decisions: does order matter, and may items repeat.

When every outcome of an experiment is equally possible, the probability of an event is an ordinary ratio: how many outcomes favour it, over how many there are. Everything else is being able to count both numbers — and knowing when the opposite event is easier to count.