Articles

What Is Place Value? Digits, Base Ten and Expanded Form

Place value is the worth a digit has because of its position in a number. Each place is worth ten times the place to its right.

Order of Operations (BIDMAS and PEMDAS) Explained

The order of operations is the agreed sequence for working out a calculation: brackets, then powers, then multiplication and division, then addition and subtraction.

Factors and Multiples: What Is the Difference?

A factor of a number divides it exactly, with no remainder. A multiple of a number is that number times a whole number. So $4$ is a factor of $12$, and $12$ is a multiple of $4$.

Prime Numbers and Prime Factorisation with Factor Trees

A prime number has exactly two factors, $1$ and itself. Prime factorisation writes a number as a product of primes, such as $360 = 2^3 \times 3^2 \times 5$.

Greatest Common Factor and Least Common Multiple (GCF and LCM)

The greatest common factor (GCF, or HCF) of two numbers is the largest number that divides both. The least common multiple (LCM) is the smallest number that both divide into.

Negative Numbers: Adding, Subtracting, Multiplying and Dividing

Negative numbers are numbers less than zero, written with a minus sign and placed to the left of zero on the number line. Together with zero and the positive whole numbers they form the integers.

Absolute Value: Distance from Zero, Equations and Inequalities

The absolute value of a number, written $|x|$, is its distance from zero on the number line. Distances are never negative, so $|-4| = |4| = 4$.

What Is a Fraction? Numerators, Denominators and Mixed Numbers

A fraction $\frac{a}{b}$ means $a$ parts of a whole that has been cut into $b$ equal parts. The bottom number is the denominator; the top is the numerator.

Equivalent Fractions and Simplifying to Simplest Form

Equivalent fractions name the same amount with different numbers, like $\frac{1}{2} = \frac{2}{4} = \frac{4}{8}$. Multiplying or dividing the top and bottom by the same number never changes a fraction's value.

Common Denominators: How to Compare, Add and Subtract Fractions

A common denominator is a denominator shared by two or more fractions. Rewriting fractions over a common denominator makes their pieces the same size, so they can be compared, added and subtracted.

Dividing Fractions: Why You Multiply by the Reciprocal

To multiply fractions, multiply the tops and multiply the bottoms. To divide by a fraction, multiply by its reciprocal, the fraction turned upside down: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.

Decimals Explained: Tenths, Hundredths and Decimal Place Value

A decimal carries place value past the units: the first place after the decimal point is tenths, then hundredths, then thousandths, each a tenth of the place before.

Percentages: Converting Fractions, Decimals and Percents

Percent means per hundred. $35\%$ is $\frac{35}{100}$, or $0.35$. Finding a percentage of an amount means converting the percentage to a decimal and multiplying.

Percentage Increase and Decrease Using Multipliers

A percentage change compares a change with the original amount. Increasing by $p\%$ means multiplying by $1 + \frac{p}{100}$, and decreasing by $p\%$ means multiplying by $1 - \frac{p}{100}$.

Ratios Explained: Simplifying and Sharing in a Ratio

A ratio compares the sizes of two or more quantities. $2 : 3$ means that for every $2$ of the first there are $3$ of the second.

Rates and Unit Rates: Speed, Best Buys and Per-One Comparisons

A rate compares two quantities measured in different units, such as $150$ km in $3$ hours. A unit rate scales it to one of the second quantity: $50$ km per hour.

Direct Proportion and Cross-Multiplication

A proportion says two ratios are equal, such as $\frac{3}{7.50} = \frac{5}{x}$. Two quantities are directly proportional when one is always the same multiple of the other: $y = kx$.

What Are Exponents? Powers, Bases and Repeated Multiplication

An exponent (or power) says how many times to multiply a number, the base, by itself: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$.

The Laws of Exponents: Product, Quotient and Power Rules

The exponent laws are shortcuts for combining powers of the same base: multiply powers by adding exponents, divide by subtracting them, and raise a power to a power by multiplying them.

Zero and Negative Exponents: Why a⁰ = 1 and a⁻ⁿ = 1/aⁿ

Any non-zero number to the power zero is $1$, and a negative exponent means a reciprocal: $a^{-n} = \frac{1}{a^n}$. So $2^{-3} = \frac{1}{8}$.

Square Roots and Surds: Simplifying Radicals Step by Step

The square root $\sqrt{n}$ is the non-negative number that squares to $n$: $\sqrt{49} = 7$. A root that cannot be written as a whole number or fraction, like $\sqrt{2}$, is called a surd (or radical).

Scientific Notation (Standard Form) Explained with Examples

Scientific notation, also called standard form, writes a number as $a \times 10^n$ where $1 \le a < 10$ and $n$ is a whole number, such as $1.5 \times 10^8$ or $4.2 \times 10^{-4}$.

Fractional Exponents: Roots Written as Powers

A fractional exponent is a root written as a power: $a^{1/n} = \sqrt[n]{a}$, and $a^{m/n} = \left(\sqrt[n]{a}\right)^m$. The bottom of the fraction is the root; the top is the power.

Algebraic Expressions: Variables, Terms, Coefficients and Like Terms

An expression combines numbers and variables (letters standing for numbers) with operations, like $3x + 5$. Like terms have exactly the same variable part and can be combined by adding their coefficients.

The Distributive Property: Expanding Brackets with the Area Model

The distributive property says $a(b + c) = ab + ac$: multiplying a sum means multiplying each part of it. It is how brackets are expanded, and, run backwards, how expressions are factorised.

How to Solve Linear Equations: The Balance Method

A linear equation like $3x - 4 = 11$ is solved by doing the same inverse operation to both sides until the variable is alone. Each step keeps the equation balanced, like a pair of scales.

Solving Linear Inequalities: Why the Sign Flips with Negatives

An inequality such as $x > 2$ describes a range of values rather than a single one. It is solved like an equation, except that multiplying or dividing both sides by a negative number reverses the inequality sign.

Interval Notation Explained: Brackets, Infinity and Unions

Interval notation writes a range of numbers using its endpoints: a square bracket includes an endpoint and a round bracket excludes it. So $-2 < x \le 5$ is written $(-2, 5]$.

Rearranging Formulae: How to Change the Subject of a Formula

Rearranging a formula means making a different letter its subject, the letter on its own on one side. You use the same balancing steps as solving an equation, treating the other letters as if they were numbers.

The Coordinate Plane: Axes, Quadrants and Plotting Points

The coordinate plane is a grid made by two perpendicular number lines, the $x$-axis and the $y$-axis. Every point has an address $(x, y)$: how far across, then how far up.

Slope (Gradient) of a Line: Rise Over Run Explained

The slope, or gradient, of a line measures its steepness: how much $y$ changes for each step of $1$ in $x$. Between two points, $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$.

Equation of a Straight Line: y = mx + b and Point-Slope Form

Every non-vertical straight line can be written $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept, the height where the line crosses the $y$-axis.

Systems of Equations: Solving by Substitution, Elimination and Graphing

A system of equations (or simultaneous equations) is a set of equations that must all be true at once. For two linear equations, the solution is the point where their lines intersect.

What Is a Function? Inputs, Outputs, f(x) and the Vertical Line Test

A function is a rule that gives exactly one output for each input. Writing $f(x)$ means "the output of the function $f$ when the input is $x$".

Domain and Range of a Function: How to Find Them

The domain of a function is the set of inputs it accepts. The range is the set of outputs it actually produces.

Composite and Inverse Functions: f(g(x)) and f⁻¹(x) Explained

A composite function $f(g(x))$ feeds the output of $g$ into $f$. An inverse function $f^{-1}$ undoes $f$: if $f(2) = 7$, then $f^{-1}(7) = 2$.

Transformations of Graphs: Shifts, Stretches and Reflections

A transformation moves or reshapes a graph. In $y = a\,f(x - h) + k$, $h$ shifts it right, $k$ shifts it up, $|a|$ stretches it vertically, and a negative $a$ reflects it in the $x$-axis.

What Is a Polynomial? Terms, Degree and the Factor Theorem

A polynomial is a sum of terms of the form $ax^n$, where each power $n$ is a whole number, such as $4x^3 - x + 7$. Its degree is the highest power present.

How to Factorise Quadratics: Sum-Product Method and Difference of Two Squares

Factorising a quadratic rewrites it as a product of brackets: $x^2 + 7x + 12 = (x + 3)(x + 4)$. For $x^2 + bx + c$, look for two numbers that add to $b$ and multiply to $c$.

Completing the Square and Vertex Form of a Quadratic

Completing the square rewrites $x^2 + bx + c$ as $(x + \frac{b}{2})^2 + k$. The result, vertex form $a(x - h)^2 + k$, shows the parabola's turning point $(h, k)$ directly.

The Quadratic Formula and the Discriminant Explained

The quadratic formula solves any equation $ax^2 + bx + c = 0$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The discriminant $b^2 - 4ac$ tells you how many real solutions there are.

Rational Functions: Asymptotes, Holes and Algebraic Fractions

A rational function is one polynomial divided by another, like $f(x) = \frac{1}{x - 2}$. Its graph can have asymptotes, lines it approaches but never reaches, and holes, single missing points.

Rationalising the Denominator: Surds and Conjugates

Rationalising the denominator rewrites a fraction so no root is left on the bottom, without changing its value. Multiply top and bottom by the root, or by the conjugate when the bottom is a sum like $3 + \sqrt2$.

Exponential Growth and Decay: Formula, Graphs and Examples

Exponential growth multiplies a quantity by the same factor every step, giving $f(x) = a \cdot b^x$. When the factor $b$ is between $0$ and $1$, the quantity shrinks instead: exponential decay.

What Is a Logarithm? Logs Explained Simply

A logarithm is an exponent. $\log_b x$ asks: what power of $b$ gives $x$? Since $2^3 = 8$, $\log_2 8 = 3$.

The Laws of Logarithms: Product, Quotient and Power Rules

The laws of logarithms turn multiplication into addition: $\log(MN) = \log M + \log N$, $\log\frac{M}{N} = \log M - \log N$ and $\log(M^k) = k\log M$. Each is an exponent law in disguise.

The Number e and the Natural Logarithm (ln) Explained

$e \approx 2.71828$ is the number that compound interest approaches when interest is added ever more often. The natural logarithm $\ln x$ is the logarithm with base $e$.

Angles Explained: Types of Angles and Angle Facts

An angle measures an amount of turn. A full turn is $360°$, a straight line $180°$ and a right angle $90°$. Angles on a straight line add to $180°$ and angles around a point add to $360°$.

Why Do the Angles in a Triangle Add Up to 180°?

The three interior angles of any triangle add up to $180°$. Isosceles triangles have two equal angles; equilateral triangles have three, each $60°$.

The Pythagorean Theorem Explained: a² + b² = c²

In a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: $a^2 + b^2 = c^2$.

Area and Perimeter: Formulas for Rectangles, Triangles and Trapeziums

Perimeter is the distance around the edge of a shape, measured in units. Area is the amount of surface it covers, measured in square units.

Circles and Pi: Circumference and Area of a Circle

For every circle, the circumference divided by the diameter is the same number, pi ($\pi \approx 3.14159$). So the circumference is $C = 2\pi r$ and the area is $A = \pi r^2$.

Volume and Surface Area of Prisms, Cylinders, Cones and Spheres

Volume is the space a solid takes up, in cubic units. For any prism or cylinder it is the cross-sectional area times the length; cones and pyramids hold exactly one third of that.

Distance and Midpoint Formulas and the Equation of a Circle

The distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ is Pythagoras on a grid. The midpoint averages the coordinates. A circle with centre $(h, k)$ and radius $r$ has equation $(x - h)^2 + (y - k)^2 = r^2$.

SOH CAH TOA: Sine, Cosine and Tangent in Right-Angled Triangles

In a right-angled triangle, sine, cosine and tangent are ratios of sides that depend only on an angle: $\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$. SOH CAH TOA is the memory aid.

What Is a Radian? Converting Between Degrees and Radians

A radian is the angle at the centre of a circle where the arc length equals the radius. A full turn is $2\pi$ radians, so $180° = \pi$ radians.

The Unit Circle Explained: Sine and Cosine for Any Angle

The unit circle has radius $1$ and centre at the origin. The point at angle $\theta$ around it has coordinates $(\cos\theta, \sin\theta)$, which defines sine and cosine for every angle, not just those under $90°$.

Graphs of Sine and Cosine: Amplitude, Period and Shifts

The graphs of $y = \sin x$ and $y = \cos x$ are waves that repeat every $2\pi$. In $y = A\sin(B(x - h)) + k$, $|A|$ is the amplitude, $\frac{2\pi}{|B|}$ is the period, and $h$ and $k$ shift the wave.

Trig Identities: sin²θ + cos²θ = 1 and the Double-Angle Formulas

A trig identity is an equation that is true for every angle. The most important, $\sin^2\theta + \cos^2\theta = 1$, is Pythagoras on the unit circle; others include $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and the double-angle formulas.

Solving Trig Equations and Inverse Trig Functions

To solve a trig equation like $\sin x = \frac12$, use an inverse trig function to find one angle, then use the symmetry of the unit circle to find every other angle in the range you need.

Arithmetic Sequences: The nth Term and Gauss's Sum Formula

An arithmetic sequence adds the same common difference $d$ every step, like $5, 8, 11, 14, \dots$ Its $n$th term is $a_n = a_1 + (n - 1)d$, and the sum of its first $n$ terms is $\frac{n}{2}(a_1 + a_n)$.

Geometric Sequences and Infinite Geometric Series

A geometric sequence multiplies by the same common ratio $r$ every step, like $3, 6, 12, 24, \dots$ Its $n$th term is $a_1 r^{n-1}$. When $|r| < 1$, adding all of its infinitely many terms gives a finite total, $\frac{a_1}{1 - r}$.

Sigma Notation (Σ) Explained: How to Read and Write Summations

Sigma notation uses the Greek capital letter $\Sigma$ to mean add up. $\sum_{k=1}^{4} (2k + 1)$ means substitute $k = 1, 2, 3, 4$ into $2k + 1$ and add the results.

The Binomial Theorem and Pascal's Triangle

The binomial theorem expands $(a + b)^n$ directly. The coefficients come from row $n$ of Pascal's triangle, or from the formula $\binom{n}{k} = \frac{n!}{k!(n - k)!}$.

Average Rate of Change and the Secant Line

The average rate of change of a function between $x = a$ and $x = b$ is the change in output divided by the change in input, $\frac{f(b) - f(a)}{b - a}$. It is the slope of the secant line joining the two points.

What Is a Limit in Maths? Limits Explained for Beginners

A limit describes the value a function approaches as its input gets closer and closer to some number, whether or not the function is defined there. We write $\lim_{x \to a} f(x) = L$.

What Is a Derivative? The Difference Quotient and the Slope of a Curve

The derivative of $f$ at $x$ is the slope of the curve at that point: the instantaneous rate of change. It is the limit of the difference quotient, $f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$.