The Maths AA glossary

The size of a real number without its sign: $|x| = x$ when $x \ge 0$ and $|x| = -x$ when $x < 0$. $|a - b|$ is the distance between $a$ and $b$ on the number line.

The rate of change of velocity with respect to time, $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$.

If a task can be done in one of several separate cases that cannot happen together, with $m$ ways in one case and $n$ ways in the other, then the task can be done in $m + n$ ways.

The situation in which two sides and a non-included acute angle of a triangle fit two different triangles, so the sine rule gives two possible values (acute and obtuse) for an angle.

Half the distance between the maximum and minimum values of a sinusoidal function; for $y = a\sin(b(x + c)) + d$ it is $|a|$.

A solution found by algebraic manipulation, which gives the exact value, such as $x = \ln 3$ or $x = \frac{1 + \sqrt{5}}{2}$, rather than a decimal approximation.

The angle between the line and its projection onto the plane (the shadow it casts straight down onto the plane). It always lies between $0°$ and $90°$.

The acute angle between two intersecting planes, equal to the acute angle between their normal vectors.

The angle between the horizontal and the line of sight from an observer down to an object below them.

The angle between the horizontal and the line of sight from an observer up to an object above them.

A function $F$ whose derivative is a given function $f$, so that $F'(x) = f(x)$. Every continuous function has infinitely many antiderivatives, which differ only by a constant.

Part of the circumference of a circle between two points on it. Its length is $l = r\theta$ when the angle $\theta$ at the centre is in radians.

The area of the region enclosed by two curves $y = f(x)$ and $y = g(x)$ between $x = a$ and $x = b$, found as $\int_a^b |f(x) - g(x)|\,dx$; the limits are often the $x$-coordinates of the points where the curves intersect.

A plane with a horizontal real axis (Re) and a vertical imaginary axis (Im), on which the complex number $a + b\mathrm{i}$ is shown as the point $(a, b)$ or as the vector from the origin to that point.

The angle $\arg z$, in radians, from the positive real axis anticlockwise to the line joining the origin to $z$. The principal argument satisfies $-\pi < \arg z \le \pi$; $\arg 0$ is undefined.

A sequence in which each term after the first is found by adding the same number, the common difference $d$, to the previous term.

The sum of the terms of an arithmetic sequence. The sum of the first $n$ terms is $S_n = \frac{n}{2}(u_1 + u_n)$.

A line that a graph approaches more and more closely, without reaching it, as $x$ or $y$ grows without bound, such as $x = 0$ and $y = 0$ for $y = \frac{1}{x}$.

The change in a function divided by the change in its input over an interval, $\frac{f(b) - f(a)}{b - a}$: the gradient of the chord joining the two points on the graph.

The fixed line about which a region is turned to form a solid of revolution. Every cross-section of the solid perpendicular to it is a circle or a ring.

The vertical line through the vertex about which a parabola is symmetric. For $y = ax^2 + bx + c$ it is the line $x = -\frac{b}{2a}$.

Solving a system in triangular form by finding the unknown in the last equation first, then substituting it into the equations above it in turn.

The number that is raised to an exponent. In $a^n$ the base is $a$; in $\log_a b$ the base $a$ is the number being raised to a power, with $a > 0$ and $a \neq 1$.

The first step of a proof by induction: a direct check that the statement $P(n)$ is true for the smallest value of $n$ it claims, usually $n = 1$.

The unit vectors $\boldsymbol{i}$, $\boldsymbol{j}$ and $\boldsymbol{k}$ in the positive directions of the $x$-, $y$- and $z$-axes. Every vector in three dimensions is a sum of multiples of them.

A formula that reverses a conditional probability: $P(B \mid A) = \frac{P(B)P(A \mid B)}{P(B)P(A \mid B) + P(B')P(A \mid B')}$.

The direction of one point from another, given as an angle measured clockwise from north and written with three digits, such as $045°$ or $210°$.

A single random trial with exactly two possible outcomes, called success (with probability $p$) and failure (with probability $1 - p$).

A systematic tendency for a sample or a method of collecting data to give results that differ from the true values for the population in the same direction.

An algebraic expression with exactly two terms, such as $a + b$, $2x - 3$ or $x^2 + \frac{1}{x}$.

The number $\binom{n}{r} = {}^nC_r = \frac{n!}{r!(n-r)!}$: the coefficient of $a^{n-r}b^r$ in the expansion of $(a+b)^n$, and the number of ways to choose $r$ objects from $n$.

The distribution of the number of successes $X$ in $n$ independent trials, each with the same probability of success $p$. Written $X \sim B(n, p)$, with $P(X = r) = \binom{n}{r}p^r(1 - p)^{n - r}$.

The result $(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$ for $n \in \mathbb{N}$, which expands any power of a binomial without multiplying out the brackets.

Data in which two variables are measured or recorded for each individual or item, giving pairs of values $(x, y)$ that can be plotted and compared.

A known value of a function, such as a point on its graph, used to find the constant of integration and so pick out one antiderivative from the family $F(x) + C$.

A diagram of a data set drawn on a scale: a box from $Q_1$ to $Q_3$ with a line at the median, and whiskers to the smallest and largest values that are not outliers.

The largest population $K$ that an environment can support in the long term; in the logistic model every positive solution approaches $K$ as time increases.

The equations $\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}$ of the line through $(x_0, y_0, z_0)$ with direction vector $\begin{pmatrix} l \\ m \\ n \end{pmatrix}$, found by eliminating the parameter.

The way of writing a complex number as $z = a + b\mathrm{i}$ with $a, b \in \mathbb{R}$, giving its real and imaginary parts. It corresponds to the point $(a, b)$ on an Argand diagram.

A relationship in which a change in one variable directly produces a change in another. Correlation alone never shows causation: both variables may depend on a third, or the link may be coincidence.

A survey that collects data from every member of the population rather than from a sample.

The rule for differentiating a composite function: if $y = g(u)$ and $u = f(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$.

The rule $\log_a x = \frac{\log_b x}{\log_b a}$, which rewrites a logarithm in any base $a$ in terms of logarithms in another base $b$, usually $10$ or $e$.

A straight line segment joining two points on a curve. On a circle the longest chord is a diameter; on the graph of a function, the gradient of a chord is the average rate of change between its two endpoints.

One of the functions $\sin x$, $\cos x$ and $\tan x$ of a real number $x$, defined from the coordinates of the point at angle $x$ radians on the unit circle.

A range of values used to group data, such as $50 \le t < 55$. Its boundaries are the values that separate it from the neighbouring classes; its width is the upper boundary minus the lower boundary.

The square $r^2$ of Pearson's correlation coefficient, with $0 \le r^2 \le 1$: the proportion of the variation in $y$ that the linear regression on $x$ accounts for.

Two lines that are the same line: their direction vectors are parallel and they share a point, so every point of one lies on the other.

Lying on the same straight line. Points $A$, $B$ and $C$ are collinear when $\overrightarrow{AC} = k\overrightarrow{AB}$ for some scalar $k$.

A selection of objects in which order does not matter. The number of combinations of $r$ objects chosen from $n$ different objects is ${}^nC_r = \frac{n!}{r!(n-r)!}$.

The fixed amount $d = u_{n+1} - u_n$ added to each term of an arithmetic sequence to get the next term. It can be positive, negative or zero.

A logarithm with base $10$. In IB notation $\log x$ with no base written means $\log_{10} x$, so $\log 1000 = 3$ because $10^3 = 1000$.

The line segment that joins two skew lines and is perpendicular to both. Its length is the shortest distance between the lines.

The fixed number $r = \frac{u_{n+1}}{u_n}$ that each term of a geometric sequence is multiplied by to give the next term.

Two angles whose sum is a right angle, $\frac{\pi}{2}$ (or $90^\circ$); the angles $\theta$ and $\frac{\pi}{2} - \theta$.

An event $A$ and the event $A'$ ("not $A$"), which contains every outcome in $U$ that is not in $A$. Exactly one of them occurs, so $P(A) + P(A') = 1$.

Rewriting a quadratic $ax^2 + bx + c$ in the form $a(x - h)^2 + k$. The result shows the vertex $(h, k)$ and can be used to solve the equation $ax^2 + bx + c = 0$.

For $z = a + b\mathrm{i}$, the number $z^* = a - b\mathrm{i}$, with the sign of the imaginary part changed. On an Argand diagram it is the reflection of $z$ in the real axis, and $zz^* = a^2 + b^2$ is real.

A number of the form $z = a + b\mathrm{i}$, where $a, b \in \mathbb{R}$ and $\mathrm{i}^2 = -1$. The set of all complex numbers is $\mathbb{C}$; it contains every real number (the case $b = 0$).

One of the numbers $v_1$, $v_2$, $v_3$ in $\boldsymbol{v} = v_1\boldsymbol{i} + v_2\boldsymbol{j} + v_3\boldsymbol{k}$, giving how far the vector goes in each axis direction. Written as a column vector, $\boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$.

The function $f \circ g$ formed by applying $g$ first and then applying $f$ to the result, so that $(f \circ g)(x) = f(g(x))$.

An identity that writes a trigonometric function of a sum or difference of two angles, such as $\sin(A + B)$, in terms of the sines, cosines or tangents of $A$ and $B$ separately.

Interest calculated on the original amount plus all the interest already added, so the balance is multiplied by the same factor each period. The balances form a geometric sequence.

The length of time between two consecutive additions of interest, for example one month for monthly compounding. The number of periods per year is written $k$.

Describes a curve on an interval where its gradient is decreasing, so it bends downwards and lies below its tangents. This happens where $f''(x) < 0$.

Describes a curve on an interval where its gradient is increasing, so it bends upwards and lies above its tangents. This happens where $f''(x) > 0$.

The direction in which a curve bends on an interval: concave up or concave down. It is decided by the sign of the second derivative.

The probability that $A$ occurs given that $B$ has occurred, $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$, where $P(B) > 0$.

A statement that appears to be true, usually from a pattern in examples, but has not yet been proved. A conjecture becomes a theorem once it is proved and is discarded if a counterexample is found.

If a polynomial equation with real coefficients has a root $a + b\mathrm{i}$ with $b \neq 0$, then its conjugate $a - b\mathrm{i}$ is also a root. Non-real roots therefore come in conjugate pairs.

Integers that follow each other without gaps, each $1$ more than the one before, such as $7, 8, 9$. In algebra they are written $n, n + 1, n + 2, \ldots$ for an integer $n$.

A system of equations that has at least one solution (a unique solution or infinitely many). A system with no solution is called inconsistent.

The arbitrary constant $C$ added to an antiderivative to give every function with the same derivative. Its value can be found from a boundary condition, a known point on the curve.

The term of an expression or expansion that does not contain the variable, that is, the term in $x^0$. In $(2x - \frac{1}{x})^6$ it is $-160$.

A condition linking the variables in an optimisation problem, such as a fixed volume or a fixed length of material. It is used to write the quantity to be optimised in terms of a single variable.

A function $f$ is continuous at $x = a$ if $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and the two are equal. Informally, its graph can be drawn through $x = a$ without lifting the pen.

Compound interest in the limit where the number of compounding periods per year increases without bound. A principal $P$ at nominal annual rate $r$ grows to $A = Pe^{rt}$ after $t$ years.

Numerical data that can take any value in an interval and are obtained by measuring, such as height, mass or time. Recorded values are rounded to some degree of accuracy.

A random variable that can take any value in an interval of real numbers, such as a time or a length. Its probabilities are areas under a probability density function, and $P(X = a) = 0$ for any single value $a$.

The distribution of a continuous random variable whose probability density function is constant, $f(x) = \frac{1}{b - a}$, on an interval $a \le x \le b$ and zero elsewhere.

A non-random sampling method that uses members of the population who are easy to reach, such as the first people met.

An infinite series whose partial sums approach a finite limit as the number of terms increases. A geometric series converges exactly when $|r| < 1$.

Lying in the same plane. Any three points are coplanar; four points or two lines may or may not be.

A relationship between two variables in which one tends to increase (positive correlation) or decrease (negative correlation) as the other increases. Linear correlation is when the points lie close to a straight line.

In any triangle $ABC$, $c^2 = a^2 + b^2 - 2ab\cos C$. It links all three sides with one angle and extends Pythagoras's theorem to triangles without a right angle.

A single specific case that satisfies the conditions of a general statement but for which its conclusion is false. One counterexample is enough to disprove the statement.

A quick way to find a partial fraction numerator: substitute the value of $x$ that makes one linear factor zero into the rest of the fraction, with that factor covered up.

A value of $x$ at which an expression in an inequality is zero or undefined. Critical values split the number line into intervals on which the sign of the expression does not change; a sign diagram records that sign on each interval.

The running total of the frequencies up to a given value or upper class boundary. Plotted against the upper boundaries, it gives a cumulative frequency graph.

The probability that a random variable takes a value less than or equal to a given value, $P(X \le x)$. A GDC's binomial cdf function gives it directly.

A point where a continuous graph changes direction sharply, so the gradients from the left and the right are different (a corner) or tend to $\pm\infty$ (a cusp, as for $y = x^{2/3}$ at $0$). The function is not differentiable there.

One complete repetition of the pattern of a periodic graph, covering an interval of $x$ equal to one period.

For any integer $n$, $[r(\cos\theta + \mathrm{i}\sin\theta)]^n = r^n(\cos n\theta + \mathrm{i}\sin n\theta)$: to raise a complex number to a power, raise the modulus to that power and multiply the argument by it.

The number of digits written after the decimal point. Rounding to 2 decimal places keeps two digits after the point, as in $3.14159 \approx 3.14$.

A function is decreasing on an interval if larger inputs give smaller outputs there: $x_1 < x_2$ implies $f(x_1) > f(x_2)$. If $f'(x) < 0$ on an interval, $f$ is decreasing on it.

The number $\int_a^b f(x)\,dx$, the limit of sums of $f(x)\,\delta x$ from $x = a$ to $x = b$. When $f(x) > 0$ on $[a, b]$ it equals the area between the curve and the $x$-axis.

Events that are not independent: knowing that one has occurred changes the probability of the other, so $P(A \cap B) \neq P(A)P(B)$.

The loss in value of an asset, such as a car or a machine, over time. With a fixed annual percentage rate the values form a geometric sequence with ratio less than 1.

The function $f'(x)$, also written $\frac{dy}{dx}$, that gives the gradient of the curve $y = f(x)$ at each value of $x$. It measures the instantaneous rate of change of $y$ with respect to $x$.

A function is differentiable at $x = a$ if $\lim_{h \to 0}\frac{f(a + h) - f(a)}{h}$ exists, so its graph has a single, non-vertical tangent there. A differentiable function is continuous, but not every continuous function is differentiable.

An equation that relates an unknown function to one or more of its derivatives, such as $\frac{dy}{dx} = 2xy$. It is first-order when the highest derivative it contains is the first derivative.

The process of finding the derivative of a function. Differentiating $y = f(x)$ with respect to $x$ gives $\frac{dy}{dx} = f'(x)$.

Finding a derivative directly from its definition $f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$, by simplifying the gradient of a chord and then letting $h$ tend to $0$.

A non-zero vector parallel to a line, written $\boldsymbol{b}$ in $\boldsymbol{r} = \boldsymbol{a} + \lambda\boldsymbol{b}$. Any non-zero multiple of it is also a direction vector of the same line.

Finding a volume of revolution by slicing the solid perpendicular to the axis of rotation into thin discs of volume $\pi r^2\,\delta x$ and adding them with an integral.

A point where a function is not continuous, such as a jump (the one-sided limits differ), a hole (the limit exists but is not $f(a)$) or a vertical asymptote.

Numerical data that can take only separate, distinct values (usually counts), such as the number of goals scored in a match.

A variable whose value is a numerical outcome of a random process and which can take only separate (countable) values, such as $0, 1, 2, \dots$. It is written with a capital letter, such as $X$.

A probability distribution in which each of the $n$ possible values of a discrete random variable has the same probability $\frac{1}{n}$, such as the score on a fair die.

The quantity $\Delta = b^2 - 4ac$ for the quadratic $ax^2 + bx + c$. The equation $ax^2 + bx + c = 0$ has two distinct real roots if $\Delta > 0$, two equal real roots if $\Delta = 0$ and no real roots if $\Delta < 0$.

The position of a particle relative to a fixed origin, measured along a line with a sign for direction, usually written $s$. A change in displacement is final position minus initial position.

The vector from one point to another, giving the change in position: $\overrightarrow{AB} = \boldsymbol{b} - \boldsymbol{a}$, where $\boldsymbol{a}$ and $\boldsymbol{b}$ are the position vectors of $A$ and $B$.

An infinite series whose partial sums do not approach a finite limit: they grow without bound or keep oscillating. A geometric series with $|r| \ge 1$ and $u_1 \neq 0$ diverges.

The set of all input values for which a function is defined, or for which it is used in a context.

The term of an expression that grows fastest as $x \to \infty$ and so decides its behaviour; in a polynomial it is the term with the highest power of $x$. For example, $3x^2$ is the dominant term of $3x^2 - 5x + 1$.

An identity expressing a trigonometric ratio of $2\theta$ in terms of ratios of $\theta$, such as $\sin 2\theta = 2\sin\theta\cos\theta$ and $\cos 2\theta = 1 - 2\sin^2\theta$.

How the values of a function behave as $x \to \infty$ and as $x \to -\infty$. For a polynomial it depends only on the degree and the sign of the leading coefficient.

Outcomes that all have the same chance of occurring, such as the six faces of a fair die. Only then does $P(A) = \frac{n(A)}{n(U)}$ apply.

Setting the coefficients of each power of $x$ on the two sides of an identity equal to each other, to find unknown constants.

A statement that two expressions are equal, which may be true for only some values of the variable. Solving an equation means finding those values; for example $2x + 1 = 7$ is true only when $x = 3$.

An equation that becomes a quadratic equation after a substitution such as $u = x^2$, $u = e^x$ or $u = \sin x$; for example, $x^4 - 5x^2 + 4 = 0$ becomes $u^2 - 5u + 4 = 0$.

The way of writing a complex number with modulus $r$ and argument $\theta$ as $z = r\mathrm{e}^{\mathrm{i}\theta}$, using Euler's formula $\mathrm{e}^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\sin\theta$.

The identity $\mathrm{e}^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\sin\theta$ for real $\theta$, linking the exponential function with the circular functions. With $\theta = \pi$ it gives Euler's identity, $\mathrm{e}^{\mathrm{i}\pi} + 1 = 0$.

A numerical method that approximates the solution of $\frac{dy}{dx} = f(x, y)$ by short straight steps along the gradient: $y_{n+1} = y_n + h f(x_n, y_n)$, $x_{n+1} = x_n + h$.

The irrational constant $e \approx 2.71828$, the limit of $\left(1 + \frac{1}{n}\right)^n$ as $n$ grows without bound. It is the base of the natural logarithm and of the exponential function $e^x$.

A function for which $f(-x) = f(x)$ for every $x$ in its domain. Its graph is symmetric in the $y$-axis, for example $y = x^2$ and $y = \cos x$.

An integer that is divisible by $2$, so it can be written as $2n$ for some integer $n$. Examples are $-4$, $0$ and $18$.

A set of outcomes from the sample space, such as "an even number" when a die is rolled. An event occurs when the outcome of the trial is one of its outcomes.

A value written without rounding, using fractions, surds or $\pi$, such as $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$ rather than $0.866$.

Events that together cover the whole sample space, so at least one of them must occur. Mutually exclusive, exhaustive events $B_1, B_2, B_3$ form a partition of $U$.

The number of times an event is predicted to occur in $n$ trials when its probability is $p$, equal to $np$.

The long-run mean value of a random variable over many repetitions, written $E(X)$ or $\mu$. For a discrete random variable, $E(X) = \sum x\,P(X = x)$.

The number written as a superscript in a power such as $a^n$, showing how many times the base is used as a factor when $n$ is a positive integer. For example, in $2^5$ the exponent is $5$.

Decrease in which a quantity is multiplied by the same factor between 0 and 1 over each equal time interval, as in $N = N_0 e^{kt}$ with $k < 0$. The time taken to halve, $\frac{\ln 2}{|k|}$, is the half-life.

An equation in which the unknown appears in an exponent, such as $2^{x+1} = 5$. It is solved by writing both sides with the same base or by taking logarithms of both sides.

A function of the form $f(x) = a^x$, where the base $a$ is a positive constant and $a \neq 1$, so that the variable is in the exponent. The natural exponential function is $f(x) = e^x$.

Growth in which a quantity is multiplied by the same factor greater than 1 over each equal time interval, as in $N = N_0 e^{kt}$ with $k > 0$. The time taken to double, $\frac{\ln 2}{k}$, is the doubling time.

The infinite series $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots$ for a negative or fractional index $n \in \mathbb{Q}$, valid for $|x| < 1$.

A value produced while solving an equation, for example after squaring both sides or splitting into cases, that does not satisfy the original equation and must be rejected.

Using a regression line to predict a value outside the range of the data used to find it. Predicting inside that range is interpolation, which is much more reliable.

For a polynomial $P(x)$, $(x - a)$ is a factor of $P(x)$ if and only if $P(a) = 0$. More generally, $(ax - b)$ is a factor if and only if $P\left(\frac{b}{a}\right) = 0$.

The product of all positive integers from 1 up to $n$: $n! = n(n-1)(n-2)\cdots 2 \times 1$ for $n \in \mathbb{Z}^+$, with $0! = 1$ by definition.

The form $f(x) = a(x - p)(x - q)$ of a quadratic function, in which $p$ and $q$ are its zeros, so its graph meets the $x$-axis at $(p, 0)$ and $(q, 0)$.

A game in which a player's expected gain is zero, so that $E(\text{gain}) = 0$: on average the player neither wins nor loses money over many plays.

A positive test result for someone who does not have the condition. A false negative is a negative result for someone who does.

A way to classify a stationary point at $x = a$ from the sign of $f'(x)$ just before and just after $a$: positive then negative gives a maximum, negative then positive a minimum, no change of sign a stationary point of inflexion.

The point on a line or plane closest to a given point, where the segment from the given point meets the line or plane at a right angle.

A shape built by repeating the same construction at smaller and smaller scales, so that parts of it look like the whole. The Koch snowflake is an example.

A table listing each value or class of values in a data set together with its frequency, the number of times it occurs.

A relation that assigns to each input in its domain exactly one output. It is written $f(x)$ or $f: x \mapsto \ldots$, where $x$ is the input.

The result linking differentiation and integration: if $F'(x) = f(x)$, then $\int_a^b f(x)\,dx = F(b) - F(a)$. Equivalently, the rate of change of the area under $y = f(t)$ from $a$ to $x$ is $f(x)$.

The value that an amount of money will have at a later date after interest has been added, written $FV$.

The family of all solutions of a differential equation, written with an arbitrary constant such as $C$; a first-order equation has one arbitrary constant in its general solution.

The expression $\binom{n}{r} a^{n-r} b^r$, which gives the $(r+1)$th term of the expansion of $(a+b)^n$ for $r = 0, 1, \ldots, n$.

A sequence in which each term after the first is found by multiplying the previous term by the same non-zero number, the common ratio $r$.

The sum of the terms of a geometric sequence. For $r \neq 1$, the sum of the first $n$ terms is $S_n = \frac{u_1(r^n - 1)}{r - 1}$.

The greatest value a function takes on its whole domain. It may occur at a local maximum or at an endpoint of the domain, and some functions have none.

The least value a function takes on its whole domain. It may occur at a local minimum or at an endpoint of the domain, and some functions have none.

The steepness of a straight line: the change in $y$ divided by the change in $x$ between any two points on it, $m = \frac{y_2 - y_1}{x_2 - x_1}$. For a curve, the gradient at a point is the gradient of the tangent there.

The single-wave form $R\cos(\theta - \alpha)$ or $R\sin(\theta + \alpha)$, with $R > 0$, of an expression $a\cos\theta + b\sin\theta$; here $R = \sqrt{a^2 + b^2}$ is its amplitude.

Half of a sphere, cut off by a plane through its centre. A solid hemisphere of radius $r$ has volume $\frac{2}{3}\pi r^3$ and total surface area $3\pi r^2$.

A derivative found by differentiating more than once. The $n$th derivative of $f$ is written $f^{(n)}(x)$ or $\frac{d^n y}{dx^n}$.

A diagram for grouped continuous data in which each class is drawn as a bar with no gaps between bars. With equal class widths, the height of each bar is the frequency.

A first-order differential equation that can be written as $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$. The substitution $y = vx$ turns it into a separable equation in $v$ and $x$.

A horizontal line $y = k$ that a graph approaches as $x \to \infty$ or $x \to -\infty$, so that $f(x)$ gets as close to $k$ as you like.

The graph of a function is one-to-one, and so the function has an inverse, when every horizontal line meets it at most once.

The curve formed by the graph of $y = \frac{k}{x}$, or of any function $\frac{ax + b}{cx + d}$ that is not constant: two separate branches, each approaching the same pair of perpendicular asymptotes.

The longest side of a right-angled triangle, opposite the right angle.

An equality that is true for every value of the variable for which both sides are defined, such as $(x + 1)^2 \equiv x^2 + 2x + 1$. The symbol $\equiv$ means "is identically equal to".

The function $I(x) = x$, which maps every input to itself. Composing a function with its inverse, in either order, gives the identity function.

The point that a given point is moved to by a transformation. Under $y = f(x - a) + b$, the image of $(x, y)$ is $(x + a, y + b)$.

For $z = a + b\mathrm{i}$ with $a, b \in \mathbb{R}$, the real number $b$, written $\mathrm{Im}(z)$. It is $b$, not $b\mathrm{i}$.

The number $\mathrm{i}$ defined by $\mathrm{i}^2 = -1$, so that $\mathrm{i} = \sqrt{-1}$. Its powers repeat in the cycle $\mathrm{i}, -1, -\mathrm{i}, 1$.

Differentiating both sides of an equation connecting $x$ and $y$ with respect to $x$, treating $y$ as a function of $x$ (so that, for example, $\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}$), then rearranging to make $\frac{dy}{dx}$ the subject.

An equation connecting $x$ and $y$ that is not written in the form $y = f(x)$, such as $x^2 + y^2 = 25$. An equation written as $y = f(x)$ defines $y$ explicitly.

An algebraic fraction whose numerator has a degree greater than or equal to that of its denominator, such as $\frac{x^2 + 1}{x^2 - 4}$. Dividing first writes it as a polynomial plus a proper fraction.

The angle between two given sides of a triangle, at the vertex where those two sides meet.

A function is increasing on an interval if larger inputs give larger outputs there: $x_1 < x_2$ implies $f(x_1) < f(x_2)$. If $f'(x) > 0$ on an interval, $f$ is increasing on it.

The family of all antiderivatives of $f(x)$, written $\int f(x)\,dx = F(x) + C$, where $F'(x) = f(x)$ and $C$ is an arbitrary constant.