The Maths AI glossary

The size of the difference between an approximate value and the exact value, $|v_A - v_E|$, measured in the same units as the quantity.

The distance of a real number from zero on the number line, written $|x|$; it equals $x$ when $x \ge 0$ and $-x$ when $x < 0$.

A state of a Markov chain that cannot be left once it is entered, so that the probability of moving from it to itself is 1.

The principle that work submitted for assessment is the student's own, with every idea, quotation, data set, image and piece of software output taken from someone else acknowledged where it is used.

The rate of change of an object's velocity with respect to time, $a = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}^2s}{\mathrm{d}t^2}$; a positive acceleration means the velocity is increasing.

The set of values of the test statistic for which the null hypothesis is not rejected; it contains every possible value outside the critical region.

The difference between a numerical estimate after many steps and the exact value of the solution, made up of the small error introduced at every step and carried forward through all later steps.

A mark, written A1 in a markscheme, awarded for a correct answer or a correct intermediate value, usually only when the method that produces it has also been credited.

The probability, assuming the null hypothesis is true, that the test statistic falls in the critical region; for a discrete test it is usually less than the nominal significance level.

An angle greater than 0° and less than 90°.

If $X \sim \mathrm{Po}(m_1)$ and $Y \sim \mathrm{Po}(m_2)$ are independent, then $X + Y \sim \mathrm{Po}(m_1 + m_2)$.

The square matrix of a graph with $n$ vertices in which the entry in row $i$, column $j$ is the number of edges from vertex $i$ to vertex $j$.

Two edges that share a common vertex.

In a right-angled triangle, the side next to a given acute angle that is not the hypotenuse.

Two vertices that are joined by at least one edge.

A transformation of the form $\boldsymbol{r}' = A\boldsymbol{r} + \boldsymbol{b}$, where $A$ is a $2 \times 2$ matrix and $\boldsymbol{b}$ is a translation vector; it applies the matrix transformation and then the translation.

The extent to which two rankings of the same items place them in the same order, measured by $r_s$, where 1 means identical orders and −1 means reversed orders.

A precise statement of what an exploration sets out to find, model, test or decide, specific enough that the conclusion can say whether, and how well, it was achieved.

A finite sequence of precisely stated steps that solves a problem or carries out a task, and that can be followed mechanically.

A pair of equal angles formed on opposite sides of a line that crosses two parallel lines, each lying between the parallels.

An electric current, or the voltage that drives it, that reverses direction periodically and is modelled by a sinusoidal function of time such as $V = A\cos(\omega t + \phi)$.

The statement, written $\mathrm{H}_1$, that is accepted if the null hypothesis is rejected; it describes a difference, an association or a departure from a model.

The situation in which two sides of a triangle and an angle that is not between them are given, and the data fit two different triangles, one with an acute angle and one with the obtuse supplement of that angle.

The repayment of a loan by equal payments at regular intervals, each of which pays the interest due for that period and repays part of the principal, so that the debt is cleared at the end of the loan term.

A table showing, for each payment of an amortized loan, the payment, the interest charged, the part of the principal repaid and the outstanding balance after the payment.

Half the vertical distance between the maximum and minimum values of a periodic function; for $a\sin(bx) + d$ and $a\cos(bx) + d$ it equals $|a|$.

The angle between a line and its projection onto the plane, measured at the point where the line meets the plane.

The acute angle (between 0° and 90°) formed by two lines, found from the angle between their direction vectors; if the angle between the direction vectors is obtuse, the angle between the lines is its supplement $180° - \theta$.

The angle $\theta$, with $0° \le \theta \le 180°$, between two non-zero vectors drawn from the same starting point; it satisfies $\cos\theta = \frac{\boldsymbol{v} \cdot \boldsymbol{w}}{|\boldsymbol{v}||\boldsymbol{w}|}$.

The GDC setting that fixes whether angles are entered and given in degrees or in radians.

The angle measured downwards from the horizontal to the line of sight of an object below the observer.

The angle measured upwards from the horizontal to the line of sight of an object above the observer.

The angle $\theta$ between a line or incline and the horizontal, measured upwards; it is related to the gradient $m$ by $\tan\theta = m$.

The angle that an object's diameter subtends at the eye of an observer, which for a distant object is approximately its diameter divided by its distance, in radians.

The angle through which an object rotates about a fixed point, usually measured in radians.

The constant $\omega$ in a sinusoidal function $A\cos(\omega t + \phi)$, measured in radians per second, equal to $2\pi f$ where $f$ is the number of cycles per second; the period is $\frac{2\pi}{\omega}$.

The angle turned through per unit time by a rotating object, usually measured in radians per second (rad s⁻¹) and written $\omega$.

The yearly cost of a loan expressed as a single percentage that includes the interest and certain compulsory fees, published by lenders so that different loans can be compared.

A series of equal payments made at equal time intervals, either paid into a fund to build up savings or paid out of a fund to provide an income.

The part of an annulus between two radii; for central angle $\theta$ degrees, outer radius $R$ and inner radius $r$ its area is $\frac{\theta}{360} \times \pi(R^2 - r^2)$.

The region between two concentric circles; for outer radius $R$ and inner radius $r$ its area is $\pi R^2 - \pi r^2 = \pi(R^2 - r^2)$.

A label in a markscheme showing that the line is the result printed in the question, so no mark is awarded for writing it; the marks are for the working that reaches it.

A function $F(x)$ whose derivative is a given function $f(x)$, so that $F'(x) = f(x)$.

A value used in place of the exact value of a quantity, such as a rounded number, an estimate or a measurement, written $v_A$.

A part of the circumference of a circle between two points on the circle.

The distance along an arc of a circle; for radius $r$ and central angle $\theta$ degrees it is $l = \frac{\theta}{360} \times 2\pi r$.

A unit of angle equal to $\frac{1}{3600}$ of a degree, written ″; a milliarcsecond is one thousandth of an arcsecond.

The area of the region enclosed by two graphs between two values of $x$, found by subtracting the area under the lower graph from the area under the upper graph over the same interval.

The area of any triangle equals $\frac{1}{2}ab\sin C$, half the product of two sides multiplied by the sine of the angle between them.

The number by which a transformation multiplies every area; for the transformation $\boldsymbol{r}' = A\boldsymbol{r} + \boldsymbol{b}$ it equals $|\det A|$.

The area of the region bounded by a curve $y = f(x)$ with $f(x) > 0$, the $x$-axis and the vertical lines $x = a$ and $x = b$.

A diagram of the complex plane in which the complex number $a + b\mathrm{i}$ is represented by the point $(a, b)$, with real parts on the horizontal real axis and imaginary parts on the vertical imaginary axis.

The angle $\theta$, measured anticlockwise in radians from the positive real axis, between that axis and the line joining the origin to the point representing $z$ on the Argand diagram; it is written $\arg z$.

The number or expression whose logarithm is taken, $x$ in $\log_a x$; it must be positive for the logarithm to be defined.

A sequence in which each term after the first is found by adding the same fixed number, the common difference, to the term before it.

The sum of the terms of an arithmetic sequence.

A tendency for the values of two variables to change together, so that knowing one gives information about the other; the pattern need not be linear.

The property that $(AB)C = A(BC)$ for any matrices $A$, $B$ and $C$ whose products are defined, so a product of several matrices can be written $ABC$ without brackets.

A statement accepted as true, without proof, in order to simplify a real situation enough for it to be modelled mathematically.

A unit of length used for distances in the solar system, defined as exactly 149,597,870,700 m, about the mean distance from the Earth to the Sun.

A straight line that the graph of a function approaches more and more closely, getting as near to it as desired, as $x$ or $y$ becomes very large in size.

A first-order differential equation $\frac{\mathrm{d}y}{\mathrm{d}x} = f(y)$ whose right-hand side depends only on the unknown $y$ and not on the independent variable.

For the differential equation $\ddot{x} + a\dot{x} + bx = 0$, the quadratic equation $\lambda^2 + a\lambda + b = 0$, whose roots are the eigenvalues of the matrix $\begin{pmatrix} 0 & 1 \\ -b & -a \end{pmatrix}$ and give the solutions $\mathrm{e}^{\lambda t}$.

The change in a quantity divided by the change in the variable it depends on over an interval, $\frac{f(b) - f(a)}{b - a}$, which is the gradient of the secant between the two endpoints.

The number of particles in one mole of a substance, fixed in the SI as exactly $6.022\,140\,76 \times 10^{23}$ per mole.

The straight line about which a region is rotated to form a solid of revolution; every cross-section of the solid perpendicular to this line is a circle centred on it.

The vertical line about which a graph is symmetrical; for the parabola $y = ax^2 + bx + c$ it is the line $x = -\frac{b}{2a}$.

The bearing of the starting point from the destination, which differs from the original bearing by 180°.

The number that is multiplied by itself in a power, written $a$ in $a^n$; in a logarithm $\log_a b$, the base $a$ is the number that is raised to a power.

The vectors $\boldsymbol{i}$, $\boldsymbol{j}$ and $\boldsymbol{k}$ of length 1 in the positive directions of the $x$-, $y$- and $z$-axes; every vector can be written as a sum of multiples of them.

The direction of one point from another, given as the angle measured clockwise from north, written with three digits from 000° to 360°.

A random trial with exactly two possible outcomes, called success and failure, where success has a fixed probability $p$.

A systematic tendency for a sample, or for the way data are collected, to give results that differ from the true population values in a particular direction.

An estimator whose expected value differs from the population parameter it estimates, so that it tends to overestimate or to underestimate.

A question whose wording, order or answer options encourage a particular response, so that the answers do not reflect respondents' true views; a question free of this is an unbiased question.

The full list of sources used in a piece of work, placed at the end and given in one consistent referencing style, with enough detail for a reader to find each source.

The number of ways to choose $r$ items from $n$ when order does not matter, written $\binom{n}{r}$ or ${}^{n}C_{r}$; in a binomial distribution it counts the sequences of $n$ trials with exactly $r$ successes.

The probability distribution of the number of successes $X$ in a fixed number $n$ of independent trials, each with the same probability of success $p$; it is written $X \sim \mathrm{B}(n, p)$.

A hypothesis test for a population proportion $p$, in which the number of successes in $n$ independent trials is modelled by $\mathrm{B}(n, p_0)$ when the null hypothesis $p = p_0$ is true.

A line, ray or segment that divides a line segment or an angle into two equal parts.

Data in which two variables are measured or observed for each individual, so that each individual gives an ordered pair of values $(x, y)$.

A joint distribution of two random variables in which each variable is normally distributed and, for any value of one, the other is normally distributed with a mean that depends linearly on it.

A known pair of corresponding values of the variables, used to find the constant of integration and so select one function from the family of anti-derivatives.

A diagram drawn against a scale in which a box spans the lower to the upper quartile with a line at the median, and whiskers extend from the box to the smallest and largest values that are not outliers; any outliers are marked separately with a cross.

A level of output at which total revenue equals total cost, so the profit is zero; graphically, a point of intersection of the revenue and cost graphs.

The way a calculator or computer displays a number in standard form, with the letter E (or a small ×10) standing for "times ten to the power", as in 2.88E4 for $2.88 \times 10^{4}$.

The volume of liquid or other material that a hollow container can hold, often measured in litres or millilitres.

The limiting value $L$ of a logistic model $f(x) = \frac{L}{1 + C\mathrm{e}^{-kx}}$, which is the horizontal asymptote $y = L$ that the output approaches but does not reach; in a population model it is the largest population the environment can support.

An equation connecting $x$ and $y$ only, found by eliminating $t$ from the parametric equations $x = x(t)$ and $y = y(t)$ of a moving object, which describes the curve the object travels along.

The form $z = a + b\mathrm{i}$ of a complex number, with $a, b \in \mathbb{R}$, in which the number is written as a real part plus a real multiple of $\mathrm{i}$.

A relationship in which a change in one variable directly produces a change in the other variable.

The region of a Voronoi diagram that contains one site and every point that is closer to that site than to any other site.

A collection of data from every member of a population.

The angle between two radii of a circle, with its vertex at the centre; it is the angle that an arc or chord subtends at the centre.

For a random sample of $n$ independent observations from any population with mean $\mu$ and finite variance $\sigma^2$, the distribution of the sample mean $\bar{X}$ approaches the normal distribution $\mathrm{N}\left(\mu, \frac{\sigma^2}{n}\right)$ as $n$ becomes large.

An equilibrium point of a linear system whose eigenvalues are purely imaginary, $\pm\beta\mathrm{i}$; the trajectories around it are closed curves (ellipses or circles), so solutions repeat periodically without approaching or leaving it.

The fixed point from which distances are multiplied by the scale factor in an enlargement; it is the only point that an enlargement with scale factor $k \neq 1$ leaves unmoved.

The fixed point about which a rotation turns the plane; it is the only point that a rotation through an angle that is not a multiple of 360° leaves unmoved.

The rule for differentiating a composite function: if $y = g(u)$ where $u = f(x)$, then $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u} \times \frac{\mathrm{d}u}{\mathrm{d}x}$.

An instant at which an object's velocity changes sign, so that it reverses its direction of motion along the line.

The equation $\det(M - \lambda I) = 0$, whose solutions are the eigenvalues of the square matrix $M$.

For a square matrix $M$, the polynomial in $\lambda$ given by $\det(M - \lambda I)$; for a 2 × 2 matrix it is a quadratic.

The continuous probability distribution, taking only non-negative values, that the chi-squared statistic follows when the null hypothesis is true; its shape is set by the number of degrees of freedom.

A test of whether observed frequencies in a set of categories are consistent with the frequencies predicted by a stated distribution.

A test of whether two categorical variables are independent in a population, using the observed frequencies in a contingency table.

The problem of finding the route of least total weight that travels along every edge of a connected weighted graph at least once and returns to its starting vertex.

A straight line segment joining two points on a circle.

A trail that starts and finishes at the same vertex, so it has no repeated edges.

Motion along a circle; for uniform circular motion about the origin, the position is $\boldsymbol{r} = \begin{pmatrix} R\cos(\omega t) \\ R\sin(\omega t) \end{pmatrix}$, the distance from the centre is always $R$, and the speed is constant.

The point that is equidistant from the three vertices of a triangle; it is where the perpendicular bisectors of the three sides meet.

The circle that passes through all three vertices of a triangle; its centre is the circumcentre and its radius is the distance from the circumcentre to any vertex.

The distance around a circle, equal to $2\pi r$ for a circle of radius $r$.

The abbreviation $\mathrm{cis}\,\theta = \cos\theta + \mathrm{i}\sin\theta$, so that a complex number in polar form is written $r\,\mathrm{cis}\,\theta$.

A short reference placed in the text at the point where a source is used, identifying that source so that the reader can find its full details in the bibliography.

A range of values grouped together as one class in a frequency distribution, written as an inequality such as $10 \le t < 15$.

An instrument for measuring an angle of elevation or depression, by sighting an object along it and reading the angle from the horizontal.

A walk that starts and finishes at the same vertex.

The moment at which the distance between two moving objects, or between a moving object and a fixed point, is smallest, together with that smallest distance.

Two angles between a pair of parallel lines on the same side of a line crossing them; they add up to 180°.

The constant that multiplies a variable, or a power of a variable, in a term of an expression or equation.

A measure of how well a regression model fits bivariate data, equal to the proportion of the variation in the dependent variable that is explained by the model; it lies between 0 and 1 for a least squares model, with values close to 1 showing a close fit.

Lines that have every point in common, so that two different equations describe the same line.

Lying on the same straight line; points $A$, $B$ and $C$ are collinear when $\overrightarrow{AC}$ is a scalar multiple of $\overrightarrow{AB}$.

The event of two moving objects being at the same position at the same time, so that $\boldsymbol{r}_A(t) = \boldsymbol{r}_B(t)$ for one value of $t$ that satisfies every component.

A vector written as a vertical list of its components in brackets, with the $x$-component at the top, then the $y$-component and, in three dimensions, the $z$-component.

An event formed from two or more events using union ("or"), intersection ("and") or complement ("not").

Merging adjacent categories of a chi-squared table, adding their observed frequencies and their expected frequencies, so that every expected frequency is greater than 5.

A word or phrase at the start of an exam question part that tells the candidate what kind of answer to give and how much working to show.

The fixed amount $d = u_{n+1} - u_n$ by which each term of an arithmetic sequence differs from the term before it; it can be positive, negative or zero.

The logarithm with base 10, written $\log_{10} x$ or simply $\log x$; it is the power to which 10 must be raised to give $x$.

The constant $r$ by which each term of a geometric sequence is multiplied to give the next term, so that $r = \frac{u_{n+1}}{u_n}$ for every $n$.

A pair of coupled differential equations for two populations that use the same resources, in which each population grows logistically on its own and the presence of the other species reduces its growth rate.

The event that $A$ does not occur, written $A'$ and read "not $A$"; it contains every outcome in the sample space that is not in $A$.

A simple graph in which every pair of distinct vertices is joined by exactly one edge; the complete graph on $n$ vertices is written $K_n$.

For a complex number $z = a + b\mathrm{i}$ with $a, b \in \mathbb{R}$, the number $z^* = a - b\mathrm{i}$, which has the same real part and the opposite imaginary part.

A number of the form $a + b\mathrm{i}$, where $a$ and $b$ are real numbers and $\mathrm{i}^2 = -1$; the set of all complex numbers is written $\mathbb{C}$.

The signed amount by which a vector moves in the direction of one of the coordinate axes; a vector in two dimensions has an $x$-component and a $y$-component, and one in three dimensions also has a $z$-component.

The function obtained by applying one function to the output of another; the composite $f \circ g$ is defined by $(f \circ g)(x) = f(g(x))$, so $g$ is applied first and $f$ second.

A transformation made by applying two or more transformations one after another, in a stated order.

Interest that is added to the balance at the end of each compounding period, so that in later periods interest is earned on the original amount and on the interest already added.

A solid formed by joining two or more simpler solids, or by removing one solid from another.

The length of time between the moments when interest is added to a balance; the number of compounding periods in a year is written $k$.

Describes a curve, or a part of one, that bends downwards, so that its gradient decreases as $x$ increases and every chord joining two of its points lies below the curve.

Describes a curve, or a part of one, that bends upwards, so that its gradient increases as $x$ increases and every chord joining two of its points lies above the curve.

The direction in which a curve bends on an interval, either concave up, where $f''(x) > 0$ and the gradient increases, or concave down, where $f''(x) < 0$ and the gradient decreases.

The probability that an event $A$ occurs given that an event $B$ has occurred, written $\mathrm{P}(A \mid B)$; it is found by treating $B$ as the new sample space.

The flat shape that folds to make the curved surface of a cone, a sector whose radius is the cone's slant height and whose arc length equals the circumference of the cone's base.

A range of values, calculated from a sample, constructed by a method that captures the true value of a population parameter in a stated proportion of all possible samples.

The percentage of all possible samples for which the method used to construct a confidence interval would produce an interval containing the true value of the parameter.

The lower and upper end points of a confidence interval.

A variable, not included in the analysis, that affects both of the variables being studied and so creates or strengthens a correlation between them.

Two complex numbers of the form $p + q\mathrm{i}$ and $p - q\mathrm{i}$, with $p, q$ real and $q \ne 0$, each the conjugate of the other.

A graph in which every vertex can be reached from every other vertex by travelling along a sequence of edges.

If a function is multiplied by a constant $a$, its derivative is multiplied by the same constant, so the derivative of $af(x)$ is $af'(x)$.

The arbitrary constant $C$ added to an anti-derivative in an indefinite integral, because a constant term disappears on differentiation and so cannot be recovered from the derivative alone.

The constant $a$ in a variation relationship $y = ax^n$, equal to $\frac{y}{x^n}$ for every pair of corresponding values.

The term of a polynomial that does not contain the variable, equal to the value of the polynomial when the variable is 0.

Motion in which the velocity vector does not change, so the object travels in a straight line at constant speed and its position at time $t$ is $\boldsymbol{r} = \boldsymbol{r}_0 + t\boldsymbol{v}$.

A fixed condition in an optimisation problem, such as a given amount of material, volume, perimeter or length, that links the variables and so allows all but one of them to be eliminated.

The extent to which a test covers all the aspects of the quantity it is intended to measure, in the right proportions; it is judged by experts who compare the content of the test with a full description of what should be measured.

A table of observed frequencies that classifies each member of a sample by two categorical variables, one for the rows and one for the columns.

The limiting case of compound interest in which interest is added at every instant, so that an amount $PV$ at a nominal annual rate $r\%$ grows to $PV \times \mathrm{e}^{rn/100}$ after $n$ years.

Quantitative data that can take any value within an interval, usually obtained by measuring.

A function whose graph can be drawn without lifting the pen from the paper; a piecewise function is continuous at a boundary when the formulae on either side give the same value there.

A random variable that can take any value in an interval of real numbers, so that its probabilities are given by areas under a curve rather than by a list of separate values.

A method in which the sample consists of members of the population who are easy to reach or who volunteer, with no random selection and no fixed numbers from each category.

An infinite series whose partial sums $S_n$ approach a finite limit as $n$ increases without bound; that limit is the sum of the series.

A relationship between two variables in which one tends to change as the other changes, described by its direction (positive or negative) and its strength.

The function whose value $\cos\theta$ is the $x$-coordinate of the point where the arm of the angle $\theta$, in standard position, meets the unit circle.

For an acute angle $\theta$ in a right-angled triangle, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.

In any triangle $ABC$, $c^2 = a^2 + b^2 - 2ab\cos C$, which links the three sides with one angle; rearranged, $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$.

A function that gives the total cost of producing or buying a quantity of goods, usually a fixed cost plus a cost for each unit.

A set of differential equations in which the rate of change of each variable depends on the other variables as well as its own, so that the equations must be solved together.

The extent to which the results of a test agree with an established external measure (the criterion) of the same quantity, taken at the same time (concurrent validity) or later (predictive validity); it is judged by a correlation coefficient between test scores and criterion scores.

The set of values of the test statistic for which the null hypothesis is rejected at a given significance level.

The boundary value of a test statistic for a given significance level; a test statistic beyond the critical value leads to rejecting the null hypothesis.

A polynomial equation of degree 3, which can be written as $ax^3 + bx^2 + cx + d = 0$ with $a \ne 0$.

A model of the form $f(x) = ax^3 + bx^2 + cx + d$, where $a \ne 0$.

Fitting a model of the form $y = ax^3 + bx^2 + cx + d$ to bivariate data by least squares.

The running total of the frequencies up to and including a given class, which gives the number of data values below that class's upper class boundary.

A graph of cumulative frequency (vertical axis) against the upper class boundaries (horizontal axis), with the points joined by a smooth increasing curve or by straight line segments.

The probability that a random variable takes a value less than or equal to a given value, $\mathrm{P}(X \le r)$, for a discrete random variable it is found by adding the probabilities of all values up to and including $r$.

The area of the curved part of the surface of a solid such as a cylinder, cone or sphere, not including any flat faces.

A walk that starts and finishes at the same vertex and repeats no other vertex, so it also repeats no edge.

A solid with two identical, parallel circular ends joined by a curved surface, so that every cross-section parallel to the ends is the same circle.

A resistance to motion, modelled by a term proportional to the velocity $\dot{x}$ in an equation of motion, that removes energy from an oscillating system so that its motion dies away.

In PageRank, the probability $d$ that at each step the random surfer follows a link from the current page; with probability $1 - d$ the surfer instead jumps to a page chosen at random from the whole network.

The result that for any integer $n$, $\big(r(\cos\theta + \mathrm{i}\sin\theta)\big)^n = r^n(\cos n\theta + \mathrm{i}\sin n\theta)$, so a power of a complex number has the modulus raised to that power and the argument multiplied by it.

The positive constant $k$ in an exponential decay model $N(t) = N_0\mathrm{e}^{-kt}$; it measures how fast the quantity decreases, as a proportion of its current value per unit of time.

The number between 0 and 1 by which a quantity is multiplied in each time period when it decreases by a fixed percentage; for a decrease of $p\%$ it equals $1 - \frac{p}{100}$.

A decrease in speed, which happens when the velocity and the acceleration have opposite signs.

The unit of a logarithmic scale of the ratio of two powers or intensities; a sound of intensity $I$ has level $L = 10\log_{10}\left(\frac{I}{I_0}\right)$ decibels, where $I_0 = 10^{-12}$ W/m² is the reference intensity, about the threshold of human hearing.

The number of digits written after the decimal point; rounding to $n$ decimal places keeps $n$ digits after the point.

A statement, fixed before the data are collected, of which observed values will lead to rejecting the null hypothesis and which will not.

A function is decreasing on an interval if, for any two values $x_1 < x_2$ in the interval, $f(x_1) > f(x_2)$, so that its graph falls from left to right.

The number $\int_a^b f(x)\,\mathrm{d}x$, the limit of the sum of $f(x)$ times the strip width over the interval from $x = a$ to $x = b$ as the strips become infinitely thin; when $f(x) > 0$ it equals the area under the curve between $x = a$ and $x = b$.

A unit of angle equal to $\frac{1}{360}$ of a full turn, written with the symbol °.

The highest power of the variable that appears with a non-zero coefficient in the polynomial.

The number of edge-ends that meet at the vertex, so each ordinary edge adds 1 and each loop adds 2; it is written $\deg(V)$.

The precision to which a value is given, stated as a number of decimal places, a number of significant figures or a unit of rounding such as the nearest 10 or the nearest cent.

The number of frequencies in a chi-squared calculation that can vary freely once the totals are fixed, written $\nu$ (or df).

The division of the region between a set of sites into triangles, formed by joining every two sites whose Voronoi cells share an edge.

A method for a lower bound for the travelling salesman problem that deletes one vertex and its edges, finds a minimum spanning tree of the remaining graph, and adds the weights of the two edges of least weight from the deleted vertex.

The output of a function, whose value depends on the value of the independent variable, usually plotted on the vertical axis.

The part of the price of an item paid at the time of purchase from the buyer's own money, so that only the remainder is borrowed.

The fall in the value of an asset over time; it is often modelled as a fixed percentage of the current value lost each year.

The function $f'(x)$, also written $\frac{\mathrm{d}y}{\mathrm{d}x}$, that gives the gradient of the graph of $y = f(x)$ at each value of $x$, equal to the instantaneous rate of change of $y$ with respect to $x$; at each point it is the limit of the difference quotient, $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.

The derivative of any constant $c$ is 0, because the graph of $y = c$ is a horizontal line, whose gradient is 0 everywhere.

A number calculated from the elements of a square matrix $A$, written $\det A$ or $|A|$; for $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ it is $ad - bc$.

The difference between a data value and the mean of the data, $x - \bar{x}$; it is positive for values greater than the mean and negative for values less than it.

A square matrix in which every entry off the leading diagonal is zero.

Writing a square matrix $M$ as $M = PDP^{-1}$, where $D$ is a diagonal matrix whose leading-diagonal entries are the eigenvalues of $M$ and $P$ is an invertible matrix whose columns are corresponding eigenvectors.

The function $f - g$ whose output at each input is the output of $f$ minus the output of $g$, so $(f - g)(x) = f(x) - g(x)$ for every $x$ in both domains.

The gradient of a secant written as $\frac{f(a + h) - f(a)}{h}$, the change in a function divided by the change in its input over an interval of width $h$; it gives an average rate of change and, for small $h$, an estimate of the gradient at $x = a$.

An equation that contains one or more derivatives of an unknown function; a solution is a function that makes the equation true for every value of the variable in its domain.

The process of finding the derivative of a function.

A relationship $y = ax^n$ with $n$ a positive integer, in which $y$ is proportional to the $n$th power of $x$, written $y \propto x^n$.

A graph in which every edge has a direction, from one vertex to another, shown by an arrow on the edge; it is also called a digraph.

A line segment with a direction from its starting point to its end point, drawn as an arrow, used to represent a vector; the segment from $A$ to $B$ is written $\overrightarrow{AB}$.

A non-zero vector parallel to a line, which gives the line's direction; any non-zero scalar multiple of it is also a direction vector of the same line.

A way of finding a volume of revolution by treating the solid as a stack of thin circular discs perpendicular to the axis of rotation, each of volume $\pi r^2$ times its thickness, and integrating.

Quantitative data that can take only particular separate values, usually whole numbers obtained by counting.

A random variable that can take only separate, countable values, such as whole numbers, with gaps between them.

A probability distribution in which a random variable takes each of a finite number of values with the same probability; with $n$ values, each has probability $\frac{1}{n}$.

For the quadratic $ax^2 + bx + c$ with $a \ne 0$, the number $\Delta = b^2 - 4ac$, whose sign determines whether the equation $ax^2 + bx + c = 0$ has two distinct real roots, one repeated real root or two non-real complex roots.

The position of an object relative to a fixed origin on a line, given as a signed number whose sign shows which side of the origin the object is on; it is usually written $s$ or $x$.

The vector $\overrightarrow{AB}$ from a point $A$ to a point $B$, which describes the change in position from $A$ to $B$; it equals $\boldsymbol{b} - \boldsymbol{a}$, the position vector of the end point minus that of the starting point.

The property that $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$ for matrices whose sums and products are defined.

An infinite series whose partial sums do not approach a finite limit, either because they grow without bound or because they never settle on one value.

The set of all input values for which a function is defined.

The replacement of a function's domain by a smaller set of inputs, usually chosen so that the function becomes one-to-one and has an inverse.

A way of writing derivatives with respect to time in which a dot over a variable denotes its first derivative and two dots denote its second derivative, so $\dot{x} = \frac{\mathrm{d}x}{\mathrm{d}t}$ and $\ddot{x} = \frac{\mathrm{d}^2x}{\mathrm{d}t^2}$.

The time taken for a quantity that increases by a constant factor to double in size.

To represent a function by an accurate, labelled graph made in pencil, on axes with a stated scale, with points correctly plotted and joined by a ruled straight line or a smooth curve.

A connection between two vertices of a graph, drawn as a line (straight or curved) joining them.

A boundary between two neighbouring cells of a Voronoi diagram; it is a line segment or ray lying on the perpendicular bisector of the two sites whose cells it separates.

The percentage by which a balance actually grows in one year once compounding within the year is included; for a nominal rate $r\%$ compounded $k$ times a year it equals $\left(\left(1 + \frac{r}{100k}\right)^{k} - 1\right) \times 100\%$.

A number $\lambda$ for which a square matrix $M$ has a non-zero vector $\boldsymbol{v}$ with $M\boldsymbol{v} = \lambda\boldsymbol{v}$; it is the factor by which $M$ scales its eigenvectors.

A non-zero vector $\boldsymbol{v}$ for which multiplying by a square matrix $M$ gives a scalar multiple of $\boldsymbol{v}$, so that $M\boldsymbol{v} = \lambda\boldsymbol{v}$ for some number $\lambda$.

One of the numbers in a matrix; the element in row $i$ and column $j$ of matrix $A$ is written $a_{ij}$.

For any normal distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two standard deviations and about 99.7% within three standard deviations.

A non-singular square matrix used to convert a message, written as columns of numbers, into code by multiplication; its inverse is the decoding matrix that recovers the message.

A transformation that maps every point $P$ to the point $P'$ on the line through a fixed centre $C$ and $P$ with $\overrightarrow{CP'} = k\,\overrightarrow{CP}$, where $k$ is the scale factor; it changes the size of a shape but not its shape.

Two matrices that have the same order and whose corresponding elements are all equal.

Vectors that have the same magnitude and the same direction; in component form, vectors whose corresponding components are all equal.

Outcomes that all have the same probability of occurring, so that with $n(U)$ such outcomes each has probability $\frac{1}{n(U)}$.

The equation of the line through $(a, f(a))$ perpendicular to the tangent there, $y - f(a) = -\frac{1}{f'(a)}(x - a)$ when $f'(a) \ne 0$, and the vertical line $x = a$ when $f'(a) = 0$.

The equation of the straight line that touches the curve $y = f(x)$ at $(a, f(a))$ with gradient $f'(a)$, which is $y - f(a) = f'(a)(x - a)$.

A boundary on which every point is the same distance from the nearest points of two coastlines or regions; between two single points it is their perpendicular bisector.

At the same distance from two or more given points or objects.

A point in the phase plane at which every rate of change of a system is zero ($\dot{x} = 0$ and $\dot{y} = 0$ simultaneously), so that a solution starting there stays there for all time.

The single numerical value that an estimator gives for one particular sample, used as a best guess of a population parameter.

An approximation to the mean of grouped data, calculated with the mean formula using the mid-interval value of each class in place of the unknown individual values.

A value of a distribution's parameter, such as a mean, a standard deviation or a probability of success, that is calculated from the sample data because it is not known or given in advance.

Finding an approximate value of a calculation or quantity, usually by rounding the numbers involved to 1 significant figure, in order to judge whether an exact result is reasonable.

A rule or formula, applied to sample data, that is used to estimate a population parameter; because the sample is random, an estimator is itself a random variable.

The length of the straight line segment joining two points, calculated in two dimensions as $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$.

The identity $\mathrm{e}^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\sin\theta$, valid for every real $\theta$ measured in radians.

The equation $\mathrm{e}^{\mathrm{i}\pi} + 1 = 0$, the case $\theta = \pi$ of Euler's formula.

A numerical method for a first-order differential equation that starts from a known point and repeatedly moves a fixed distance along a straight line whose gradient is given by the differential equation at the current point.

The irrational constant $\mathrm{e} = 2.718\,281\,828\ldots$, the value that $\left(1 + \frac{1}{n}\right)^n$ approaches as $n$ becomes very large; it is the base of the natural logarithm.

A circuit that uses every edge of a graph exactly once, starting and finishing at the same vertex.

A connected graph that has an Eulerian circuit, which happens exactly when every vertex has even degree.

A trail that uses every edge of a graph exactly once.