NOTE: This is the medium level dictionary. Words which are useful for A-level, but not for GCSE, are in the advanced dictionary. The advanced dictionary also uses words more precisely in some cases. If you find this dictionary too difficult, try the first level dictionary. Some words that you might want to use to get you started are:
A frame with rods and beads that was used for counting in ancient times. The beads slide along the rods.
The acceleration of an object is its rate of change of velocity; that is, it is the change in velocity per unit of time. The acceleration of an object at any time can be found using a velocity /time graph, by finding the slope of the tangent to the graph.
An angle which is less than a right angle.
Addition is the combining ( adding ) of numbers to make a total, called the sum. The sign for adding two numbers is a "+" between them. For example, 2 + 3 is 5. To add is to do an addition. Addition can also be done on sets or matrices. Two matrices of the same order are added by adding together the corresponding pairs of elements to make a new matrix; for example (2 3) + (4 5) = (6 8) - the 2 is added to the 4 to make 6, and the 3 is added to the 5 making 8.
Algebra is like arithmetic, but with letters used to stand for numbers that we do not know. These letters used are usually x and y. The letters, and the signs used for adding (+), subtracting (-) and so on are called symbols. [ Algebraic is to-do-with-algebra]. For example, (x-4)=2 is an equation in algebra with one solution, x=6.
[The word comes from the name of an old arabic maths book].
A step-by-step way of solving a problem. Algorithms are useful when a computer is being used. See Logo.
Two angles made where a line (called the transversal ) intersects (crosses) two other lines so that one angle is made with each of the intersected lines, and the two angles are on opposite (alternate) sides of the transversal.
AMPLITUDE: see advanced dictionary.
An analogue clock is one with hands, as opposed to a digital clock. The time is shown by the positions of the hands. Any device that uses a pointer that can point to anywhere on a scale is an analogue device, so a magnetic compass, for example, is analogue.
ANALYSIS: See advanced dictionary.
ANALYTIC GEOMETRY : See advanced dictionary.
An angle is made when two straight lines, that go in different directions, meet at a point. The size of the angle is a measure of how different the directions of the lines are. A square has four angles, all the same size, where the sides of the square meet at the corners. An angle which is the same size as a corner of a square is called a right angle. Angles are measured in degrees (or radians - but you don't need to know about radians for GCSE). There are 90 degrees in a right angle. A triangle has three angles. Select -pic- for a picture. See also trigonometry.
A way of showing the position of a point. Given a starting position and a starting direction, if we turn through the given angle and move forward the given distance, we should end up at the right point. Using LOGO, for example, if the turtle is pointing straight up and we want it to go 20 units to the right, we can tell it to RIGHT 90 FORWARD 20 and it will turn 90 degrees to the right and move forwards 20 units. Compass bearings can also be used; for example, in orienteering you might be told to go east one mile.
ANGULAR VELOCITY : see advanced dictionary.
ANNULUS: See advanced dictionary.
The opposite of clockwise.
ANTILOGARITHM: see advanced dictionary.
1. The highest point of a plane figure, such as a triangle, when that figure is resting on some base line. 2. The highest point of a solid figure, such as a cone or pyramid, when that figure is resting on some base plane.
Making an approximation.
An estimate of a given quantity, to a certain degree of accuracy, for example, to the nearest thousand, or to two decimal places.
A part of a curve. In a network, an arc joins two nodes.
A famous Greek mathematician, who lived from 287 BC to 212 BC. He was the leading mathematician of his time.
The measure of the size of a flat shape (for instance, the lid of a box), or of a shape that can be folded or bent until it is flat (for instance, the whole of a box). Knowing the area of a box is useful because it tells us how much material we will need to make another box like it. The following formulae are useful for calculating areas. The area of a triangle = base × height /2 The area of a parallelogram = base × height The area of a trapezium = 1/2 ( sum of parallel sides × distance between) The area of a circle = pi × the radius squared. The surface area of a sphere = 4 × pi × the square of the radius The area under a curved shape can be approximated by tracing it onto squared paper and counting the squares. The area "under a graph " means the area below the line of the graph but above the x-axis (assuming that the graph is positive ), and between two points on the axis. It can be approximated using trapezia and the trapezium rule. The area of a complicated shape can be found by dividing it into simpler shapes.
*ARGAND DIAGRAM: see advanced dictionary
*ARGUMENT: See advanced dictionary
The studying of numbers, and applying simple operations ( adding, subtracting, multiplying and dividing) to solve problems with numbers.
An average of a set of values, made by adding them all together, and dividing by the number of values. It is sometimes just called the mean, and sometimes it is called "the average ", but there are other kinds of average, and at least one other kind of mean - see geometric mean.
A sequence where the difference between any term and the one immediately before it is always the same, and is called the common difference. for example, 3,8,13,18,23... is an arithmetic progression, and the common difference in this case is 5 (8 - 3 = 5, 13 - 8 = 5, and so on).
An operation * is associative if a * (b * c) = (a * b) * c for any a, b and c. Ordinary adding and multiplying are associative; For instance, you can see that (4 + 2) + 3 = 4 + (2 + 3). However, subtraction is NOT associative. The brackets show which operation is done first, so if an operation is associative it means that the order it is done in is not important.
A single number, derived from a collection of numbers, that is used as a representative of those numbers. Common averages are the arithmetic mean, the mode and the median. Of these, the arithmetic mean is most commonly used, so that when people talk about an average in mathematics they are usually referring to the arithmetic mean.
An axis is a line chosen as a base, so that other lines, points and figures can be described by saying how far they are from the axis. The plural is axes. Axes are important in a cartesian coordinate system. An axis of symmetry is a line which can be drawn through an object, so that the object is unchanged by reflection in the line.
A graph in which bars of equal width are shown side by side, and the heights of the bars give us some information.
A bar-line graph is like a bar chart, but with vertical lines instead of bars. It can be used when you want to make it clear that each height represents a single value for a discrete variable - not an interval.
In navigation and surveying, the angle between a given line and the north line, measured clockwise. Some bearings have special names, for instance, north, south, east and west. North is the direction of the earth's north pole. It has a bearing of zero degrees. East has a bearing of 90 degrees, south is 180 degrees, and west is 270 degrees.
A thousand million. In Britain, it used to refer to a million million, but now the American meaning is used.
A distribution is bimodal if it has two modes.
A rule for taking two members of a set and using them to produce a third member of the set. For example, in adding 3 and 4 we take two numbers (3 and 4) and produce a third number (7). Addition is a binary operation, and so are subtraction, multiplication and division.
See advanced dictionary.
A statistical distribution which appears when an experiment is performed repeatedly and the number of times that a particular event occurs during the trials is totalled. For example, if a coin is tossed 5 times, the total number of heads has a binomial distribution.
To bisect is to divide something into two equal parts. A bisector divides something into two equal parts. Often applied to lines and angles.
See Upper bound, Lower bound.
The curves which comprise the outer edges of a figure.
A function f(x) is bounded over an interval if there is a value n for which |f(x)| < n for all x in the interval; in this case, n is called an upper bound, and -n is a lower bound for the function.
brackets are symbols which enclose an operation to show that it is to be performed before other operations. For example, 3 × (2 + 4) = 3 × 6 = 18 the brackets round the 2 + 4 show that the addition is to be done before the multiplication.
To perform some operation or operations witn numbers.
A device, usually small and hand-held, with a keyboard and display, which allows the entry of numbers and operations, and displays the result. Simple calculators will perform addition, subtraction, multiplication and division. Scientific calculators will calculate sines, cosines, tangents, logarithms, square roots, arithmetic means, variances, standard deviations etc. They are very versatile, but they can produce rounding errors, for example, displaying 3.9999999 when the correct answer is 4.
The branch of mathematics dealing with continuously changing quantities. There are two types of calculus, involving differentiation and integration.
Simplifying fractions by dividing the numerator and the denominator by a common factor.
A number that describes "how many" are in a set of things. For example, "four" (4) and "five" (5) are cardinal numbers. Compare with ordinal number.
The locus of a point on a moving circle which rolls around another circle of the same diameter.
See coordinate. Cartesian coordinates are named after Rene Descartes (1596-1650), a French mathematician and philosopher.
The shape of a curve made by a flexible rope hanging between two points. The graph of y=cosh(x) (called the hyperbolic cosine, and usually included in books of table s) forms a catenary.
A prefix meaning hundredth. See metric.
A scale for measuring temperature, in which 0° is the temperature at which water freezes and 100° (read as "a hundred degree s") is the temperature at which it boils, given normal air pressure. 20 degrees centigrade is written as 20°C. See also fahrenheit.
A hundredth of a metre.
The integer part of a common logarithm.
A line segment that joins two points on a curve. The longest chord on a circle passes through the centre and is called a diameter.
The set of points in a plane that are all the same distance from a given fixed point. The plane figure enclosed by the circle is called a disc. The " area of a circle " is the area of the disc enclosed by it.
To draw a geometric figure around another so that the two are in contact but do not intersect.
A grouping of statistical data, so that similar values are placed in the same class.
A clockwise rotation is one that is in the same sense as the movement of the hands of a clock.
In algebra, the numerical part of a term.
Lying in a single straight line.
A vertical array of elements, which forms part of a matrix.
The possible selections of items from a set, regardless of order. For instance, when selecting three diferent letters from the alphabet, ABC, CBA, ACB etc. all represent the same combination. See also permutation.
The probability that two or more events will happen. How it is calculated depends on whether the events are independent or not.
A whole number that is a multiple of the denominators of two or more fractions.
The difference between successive terms in an arithmetic progression.
An operation is commutative if varying the order of the operands does not affect the result. For numbers, addition and multiplication are both commutative, but subtraction and division are not.
The set of all elements that are not in a particular set.
An angle is complementary to another if the sum of the two angles is 90°.
A method of solving a quadratic equation by reducing it to the form (x+h)²=k.
*COMPLEX NUMBER: See advanced dictionary.
The function produced by applying two component functions, one after the other. The domain of the second function must include the range of the first.
Interest charged or earned on money, when the interest earned during a period is added to the original sum for use in calculating the interest for the next period.
An electronic device used to process information. There are two types, analogue and digital, but analogue computers are hardly ever used now. You are using a digital computer to read this text. A computer has input devices (for example, the keyboard or mouse), output devices (such as the screen, or a printer), storage devices (disk drives) and a central processor.
A region is called concave if it is possible to draw a straight line segment from one part of the region to another, so that the line segment lies partly outside the region.
Two or more circles having the same centre.
A number of lines are concurrent if they all meet at a single point.
A solid bounded by a conical surface and a plane base. The conical surface tapers to a point called the vertex. Any straight line joining the vertex to an edge of the base lies on the conical surface. If the base is a circular disc then the cone is called a circular cone. It is usual to assume that a cone is circular, unless stated otherwise.
Having the same shape or size.
A curve obtained by taking a plane section through a conical surface. There are four types of conic - circles, ellipses, hyperbolas and parabolas.
See cone.
See advanced dictionary.
A fixed quantity in an expression; for instance, in Y= 3X + 4 the 3 and 4 are constants. X and Y are variables.
A variable is continuous if between any two values of the variable there are an uncountable number of other possible values.
An infinite sequence that has a limit is said to converge to that limit. See also diverge.
The process of changing from one system of units to another, for example, from degrees to radians, or from fahrenheit to centigrade.
A region is convex if every line segment drawn from one point in the region to another lies wholly within the region.
Coordinates are measurements used to define the position of a point in a space or on a surface. For instance, latitude and longitude are coordinates used to define the position of a point on the surface of the earth. On a flat surface, if two orthogonal lines are drawn, called the axes and labelled x and y, then the position of any point can be represented as a pair of numbers, the distances of the point from the two axes. These are called Cartesian coordinates. The first coordinate is called the x-coordinate and the second is called the the y-coordinate. In a three dimensional space, three axes and three coordinates are needed. See also polar coordinate.
Lying in the same plane. A set of lines or planes is coplanar if one plane can be found which includes them all.
Extent to which variation in one set of figures is accompanied by variation in another set.
A relationship between pairs of elements from two sets.
In a transformation, features of the first figure are said to correspond to their images after the transformation.
A trigonometric function. The reciprocal of the sine.
A trigonometric function. In a right angled triangle, the cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse.
A trigonometric function. The reciprocal of the tangent.
Counting is the matching of elements from a set with the names of the positive integers, on a one-to-one basis, starting at one and continuing in order. For example, we count the elements in a set as "one, two, three". If at this point all the elements are matched, we say that there are three elements in the set.
A technique used to find the fastest way of completing a big project given that some activities depend on the completion of others. It finds the critical path, that is, the set of activities which must be completed on time, because otherwise the project will be delayed.
A way of simplifying equations with fractional terms, by multiplying both numerators or both denominators by the same amount.
A plane shape that results from cutting through a solid. For example, a cross section of a sphere is always a circular disc. Cylinders and prisms have a uniform cross-section.
If a number X is the cube of Y, then Y is the cube root of X. See also root.
A solid with six rectangular faces. A rectangular box.
In statistics, when the observations have been divided into classes, and these classes can be ordered, then the cumulative frequency for each class is the total frequency for that class and all the classes that come before it. A cumulative frequency graph shows the classes along the x axis and the cumulative frequency along the y axis.
A line which need not be straight. A continuous path traced by a moving point.
A point on a curve where the curve turns back on itself. For example, there is a cusp in a cycloid.
A cyclic quadrilateral is one that can be inscribed in a circle, so that all its vertices touch the circle. The opposite angles in a cyclic quadrilateral are supplementary, that is, their sum is 180°.
The curve traced by a point on the circumference of a circle that rolls along a straight line. It is the locus of the point. There is a cusp where the point comes in contact with the line. See also epicycloid, hypocycloid.
A solid with one axis of symmetry and a uniform circular cross section. See area and volume, for formulae.
Information which is given; especially, information given to a computer for using in calculations or for storing in a database.
A collection of information, or data, especially when it is inside a computer. The data is stored as records. A relational database is one in which the data is treated as a set of relations.
A prefix meaning ten. See metric.
A ten-sided polygon.
The amount by which a given quantity decreases in a given time.
A prefix meaning one tenth of. See metric.
A number system with a base of 10.
A way of representing a fraction by dividing the unit into ten equal parts, then each of these tenths into ten further parts, and so on. Positional notation is used, with a point after the units digit. The next position represents tenths, the position after that represents hundredths, then thousandths, and so on.
A diagram which helps in making a decision, by showing a set of questions each with two branches, one for yes and one for no, which may lead to other questions.
The bottom part of a fraction. See also common denominator
If f(x) is a function of x, then its derivative, called f'(x), is the rate of change of f(x) with respect to x. If y=f(x) then the derivative may also be written as dy/dx. Note that this may look like a division, but it isn't. For the process of finding a derivative, see differentiation.
A determinant is a number associated with a given square matrix which is useful in solving equations involving that matrix. For a 2 × 2 matrix / a b \ \ c d / the determinant is (a × d) - (c × b).
A line joining two non-adjacent vertices of a polygon.
A chord of a circle that passes through the centre.
A die is an object, usually in the form of a cube, which has numbered or marked faces and can be thrown to produce a random outcome. The plural is dice.
The difference of two numbers is the result of a subtracting one from the other.
See advanced dictionary.
A single symbol (0,1,2..9) used in the writing of numbers. Other symbols may also be used when more than ten different symbols are needed; for instance, in a hexadecimal number the letters a to f are used as digits.
Making use of digits. A digital computer is one that represents numbers using a digital code (usually binary inside, but converted into decimal for printing on the screen).
The dimension of a space is the number of coordinates needed to represent a point in that space. Ordinary space is three-dimensional. A plane is two-dimensional.
A signed number ( positive or negative ), shown on a horizontal
line so that positive
numbers are to the right of zero, and negative numbers to the left. Given a number, it is
possible to add to it by moving towards the right and to subtract by moving left. Given two directed numbers a and b, if a is to the left of b
we say that a is less than b, written as a
An indication of how to get from one place to another, such as up or down. A compass direction is one of the four main directions, north, south, east or west, or else it lies between two of the main directions. Northeast is midway between north and east, and so on. Two vectors have the same direction if they are parallel to one another, and one is a positive multiple of the other.
A line used in the construction of a conic. For any point on the conic, the ratio of the distance to the focus and the distance to the directrix is a constant, called the eccentricity.
The plane figure enclosed by a circle.
A variable is discrete if the number of values that it can have is countable.
For a quadratic equation y=ax² + bx +c the quantity b²-4ac is called the discriminant. if this is greater than zero, the equation has two real solutions. if it is equal to zero, the equation has one real solution if it is less than zero, the equation has no real solutions.
Two sets are disjoint if no element is a member of both sets. Their intersection is the empty set.
See measure of dispersion.
A vector quantity representing the difference in position of two points. Note that the velocity of an object is the rate of change of its displacement from another "fixed" point.
The set of values and associated frequencies of a statistical variable. There are many distributions which are encountered frequently. See binomial, normal distribution.
When two binary operations, * and +, are defined on a sets, and for any 3 elements a,b,c ins a * (b + c) = (a * b) + (a * c) then * and + are said to obey the distributive law, and * is said to distribute over +. In ordinary arithmetic, multiplication distributes over addition but not the other way round.
An infinite sequence is said to diverge if it has no limit. See also converge.
To divide is to perform a division.
in a division, the quantity that is divided
A whole number A is divisible by a whole number B if A can be divided by B without leaving any remainder. If A is divisible by B, then A is a multiple of B. Sometimes it is easy to test if a number is divisible by another. For instance, a decimal number is divisible by 5 if it ends in 0 or 5.
Given an amount, finding a set of smaller amounts that add up to it. The smaller amounts should all be equal, except perhaps for a remainder. The symbol for division is ÷, or /. Division is the inverse of multiplication. For example, because 4 × 5 = 20, it follows that 20 ÷ 5 = 4. (Read as "twenty divided by five equals four") In this equation, 20 is the dividend, 5 is the divisor, and 4 is the quotient.
In a division, the quantity that the dividend is divided by. for instance, in 20 / 4 = 5, 4 is the divisor.
A polygon having twelve sides.
A polyhedron having twelve faces. In a regular dodecahedron, the faces are all regular pentagons.
The set which is operated on by a function.
A solid that can be obtained from another by replacing each vertex with a face, and vice versa. For example: A cube has six faces and eight vertices. An octahedron can be produced from a cube by cutting off each of the vertices to produce a face. The octahedron is the dual of the cube and has eight faces and six vertices. Similarly, the icosahedron is the dual of the dodecahedron, and vice versa. The dual of a tetrahedron is another tetrahedron.
A number system with base 12.
The symbol for the number which is the base of natural logarithms. E is approximately 2.71828
In a conic, the ratio of the distance from any point on the conic to the focus and the distance from the same point to the directrix. An ellipse has eccentricity less than 1. A parabola has eccentricity 1. A hyperbola has eccentricity greater than 1.
A line segment consisting of the points where two faces of a polyhedron meet.
A closed curve resembling a flattened circle. It has two axes of symmetry, unlike an oval, which has one end larger than the other. An ellipse can be produced as the intersection of a plane with a cone; therefore, it is a conic. Alternatively, it can be produced as the locus of a point in a plane that moves so that the sum of its distances from two fixed points in the plane is constant.
A set that contains no members, denoted by the symbol ø.
A transformation of plane or solid shapes that maps them onto larger but similar shapes.
A plane curve that is tangential to (touches) a whole family of curves.
The path traced by a point p on the circumference of a circle which rolls around the circumference of another circle. See also cycloid, hypocycloid.
Things which are identical in some respect are said to be equal in that respect. Two numbers are equal if they are identical. Two expressions are equal if they both produce the same result in all cases, or if one can be transformed into the other using various rules. The symbol for equality is "=", first used by Robert Recorde in 1557 Examples: 2 = 2. 3+4=4+3 because both additions produce 7 as a result. 3+X=X+3 by the commutative rule of addition.
A set of random events are equally likely if none of them can be expected to occur more often than any other in a long set of trials. This usually happens where there is a symmetry about the event s; for instance, in the tossing of an ordinary coin, heads and tails are equally likely events because, although there are differences between the two faces, these do not affect the outcome. (Of course, it is almost certain that one face will come up more often than the other, but there is no way to know in advance which one.) See unfair.
A mathematical statement in which two quantities or expressions are said to be equal, using an " equals sign ", =
Having the same distance.
An equilateral polygon is one which has all its sides of the same length. In an equilateral triangle the angles must necessarily be equal too.
A state of balance, when the forces acting on a body have a zero resultant. The equilibrium is unstable if some small movement of the body will cause the forces to have a non-zero resultant which takes the body further away from the equilibrium position. Otherwise, it is stable.
A relation R which groups elements so that if aRb then a and b belong to the same group. An equivalence relation is reflexive, symmetric and transitive.
Two fractions are equivalent if they both represent the same number.
A value which is close to the true value of some variable. An estimate can be used when an exact value is not needed, or as a check that a calculation has been done correctly, or as a starting point for a procedure which produces a better estimate. In statistics, we sample a population to get an estimate for some property of the population.
A Greek mathematician (330-275 BC), who published a set of 13 books called the Elements. These contained a systematic account of geometry and number theory.
The geometry of ordinary space of two or three dimensions is called Euclidian, after Euclid.
A Swiss mathematician famous for his work in geometry, trigonometry, calculus and topology. See Euler's formula, Konigsberg bridge.
A whole number that can be divided exactly by 2. See also odd number.
Something which can be said to have happened or not, depending on the outcome of a trial. For instance, when a die is thrown, the outcome will be 1, 2, 3, 4, 5 or 6. " Odd number " is an event which occurs if the outcome is 1, 3 or 5.
The result of multiplying the terms of an expression, especially a binomial. For instance, x³+3x²+3x+6 is the expansion of (x+1)³.
A symbol denoting the number of times a quantity is to be multiplied by itself.
An exponential function is one in which the rate of change of the value is proportional to the value.
An expression is a written collection of numbers and symbols which has a value that can be calculated. For example, 2 + 2 is an expression. If some of the numbers are unknown, and are shown as letters, then the expression is algebraic.
To use a set of given data within a range to estimate values outside the range.
A maximum or a minimum. The plural is extrema.
A flat surface of a polyhedron.
A number or expression which can be divided into another a whole number of times, with no remainder. A number or expression X is a factor of another number or expression Y if Y is an exact multiple of X.
For a polynomial P(x), if P(a)=0 then (x-a) is a factor of P(x).
Factorial n, written as n!, is the product of multiplying the first n positive integers together. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
To express a number or a polynomial as the product of some of its factors. For example, 12 = 3 × 4 x²-x-6 = (x + 2)(x - 3)
A scale for measuring temperature, in which 32° is the point at which water freezes, and 212° is the boiling point, given normal air pressure.
A die or coin is fair if the outcomes of throwing it are equally likely. See also unfair.
A sequence of numbers in which, apart from the first two, every number is the sum of the two immediately before it. It begins 1, 1, 2, 3, 5, 8,... and is named after Leonardo Fibonacci (1175? - 1250?), a mathematician of Pisa.
A bounded set of points in a plane, such as a triangle or a disc, or a solid.
Not infinite. A finite quantity will have an upper bound which it never exceeds.
A diagram showing the procedure, or algorithm, to be used in solving a problem.
Plural foci. A point associated with a conic, and used in the production of that conic. An ellipse and a parabola both have two foci. A parabola has one focus and a directrix.
A foot is a unit of length. 1 foot =12 inches.
A mathematical statement, made using symbols. The plural is formulae. Some formulae are very useful and should be learned. See area, volume, surface area, trigonometric formula, Pythagoras' theorem, quadratic equation.
A geometric shape that is complex and detailed in structure at any level of magnification. Fractals are often self-similar, i.e. they have the property that each small portion of the fractal can be viewed as a reduced-scale replica of the whole. The French mathematician Benoit B. Mandelbrot was responsible for the discovery of fractal geometry in the 1970's.
Part of an integer, or whole number. It can be shown as one whole number (the numerator ) over another (the denominator ). In this case the numerator is divided by the denominator. See also decimal fraction, proper fraction.
A diagram showing, for a set of events, the number of times each event has occurred.
A solid bounded above and below by two parallel planes, the base and the top. A frustrum can be obtained from another solid, as the part of that solid that lies between two parallel planes cutting the solid.
A relation between two sets, called the domain and the range of the function, in which each member of the domain is related to exactly one member of the range. If f is the name of the function, and x is a member of the domain, then f(x) is the corresponding member of the range. A function can be defined using an expression : for example f(x)= x² + 2x.
Eight pints. A measure of volume. One gallon is approximately 4.5 litres.
See geometric mean, geometric progression, geometry.
The geometric mean of a set of n numbers is the nth root of the product of all the numbers.
A sequence of numbers in which each number except the first is a constant multiple of the number preceding it.
The mathematical study of space, including the relationships between angles, lengths, areas and volumes. Euclidian geometry is the geometry of normal space, in which only one line is allowed parallel to a given line and passing through a given point. Non-Euclidian geometry is any geometry in which it is assumed that this rule is not true. See also analytic geometry.
A prefix meaning one billion (one thousand million ).
A transformation composed of a reflection and a translation.
A measure of steepness of a line or curve.
A metric unit of mass. There are 1000 grams (or gramme s) in 1 kilogram.
A diagram showing the relationship between variable quantities. The graph of a function shows the relationship between points in the domain and corresponding points in the range of the function. See bar chart, line graph, pictogram.
A circle drawn on the surface of a sphere which has the same radius as that sphere. The largest circle that can be drawn on a given sphere. See also meridian.
See advanced dictionary.
The rate at which a particular quantity increases with time (it will be negative if the quantity is decreasing).
See highest common factor.
A measure of area, equal to 10,000 square metres.
A curve which lies on a cylindrical or conical surface and makes a constant angle with the axis.
One half of a sphere.
A formula for the area of a triangle, given the lengths of its sides. Named after Heron of Alexandria (1st century AD).
A polygon having six sides. A regular hexagon has all its interior angles equal to 120°, and all its sides of equal length. A tesselation can be made with regular hexagons.
The largest number which is a factor of two given numbers.
A chart, like a bar chart but with the frequency of a bar represented by area rather than height, and the width of a bar proportional to the size of interval it represents.
A unit of time. There are sixty minutes in one hour, and twenty-four hours in a day.
The curve formed when a plane cuts a double cone into two parts, with each the mirror image of the other. A conic.
The path traced by a point p on the circumference of a circle which rolls around inside the circumference of another circle. See also cycloid, epicycloid.
The side opposite the right angle in a right-angled triangle.
An assumption which needs to be tested, especially in statistics. See null hypothesis.
A polyhedron having 20 faces. In a regular icosahedron, the faces are all equilateral triangles.
An element of a set which, when combined with any other element according to some binary operation, leaves it unchanged. For instance, zero is the identity element for addition; a+0=a for all a. 1 is the identity element for multiplication.
The result of applying a function to a particular element of the domain. For instance, if f(x)=x² then the image of 3 is 9.
See advanced dictionary.
A standard system of measuring, in which measures are defined by United Kingdom laws; gradually being replaced by the metric system. Weight is measured in stones, pounds and ounces, length is measured in miles, yards, feet and inches, volume is measured in gallons, pints and fluid ounces, and area is measured in square miles, square yards, square feet, or square inches, or sometimes in acres.
A fraction in which the numerator is larger than the denominator.
One of the centres of a triangle. It is where the three angle bisectors meet. It is also the centre of the incircle of the triangle.
An imperial unit of measure. There are 12 inches in one foot.
A circle drawn within a triangle so that it just touches the 3 sides of the triangle. The centre of this circle is called the incentre.
A variable allowed to vary over the domain of a function is called an independent variable.
Represents a difference between a pair of expressions, which allows them to be ordered. Inequalities can be "less than " ( <) or "greater than" ( > ). An inequality without variables must be either true or false: For example, 4 < 2 is false. See also directed numbers and unconditional inequality.
A quantity is said to be infinite if it always exceeds any fixed limit. For example, the number of integers is infinite, and so is the series 1+2+3+4+5...
See advanced dictionary.
A circle drawn inside a polygon which touches all sides. Not all polygons can have an inscribed circle, but all triangles can. See incircle.
A number, positive, negative, or zero, with no fractional part.
See advanced dictionary.
The distance from the origin of a point where a line cuts through an axis.
Estimating the value of a function at a point between two points at which the value is known.
A measure of dispersion. The difference between the upper and lower quartiles of a distribution (that is, upper quartile - lower quartile ). It is twice the value of the semi-interquartile range.
The set of points common to two lines, curves, regions, planes, solids or sets.
A range of values that a variable is allowed to take.
something which is not affected by a transformation. For instance, the area of a geometric figure is invariant when the figure is rotated around a point.
If a function y=f(x) maps values of domain d onto values of range r, then the inverse function x=g(y), if it exists, is one that maps values of r onto values of d, so that x=g(f(x)) for all x. In other words, applying an inverse function reverses the effect of applying the original function. 2)The inverse A' of a matrix A is one which, under matrix multiplication, reverses the effect of multiplying by A, so that A' × (A x B) = B for any B.
Two variables are inversely proportional to one another if their product is a constant, so that one one increases the other must decrease and vice versa. One variable is proportional to the reciprocal of the other.
A kind of spiral. The curve followed by the end of a thread as it is unravelled from the end of a fixed spool, being kept taut at all times.
A number that cannot be expressed as a fraction, or as a finite decimal
An isometry, or isometric mapping, is a transformation that leaves lengths unchanged; for instance, a rotation, translation, reflection or glide reflection.
Having the same structure.
A triangle with two equal sides and two equal angles is an isosceles triangle. (If it has two equal sides, it must have two equal angles, and vice-versa). An isosceles trapezium is one in which the non-parallel sides are equal in length. In an isosceles trapezium the diagonals are equal and the interior angles at the base are equal. An isosceles triangle has an axis of symmetry which bisects the angle between the two equal sides.
To repeat. For example, if x is an approximation for the square root of 2, (x + (2/x))/2 will be a closer approximation, so iteration will allow us to get ever closer to the true value ( 1 -> 1.5 -> 1.41667 -> 1.41422 -> 1.41421....). See trial and improvement.
Involving iteration.
A prefix meaning one thousand.
A thousand grams. See metric.
A thousand metres. See metric.
A quadrilateral having two pairs of adjacent sides equal.
A unit of speed. One nautical mile per hour.
The townspeople of Konigsberg wondered, in the eighteenth century, if it was possible to cross each of the town's seven bridges just once and return to the starting point, without using a boat or getting wet. Leonhard Euler showed that it could not be done, by analysing the network of routes.
A circle of latitude is a circle on the surface of the earth, with a centre on the line joining the north and south poles. The largest circle of latitude is a great circle, called the equator. The latitude of a point on the earth is the angle formed by a line from the centre of the earth to the point, and another line from the centre of the earth to a point on the equator with the same longitude. See also longitude, coordinate.
For a straight line segment, the distance between the two end points.
A sequence tends to a limit if the terms get progressively closer to some value without ever reaching that value. This value is called the limit. For example, 1, 1/2, 1/4, 1/8... is a sequence that tends to zero (the numbers get closer and closer to zero ). We say that zero is the limit of this sequence. Similarly, a series tends to a limit if the sum of the first n terms approaches some value as n gets larger, without ever reaching that value. For example, 1 + 1/2 + 1/4 + 1/8... is a series with 2 as its limit. The more numbers you add together from the series, the closer you get to 2.
In mathematics, usually refers to a straight line extending indefinitely in both directions. A part of a line is called a line segment. See also curve. For ` line of longitude ` - see longitude.
A graph in which a set of points are joined by line segments.
The line which fits a set of data most accurately. Often, it can be drawn precisely enough by eye, drawing a scatter graph for the data, and a straight line that follows the dots as closely as possible. Alternatively, It can be calculated using the method of least squares (see advanced dictionary).
A part of a straight line.
Associated in some way with a straight line. A linear relationship between two variables is one that appears on a graph as a straight line.
A technique used to find out how to get the most out of limited resources. For example, if a factory has only a limited amount of materials, workers, and machines, and the management want to know what to make to get as much profit as possible, they can use linear programming to help work out the answer.
A metric unit of volume. One litre = 1000 cubic centimetres.
The locus of a point is the curve followed by the point when it moves according to certain rules. For example, a cycloid is the locus of a point on the circumference of a circle that rolls along a line.
The logarithm of a number to a given base is the power to which the base must be raised to give that number. Note that, for a positive base, only positive numbers have a logarithm. Logarithms can be used to simplify calculations because multiplication and division can be converted to the addition and subtraction of logarithms.
An easy-to-use computer programming language. With logo you can write a program, or algorithm, to tell a computer to draw a shape. This is done using a pretend "turtle" which has to be told when to move forward and when to turn. The "turtle" (which might be a robot buggy with a pen, or might just be a mark on the computer screen) can leave a line behind it showing where it has been.
A line of longitude is a half of a great circle on the surface of the earth, joining the north and south poles. The longitude of a point on the earth is the number of degrees east or west of Greenwich of a line of longitude passing through the point. See also latitude, meridian, coordinate.
A number that is less than or equal to every member of a given set.
The smallest number that will divide into two given numbers exactly.
The smallest number that two given numbers will divide into exactly.
A square array of numbers in which every row, every column, and both diagonals all add up to the same total.
The length of a vector, when represented as a directed line segment.
The larger of two quantities. The major axis of an ellipse is the longest of the two axes. A major arc of a circle consists of more than half of the circle.
Another name for a function. A function maps members of its domain onto members of its range. The word " mapping " is usually applied when a diagram is used to show the connection between the two sets, with arrows connecting members of the domain to members of the range.
A rectangular array of numbers. Matrices is the plural. Matrices with the same number of rows and columns are said to be of the same order. They are added by adding corresponding elements. A matrix with m rows and n columns is called an m × n matrix. If m = n, we say that it is a square matrix.
A maximum is a point on a graph where a function has a higher value than for neighbouring points. The plural is maxima.
See arithmetic mean, geometric mean.
A measure of dispersion. See advanced dictionary.
To find the size of something by comparing it to a standard. There are different measures for different things - area, weight, length and so on. Some things are measured by counting. See imperial and Metric. See also measure of dispersion.
An indicator of how much spread there is in the values of a random variable. Just as there are several types of average, so also dispersion can be measured in several ways. There is the variance, the standard deviation, the semi-interquartile range, and the range.
A prefix indicating one million.
An item that belongs to a set. Also called an element.
The measuring of geometrical quantities such as lengths and areas.
A great circle on the earth's surface which passes through the north and south poles. A line of longitude is one half of a meridian.
A unit of length or distance in the metric system.
A standard system of measuring. The units used in the metric standard are: metres for length, grammes or tonnes for weight, litres for volume and square metres or hectares for area. Larger or smaller units can be made by adding prefixes. To multiply by a thousand, the prefix is kilo- : 1 kilometre = 1000 metres. To divide by a hundred, the prefix is centi- : 1 centilitre = 1/100 litres. See also mega-, deca-, deci-, milli-, micro-.
A prefix meaning one millionth of. See metric.
A unit of length or distance in the imperial system. 1 mile = 1760 yards.
a prefix meaning a thousandth. See metric.
A thousandth of a metre.
A thousand thousands.
A minimum is a point on a graph where a function has a lower value than for neighbouring points.
The smaller of two quantities. The minor axis of an ellipse is the shorter of the two axes. A minor arc of a circle consists of less than half of the circle.
The sum of a whole number and a proper fraction. Any rational number can be expressed as a mixed number.
The value that occurs most often in a set of numbers. Used as an average when we want to cater for the most common or most popular value.
Arithmetic in which all numbers are replaced by their remainder on dividing by n, so that the largest number that can appear is n-1. For example, in modulo 9, 3 + 7 = 1.
Another name for absolute value.
A number or expression is a multiple of another number or expression if the first can be divided by the second. It is an integer multiple, or exact multiple, if the result of the division is an integer, with no remainder.
Multiplication is one of the basic operations of arithmetic. It involves taking a number of copies of an amount, and seeing what new amount this gives us. The sign for multiplication is ×, or sometimes *. For whole numbers, multiplication is repeated addition. To multiply is to perform a multiplication.
Two or more events are mutually exclusive The combined probability of mutually exclusive events is the sum of the individual probabilities.
A number used for counting or ordering a whole number of items. A positive integer, or zero.
A distance of 6080 feet, significantly longer than a mile on land. See knot.
A negative number is a directed number that is less than zero. Integers and real numbers can be negative, but not natural numbers. Note that zero is neither positive nor negative, though sometimes it is written as ±0 (plus or minus zero ) and can be thought of as either.
A surface which can be folded to form a solid.
A system of points in a plane joined by arcs. The arcs divide the plane into regions. Euler's formula for a network is nodes + regions = arcs + 2.
An iterative method of finding the roots of an equation, developed by Isaac Newton.
A point in a network, connected to other nodes by arcs.
A nine-sided figure.
A continuous distribution of a random variable which is found very often in nature. The graph of the function is a bell-shaped curve.
Symbols representing quantities and operations, which are used according to given rules. For example, 4! is the notation for the factorial of 4, that is 4 × 3 × 2 × 1.
A hypothesis that the difference between observed and expected results is due to chance. See significance.
A line with points on the line, or distances along the line, representing numbers. See directed numbers, and ordering.
A single symbol used to represent a number. For example:
The top part of a fraction.
A form designed to simplify the collection of data.
An obtuse angle is one which is larger than a right angle, but less than a straight angle.
An eight sided polygon.
An eight-faced polyhedron. A regular octahedron has eight equilateral triangles as faces.
A whole number that cannot be divided exactly by 2. See also even number.
Another name for a cumulative frequency curve.
A set of values can be ordered by placing them on a number line so that, for any a and b in the set, if a is to the right of b then a > b ("a is greater than b") is true, and so is b inequality.
A pair of items (a,b) where the order is significant, so that (a,b) is not the same as (b,a).
A number used to order the members of a set. For example, "fourth" (4th) and "fifth" (5th) are ordinal numbers. Compare with cardinal number.
The y coordinate of a pair of Cartesian coordinates. See also abscissa.
The point where the axes meet in a cartesian coordinate system. It has coordinates (0,0).
One of the centres of a triangle. The point where the altitudes meet.
An imperial unit of weight. There are sixteen ounces in a pound.
The result of a statistical experiment, or trial. Each experiment or trial results in exactly one outcome, though this may be describable in terms of several events.
The locus of points in a plane that are equidistant from a give point and a given line. The line is called the directrix, and the point is called the focus. A parabola is a conic. The path of a projectile is approximately parabolic.
Having the form of a parabola.
Two lines or line segments are parallel if they have the same direction. In Euclidian geometry, two distinct parallel lines do not meet anywhere.
A polyhedron whose faces are all parallelograms.
A quadrilateral in which opposite pairs of sides are parallel to one another.
A quantity which, when varied, affects the value of another.
A pattern of numbers in which each number is the first number is 1, and the other numbers are each obtained by adding the numbers that are immediately above and to the left or right. It is useful because the numbers in each row are the coefficients of a binomial expansion.
A unit of currency (money). The plural is pennies or pence. There are 100 pennies in one pound. Often shortened to P, so 50p is 50 pence.
A five-sided figure.
A fraction of some total quantity, given as a number of hundredths. The symbol for percentage is %, so that, for instance, 50% of 12 is 50 hundredths of 12, = 1/2 of 12, = 6.
The nth percentile of a set of data is a value that divides the set into two parts, so that n% of the data is in the first part. If the set does not divide exactly, then a weighted average of the values immediately above and below the percentile is used. A percentile can be found graphically using a cumulative frequency graph.
The length of the path around the edge of a figure.
A function f(x) is periodic, with period p, if f(x)=f(x+p) for all x. This means that the graph of the function forms a pattern which is repeated every p units along the x axis.
The possible selections of items from a set, when the order is significant. For instance, when selecting three diferent letters from the alphabet, ABC, CBA, ACB etc. represent different permutations. See also combination.
A line or plane is perpendicular to another line or plane if they meet at right angles. A line which is perpendicular to a plane is said to be normal to it.
The ratio of the circumference of a circle to its diameter
A graph in which pictures are used.
A chart in which a disc is divided into sectors, and the area of each sector is proportional to some value to be depicted, so that the area of the circle represents the total of all the values depicted.
An imperial unit of volume. There are eight pints in a gallon, and twenty fluid ounces in a pint.
The view of an object when looking at it from above. A plan is often drawn together with an elevation.
A flat surface. A figure which can be drawn on a plane, like a triangle or a square for instance, is called a plane figure. For a plane which is a part of a three-dimensional space with cartesian coordinates, the equation is ax + by + cz + d = 0
A single location in a space or on a surface or line. The position of a point can be given using coordinates.
If the position of a point is given using a distance and an angle, then these are called polar coordinates. These are the distance of the given point from the origin, or pole, and the angle between the line from the pole to the given point and some fixed line through the pole.
A closed plane figure bounded by straight edges. Examples are: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon. Select -pic- for a picture. See also regular.
A polyhedron, plural polyhedra, is a 3-dimensional shape bounded by plane faces. Examples are: pyramid, cube, cuboid, prism, tetrahedron, octahedron, dodecahedron, icosahedron. Select -pic- for a picture. See also solid, dual, regular.
In statistics, the set of items from which a sample is taken.
A system of writing numbers where the value of a digit depends on its position in a list of digits. See base, decimal.
Having a value greater than zero. See negative.
The number of times a value is to be multiplied by itself. When we square a number, the power is 2. when we cube it, the power is 3. For example, 5 raised to the power 3 (or 5 cubed, written as 5³) is 5x5x5, = 125. It is possible to have negative or fractional powers, but that is outside the scope of this dictionary.
In the set of natural numbers, a number that is only divisible by one and by itself.
A polyhedron having the same (polygonal) cross section throughout its length. The volume of a prism is the area of the cross section, multiplied by the length.
The indication of how likely an event is. It is measured on a scale from 0 to 1. Zero signifies that an event is impossible, and one signifies that the event is certain. Often we have to estimate a probability, for instance, the probability that a football team will win a particular match depends on many things that we cannot measure. See also combined probability, dice, equally likely, fair, random, trial
The result of a multiplication.
See arithmetic progression, geometric progression.
An object that is thrown, or projected. Near the earth the trajectory of a projectile is roughly a parabola. If v is the initial velocity, a the angle of projection and g gravity
A transformation which reduces the number of dimension s; for instance, any 3-dimensional shape can be projected onto a plane in a variety of ways.
The logical argument used to establish that a statement is true. Proof by contradiction is a method which involves assuming that the statement is not true, and showing that this leads to a contradiction. Proofs generally consist of several steps, and in each step deductions are made from axioms or from theorems that have already been proved.
A fraction in which the numerator is smaller than the denominator. See improper fraction, mixed number.
A subset x of set y is proper if it does not consist of the whole of y.
Varying in a constant ratio to another quantity.
A device used for measuring angles, usually consisting of a semicircular piece of plastic marked in degrees.
A polyhedron with a base consisting of a polygon, and triangles with a common vertex for the other faces. For example, a square pyramid has a square base and four triangular faces.
A mathematician of Samos, famous for his theorem. He believed that numbers had mystical significance. See Pythagoras' theorem.
The theorem that in a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.
One of the four parts into which a plane is divided by rectangular coordinate axes. In the first quadrant, the x and y values are both positive. In the second, x is negative and y is positive. In the third, x and y are both negative. In the fourth, x is positive and y is negative.
A polynomial in which the highest power of x is 3.
A polynomial equation of the second degree. If it has one variable, and if the variable is called x, say, then "of the second degree " signifies that the highest power of x is 2.
A four sided polygon. See square, rectangle, parallelogram, rhombus, kite and trapezium, all of which are particular kinds of quadrilateral.
The result of dividing a value by four. One part of a shape divided into four equal parts.
A polynomial in which the highest power is a fourth power.
The 25th and 75th percentiles of a sample are sometimes called the first and third quartiles. See also semi-interquartile range.
The result of a division.
See advanced dictionary.
Any straight line from the centre of a circle to its edge. Also, the length of this line. The plural is radii.
A vector from the origin of a coordinate system to a given point.
A random experiment or trial is one that has an outcome which cannot be known beforehand.
The set of values that are mapped to by a function.
One number divided by another. For example, the ratio of 6 to 2 is 3, because 6 divided by 2 is 3. PI is one important ratio.
A number is rational if it can be expressed as a fraction with an integer numerator amd denominator. There are many ways of writing any given fraction. See mixed number.
To simplify an expression without changing its value, by eliminating sub-expressions involving square or cube roots etc.
Those points of a line that lie in one direction from a point O of that line. Also called a half-line.
A polygon that bounds a concave region. One of the angles of a re-entrant polygon must be more than 180°.
The number 1 divided by a quantity. For instance, the reciprocal of 3 is 1/3.
A parallelogram with four equal angles. These must each be 90°.
A rectangle number is a whole number that has 2 or more factors.
Motion along a straight line.
A recurring decimal contains a group of digits that is repeated infinitely. For example, 1/7 =.142857142857142857..... The digit sequence 142857 recurs.
A transformation that flips a shape
An angle which is larger than 180° but smaller than 90°.
A relation R is reflexive, if aRa for all a.
An area of a plane bounded by arcs of a network.
Having angles and sides or faces that are all equal. A regular polygon has all its sides and interior angles equal. A regular triangle is also called equilateral, and a regular quadrilateral is a square. In a regular polyhedron, all the faces are equal and congruent regular polygons. There are five regular poyhedra, the tetrahedron, cube, octahedron, dodecahedron and icosahedron.
A relation, in mathematics, is a connection between pairs of members of a set, or members of one set and members of another. It can be thought of as a set of ordered pairs, and given a relation R we say that a is related to b if (a,b) is in this set of pairs. We write this as aRb. A function is a relation where each a is related to exactly one b. See also equivalence relation.
What is left when one number is divided by another.
A root of an equation which appears more than once.
A residue class is the set of all numbers which have the same remainder when divided by a given number (the modulo).
To find the component of a vector in a particular direction.
To rotate about an axis or point.
A parallelogram with all four sides equal. Opposite angles are equal in a rhombus, and there are two lines of symmetry, each passing through pairs of opposite vertices.
An angle of 90°, or one quarter of a full turn.
A system used by the romans for writing natural numbers.
A geometric transformation where every point turns through the same angle. In a plane, rotation is about a point, called the centre of rotation, and is clockwise or anticlockwise. In a 3-dimensional space it is about a line, called the axis of rotation. See also solid of revolution.
A way of approximating a number, by setting non-significant digits to zero and rounding the last significant digit either up or down, depending on the value of the first insignificant digit. If this is 0 to 4 we round down (we leave the last significant digit unchanged),otherwise we round up (we add 1 in the position of the last significant digit ).
When we round off a number to a certain number of digits we create a rounding error. This is the difference between the original number and the rounded number. Even small rounding errors can have a large effect, especially when one large rounded number is subtracted from another number of similar magnitude. Some calculators are prone to rounding errors.
A straight-edged instrument marked with a scale and used for measuring distances and drawing straight line segments.
A finite selection from a population (which may be infinite ).
To fulfil the condition of.
A quantity that has a size, but no direction (as opposed to a vector ). For example, velocity is a vector but speed is a scalar.
A set of marks on a ruler used to measure distances, or to perform calculations in a slide rule.
A measure of the size change in an enlargement.
In a scalene triangle, no two sides or angles are equal.
A diagram showing one point for each observation of a pair of related variables.
A trigonometric function. The reciprocal of the cosine.
See cross-section.
A part of a disc enclosed between 2 radii. See also pie chart.
An element of a set is self-inverse under some operation if it combines with itself under that operation to produce the identity element.
One half of a circle.
One half of the difference between the upper and lower quartile of a distribution or sample. Used as a measure of dispersion.
An ordered set of quantities. In an infinite sequence, one element of the sequence is associated with each natural number. A sequence can be defined by a formula, or inductively.
The sum of a finite or infinite number of terms, a1+a2+a3...
A collection of distinct objects, called elements. See also: empty set, subset, universal set, Venn diagram.
A number system with base 60. For instance, hours, minutes and seconds are a sexagesimal system for measuring time. There are 60 minutes in one hour, There are 60 seconds in one minute.
In 2 dimension s: A transformation in which one line remains fixed while other points move parallel to the line by an amount proportional to their directed distance from the line. In 3 dimension s: As for 2 dimensions, but with a plane replacing the line.
One of the line segments forming the borders of a polygon.
A way of finding prime numbers by writing down a set of natural numbers (from 1 to the largest number to be checked), and crossing out all the multiples of 2, then all the multiples of 3, then 5, and so on, until all the multiples of prime numbers are crossed out. The numbers that are left uncrossed are prime.
A letter of the Greek alphabet, often used in upper case to represent a summation, and in lower case to represent a standard deviation.
Two or more figures are similar if they have the same shape, but not necessarily the same size.
A curve that divides a plane into two parts, an inside and an outside. If it is drawn with a pencil, the ending point must be the same as the starting point, and the curve must be continuous and not cross itself.
The motion of an object along a line, when the acceleration of the object is towards a point on the line and proportional to the distance from the point. The motion is periodic and has the form of a sine wave.
To reduce the complexity of an expression using algebraic rules.
A method for finding the approximate area under a curve.
A set of equations is called simultaneous if we need to find those solutions which satisfy all the equations simultaneously. Solutions can be found by algebraic manipulation, graphically, or by the use of matrices.
A trigonometric function. In a right angled triangle, the sine of an angle is the length of the opposite side divided by the length of the hypotenuse.
A matrix that has a determinant of zero. A singular matrix has no inverse.
see skew lines, skew distribution.
A distribution which is not symmetrical about the mean.
In three dimensions, lines that are not parallel and do not intersect.
A ruler used for performing calculations, which consists of three parts, the rule, the slide, and the cursor or runner. The slide and the rule are marked with logarithmic scales, and the cursor is transparent with a line marked on it to aid in reading off values. It has largely been replaced by the electronic calculator.
In geometry, a solid is the set of all points of three-dimensional space that lie within a complete closed part of the space, that is, within the bounding surfaces of the solid. It is not " solid " in the more normal sense, not having any hardness, so it is not unusual for two solids to have an intersection.
A solid bounded by a curve which is rotated about an axis of revolution.
The rate of change of displacement with time.
A solid bounded by a spherical surface. See area and volume for useful formulae.
A set consisting of all the points in 3-dimensional space that have a given distance, called the radius, from a given fixed point, called the centre. The equation of a spherical surface with radius r and centre at (a,b,c) is (x-a)² + (y-b)² + (z-c)² - r² = 0.
A curve whose radius vector r is some function of the angle turned through by r. Spirals are most easily worked with using polar coordinates.
The dispersion of data from some measure of their average. See measure of dispersion.
If X is the square of Y and if Y is positive, then Y is the square root of X. See also root.
A way of working or a measure that is generally agreed. Standards are needed so that different results can be compared sensibly. A standard should be the same for everybody, so handspans are not used as a standard because different people have differently sized hands. See metric and imperial. Days, hours, minutes and seconds are used for measuring time.
A measure of dispersion. The square root of the variance.
Also known as scientific notation; the writing of a number as the product of a number between 1 and 10 (not including 10) and a power of 10. For example, 372.4 is 3.724 × 10². Useful for representing very large and very small numbers.
A point on a graph of a function where the derivative of the function is zero. The tangent to the graph is therefore horizontal. Any stationary point is a maximum, a minimum or a point of inflection.
A statistic is a property of data that is to be measured. Statistics is the study of ways of measuring and analysing large quantities of data by conducting experiments and deducing properties of a population from properties of a sample.
See advanced dictionary.
if all the elements of a set A are also members of another set B, then A is a subset of B. It is a proper subset if it does not contain all the members of B. The empty set is a subset of all sets. All sets are subsets of the universal set.
The replacement of one expression by another in order to simplify.
An arc b subtends an angle ß at a point p if lines from p to the end points of b form an angle ß at p.
Taking one amount away from another, to see what is left. Subtraction is the inverse of addition, with sign "-".
A statements is a sufficient condition for a statement T if, whens is true it follows that T must be true. We say thats implies T.
The result of an addition.
Two angles are supplementary if they add up to 180°.
An expression involving roots of numbers. It is often possible to simplify the expression so that it no longer contains any roots.
A set of points whose coordinates satisfy an equation of the form F(x,y,x)=0. Examples are spherical surfaces, planes, and surfaces of revolution. Surfaces form the boundaries of a solid.
A surface formed by rotating a curve around an axis. The equation for a surface of revolution about the z axis is x² + y²= f(z)²
A sign with a standard meaning in mathematics, such as =, % and so on. In algebra, a letter that stands for a number that is unknown.
There are two types of symmetry, mirror symmetry and rotational symmetry. A figure has mirror symmetry if it is symmetric under some reflection. It has rotational symmetry if it is symmetric under some rotation. Select -pic- for a picture.
A table is a list of numbers which can be used to look up the value of a function. A multiplication table shows the result of multiplying different pairs of numbers. A book of tables might contain tables for logarithms, antilogarithms, sines, cosines, tangents and so on. A table of random numbers is useful for simulating a random experiment.
A regular pattern of tiles covering a surface.
A polyhedron with four (necessarily triangular ) faces. A pyramid with a triangular base. In a regular tetrahedron the faces are equilateral triangles.
A significant general conclusion obtained by deduction from certain axioms. For example: the binomial theorem, factor theorem, Pythagoras' theorem. See also proof.
A metric measure of mass, equivalent to 1000 kilograms.
Study of shapes and surfaces without regard to their shape or area. The branch of mathematics dealing with the properties of a shape that are unchanged by a continuous deformation, such as "connectedness". Topology is closely connected with network theory.
A doughnut shaped surface of revolution produced by rotating a circle around an axis.
The path followed by a moving point or body.
A transcendental number is an irrational number that cannot be obtained as the solution to a polynomial equation with rational coefficients. Examples are e and pi.
A mapping of points in a space to other points, which involves changing the shapes or positions of objects in the space; or The mapping of one expression to another, often by substitution.
A relation R between members of a set is transitive if for any 3 elements x, y and z, if xRy and yRz then xRz.
Movement in a straight line. A translation is a geometric transformation where every point is moved by the same distance and in the same direction.
The transpose of a matrix is the matrix obtained by interchanging all the rows and columns of the given matrix.
A transversal is a line that intersects two or more other lines. A transversal of a triangle is any line that intersects the triangle.
A quadrilateral with one pair of parallel sides.
A diagram in which possible events are shown by points joined by lines.
A repeatable action that has a random outcome.
The technique of solving a problem approximately by starting with an estimate and repeatedly applying a procedure that gives successively better approximations, until one is found that is sufficiently close to the correct answer. See iterate.
A triangle is a figure that is bounded by three line segments, the sides, and has three angles. The branch of mathematics that deals with measurement of triangles is called trigonometry. A triangle with the lengths of the sides all different is called scalene. A triangle with two equal sides is called isosceles. A regular triangle (one with three equal side s) is called equilateral. An equilateral triangle must also have three equal angles. The internal angles of a triangle add up to 180 degrees, or pi radians. Centre of a triangle : see centre. See also trigonometric function.
A function which is used to work out unknown sides or angles in a triangle. The trigonometric functions are sine, cosine, tangent and their reciprocals cosecant, secant and cotangent. These all give the ratio of two sides of a right-angled triangle given one of the angles. The inverses of these functions are called arcsine, arccosine, and so on.
The branch of mathematics dealing with triangles. See also trigonometric function.
A truncated solid is one created by removing from the original solid everything that lies on one side of a plane that intersects the solid.
A function f(x) is unbounded over an interval if there is no value n for which |f(x)| < n for all x in the interval.
An inequality which is always true, for instance x > x-1, which is true for all x.
A die or coin is unfair of the outcomes are supposed to be equally likely but aren't.
The same in all places. For example, a prism has a uniform cross section. This means that a cross section from one end, one from the middle and one from the other end are all the same. An object with a uniform velocity doesn't speed up or slow down.
The union of two sets A and B contains all the elements of set A, and all the elements of set B, but no other elements.
A vector of length 1. When vectors are used with cartesian coordinates, the letters i,j and k are used to refer to unit vectors in the x,y and z directions, so, when the dimension is 3: i = (1,0,0) j = (0,1,0) k = (0,0,1) and when the dimension is 2: i = (1,0) j = (0,1).
A set containing all the elements of all the sets being considered. All sets are subsets of the universal set.
An upper bound of a set of numbers is a number which is larger than any number in the set. The smallest such number is called the least, or lowest upper bound.
The value of an independent variable can be any number that is the right type and in the allowed range. The value of an expression is the amount or quantity represented by the expression, given values for the variables in it. The value of a function is the member of the domain associated with a given member of the range. See also Absolute value.
A quantity that can take on a range of values. A dependent variable is one whose value depends on the value of another variable.
A measure of dispersion. For a given collection of data, if we take the square of the deviation from the mean for each member, and average it (that is, take the mean of it), then we obtain the variance. So, the variance = the mean of the squares of the deviations from the mean of the data. The standard deviation is the square root of the variance.
A measure of speed in a particular direction. It is the rate of change of position (or displacement ) with time. Velocity is a vector quantity. The rate of change of velocity with time is called acceleration.
A diagram in which sets are represented as regions enclosed by simple closed curves, so that the intersection of two sets is represented as the intersection of two regions.
A point at which edges meet in a polygon or polyhedron. The plural is vertices.
The space enclosed by a solid. For some solids there is a formula that can be used to work out the volume. Solid volume sphere 4/3 × pi × the cube of the radius. pyramid 1/3 × area of the base × the height. prism the area of a cross section × the height. cuboid (box) length × width × height. circular cylinder pi × (the square of the radius ) × the height. circular cone 1/3 × pi × (the square of the radius ) × the height.
A simple fraction with integer numerator and denominator.
A number which has no fractional part. Another name for an integer.
The middle section of a sphere cut by two parallel planes.
© 1997 M.T.Gradwell, Ambition Data Services Ltd.
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