Oxford concise dictionary of mathematics(1)

acceleration Suppose that a particle is moving in a straight line, with a point O on the line taken as origin and one direction taken as positive.. If the velocity is positive that is, i

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The Concise Oxford Dictionary of Mathematics

Some entries in this dictionary have recommended web links.When you see the symbol at the end of an entry go to thedictionary’s web page at http://www.oup.com/uk/reference/resources/mathematics, click on Web links in the Resourcessection and click straight through to the relevant websites

Christopher Clapham was until 1993 Senior Lecturer in

Mathematics at the University of Aberdeen and has alsotaught at universities in Nigeria, Lesotho, and Malawi He is

the author of Introduction to Abstract Algebra and

Introduction to Mathematical Analysis He lives in Exeter.

James Nicholson has a mathematics degree from Cambridge,

and taught at Harrow School for twelve years beforebecoming Head of Mathematics at Belfast Royal Academy in

1990 He lives in Belfast, but now works mostly with theSchool of Education at Durham University He is co-author of

Statistics GCSE for AQA.

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The Concise Oxford Dictionary of Mathematics

FOURTH EDITION

CHRISTOPHER CLAPHAM

JAMES NICHOLSON

Great Clarendon Street, OxfordOX2 6DP

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© Christopher Clapham 1990, 1996

© Christopher Clapham and James Nicholson 2005, 2009First edition 1990

Second edition 1996

Third edition 2005

Fourth edition 2009

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of Oxford University Press, or as expressly permitted by law,

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ISBN 978–0–19–923594–0

1 3 5 7 9 10 8 6 4 2

Preface

Dictionary

APPENDICES

1 Table of areas and volumes

2 Table of centres of mass

3 Table of moments of inertia

8 Table of convergence tests for series

9 Table of common inequalities

10 Table of trigonometric formulae

11 Table of symbols

12 Table of Greek letters

13 Table of Roman numerals

14 Table of Fields Medal Winners

Preface to Second Edition

*

This dictionary is intended to be a reference book that givesreliable definitions or clear and precise explanations ofmathematical terms The level is such that it will suit, amongothers, sixth-form pupils, college students and first-yearuniversity students who are taking mathematics as one oftheir courses Such students will be able to look up any termthey may meet and be led on to other entries by following upcross-references or by browsing more generally

The concepts and terminology of all those topics that feature

in pure and applied mathematics and statistics courses at thislevel today are covered There are also entries onmathematicians of the past and important mathematics ofmore general interest Computing is not included Thereader’s attention is drawn to the appendices which giveuseful tables for ready reference

Some entries give a straight definition in an opening phrase.Others give the definition in the form of a complete sentence,sometimes following an explanation of the context Anasterisk is used to indicate words with their own entry, towhich cross-reference can be made if required

This edition is more than half as large again as the firstedition A significant change has been the inclusion of entriescovering applied mathematics and statistics In these areas, I

am very much indebted to the contributors, whose names aregiven on page v I am most grateful to these colleagues for

their specialist advice and drafting work They are not,however, to be held responsible for the final form of theentries on their subjects There has also been a considerableincrease in the number of short biographies, so that all themajor names are included Other additional entries havegreatly increased the comprehensiveness of the dictionary.The text has benefited from the comments of colleagues whohave read different parts of it Even though the names of all ofthem will not be given, I should like to acknowledge heretheir help and express my thanks.

Christopher Clapham

Preface to Third Edition

Since the second edition was published the content andemphasis of applied mathematics and statistics at sixth-form,college and first-year university levels has changedconsiderably This edition includes many more appliedstatistics entries as well as dealing comprehensively with thenew decision and discrete mathematics courses, and a largenumber of new biographies on 20th-century mathematicians I

am grateful to the Headmaster and Governors of BelfastRoyal Academy for their support and encouragement to take

on this task, and to Louise, Joanne, and Laura for transcribing

my notes

James Nicholson

Preface to Fourth Edition

Since the third edition was published there has been adramatic increase in both access to the internet and theamount of information available The major change to thisedition is the introduction of a substantial number ofweblinks, many of which contain dynamic or interactiveillustrations related to the definition

James Nicholson

a- Prefix meaning ‘not’ For example, an asymmetric figure is

one which possesses no symmetry, which is not symmetrical

A The number 10 in hexadecimal notation.

abacus A counting device consisting of rods on which beads

can be moved so as to represent numbers

• A description of how one abacus works

abelian group Suppose that G is a *group with the operation

Then G is abelian if the operation is commutative; that is,

if, for all elements a and b in G, a b = b a.

Abel, Niels Henrik (1802–29) Norwegian mathematician

who, at the age of 19, proved that the general equation ofdegree greater than 4 cannot be solved algebraically In otherwords, there can be no formula for the roots of such anequation similar to the familiar formula for a quadraticequation He was also responsible for fundamentaldevelopments in the theory of algebraic functions He died insome poverty at the age of 26, just a few days before hewould have received a letter announcing his appointment to aprofessorship in Berlin

Abel’s test A test for the convergence of an infinite series

which states that if ∑a n is a convergent sequence, and {b n} is

monotically decreasing, i.e b n+1 ≤ b n for all n, then ∑a n b n isalso convergent

above Greater than The limit of a function at a from above is

the limit of f(x) as x → a for values of x > a It is of particular importance when f(x) has a discontinuity at a, i.e where the

limits from above and from below do not coincide It can be

written as f(a+) or

abscissa The x-coordinate in a Cartesian coordinate system in

the plane

absolute address In spreadsheets a formula which is to

appear in a number of cells may wish to use the contents ofanother cell or cells Since the relative position of those cellswill be different each time the formula

appears in a new location, the spreadsheet syntax allows anabsolute address to be specified, identifying the actual rowand column for each cell When a formula is copied andpasted to another cell, a cell reference using an absoluteaddress will remain unchanged A formula can contain amixture of absolute and *relative addresses

absolute error SeeERROR.

absolute frequency The number of occurrences of an event.

For example, if a die is rolled 20 times and 4 sixes areobserved the absolute frequency of sixes is 4 and the *relativefrequency is 4/20

absolute measure of dispersion =MEASURE OF DISPERSION.

absolutely convergent series A series {a n} is said to beabsolutely convergent if is *convergent For

example, if then the series is convergent

is absolutely convergent

absolutely summable =ABSOLUTELY CONVERGENT.

absolute value For any real number a, the absolute value

(also called the *modulus) of a, denoted by |a|, is a itself if a

≥ 0, and −a if a ≤ 0 Thus |a| is positive except when a = 0.

The following properties hold:

(i) |ab| = |a||b|.

(ii) |a + b| ≤ |a| + |b|.

(iii) |a − b| ≥ ||a| − |b||.

(iv) For a > 0, |x| ≤ a if and only if −a ≤ x ≤ a.

absorbing state SeeRANDOM WALK.

absorption laws For all sets A and B (subsets of some

*universal set), A ∩ (A∪B) = A and A∪(A ∩ B) = A These

are the absorption laws

abstract algebra The area of mathematics concerned with

algebraic structures, such as *groups, *rings and *fields,involving sets of elements with particular operationssatisfying certain axioms The purpose is to derive, from theset of axioms, general results that are then applicable toany particular example of the algebraic structure in question.The theory of certain algebraic structures is highly developed;

in particular, the theory of vector spaces is so extensive thatits study, known as *linear algebra, would probably no longer

be classified as abstract algebra

abstraction The process of making a general statement which

summarizes what can be observed in particular instances For

example, we can say that x2 < x for 0 < x < 1 and x2 > x for x

< 0 or x > 1 Mathematical theorems are essentially

abstraction of concepts to a higher level

abundant number An integer that is smaller than the sum of

its positive divisors, not including itself, For example, 12 isdivisible by 1, 2, 3, 4 and 6, and 1 + 2 + 3 + 4 + 6 = 16 > 12

acceleration Suppose that a particle is moving in a straight

line, with a point O on the line taken as origin and one direction taken as positive Let x be the *displacement of the particle at time t The acceleration of the particle is equal to ẍ

or d2x/dt2, the *rate of change of the *velocity with respect to

t If the velocity is positive (that is, if the particle is moving in

the positive direction), the acceleration is positive when theparticle is speeding up and negative when it is slowing down

However, if the velocity is negative, a positive accelerationmeans that the particle is slowing down and a negativeacceleration means that it is speeding up.

In the preceding paragraph, a common convention has been

followed, in which the unit vector i in the positive direction

along the line has been suppressed Acceleration is in fact avector quantity, and in the one-dimensional case above it is

equal to ẍi.

When the motion is in two or three dimensions, vectors are

used explicitly The acceleration a of a particle is a vector

equal to the rate of change of the velocity v with respect to t Thus a = dv/dt If the particle has *position vector r, then

When Cartesian coordinates are used, r = xi

If a particle is travelling in a circle with constant speed, it stillhas an acceleration because of the changing direction of thevelocity This acceleration is towards the centre of the circleand has magnitude where v is the speed of the particle and

r is the radius of the circle.

Acceleration has the dimensions LT−2, and the SI unit ofmeasurement is the metre per second per second, abbreviated

to ‘ms−2

acceleration–time graph A graph that shows acceleration

plotted against time for a particle moving in a straight line

Let v(t) and a(t) be the velocity and acceleration, respectively,

of the particle at time t The acceleration–time graph is the graph y = a(t), where the t-axis is horizontal and the y-axis is

vertical with the positive direction upwards With theconvention that any area below the horizontal axis is negative,the area

under the graph between t = t1 and t = t2 is equal to v(t2) −

v(t1) (Here a common convention has been followed, in

which the unit vector i in the positive direction along the line

has been suppressed The velocity and acceleration of the

particle are in fact vector quantities equal to v(t)i and a(t)i,

respectively.)

acceptance region SeeHYPOTHESIS TESTING.

acceptance sampling A method of quality control where a

sample is taken from a batch and a decision whether to acceptthe batch is made on the basis of the quality of the sample.The most simple method is to have a straight accept/rejectcriterion, but a more sophisticated approach is to take anothersample if the evidence from the existing sample, or a set ofsamples, is not clearly indicating whether the batch should beaccepted or rejected One of the main advantages of thisapproach is reducing the cost of taking samples to satisfyquality control criteria

accuracy A measure of the precision of a numerical quantity,

usually given to n *significant figures (where the proportional accuracy is the important aspect) or n *decimal places (where

the absolute accuracy is more important)

accurate (correct) to n decimal places Rounding a number

with the accuracy specified by the number of *decimal places

given in the rounded value So e = 2.71828 … = 2.718 to

three decimal places and = 2.72 to two decimal places

is 9.30 correct to two decimal places.Where a number of quantities are being measured and added

or subtracted, using values correct to the same number ofdecimal places ensures that they have the same degree ofaccuracy However, if the units are changed, for examplebetween centimetres and metres, then the accuracy of themeasurements will be different if the same number of decimalplaces is used in the measurements

accurate (correct) to n significant figures Rounding a

number with the accuracy specified by the number of

*significant figures given in the rounded value So e =

2.71828 … = 2.718 to four significant figures and = 2.72 to

three significant figures e−3 = 0.049787 … = 0.0498 correct

to three significant figures Rounding to the same number ofsignificant figures ensures all the measurements have aboutthe same proportionate accuracy If the units are changed, forexample between centimetres and metres, then the accuracy

of the measurements will not be changed if the same number

of significant figures is used in the measurements

Achilles paradox The paradox which arises from considering

how overtaking takes place Achilles gives a tortoise a headstart in a race To

overtake, he must reach the tortoise’s initial position, then

where the tortoise had moved to, and so on *ad infinitum The

conclusion that he cannot overtake because he has to cover aninfinite sum of well-defined non-zero distances is false, hencethe paradox

acre An imperial unit of surface area, which is 4840 square

yards This is the area of a furlong (220 yards) by a chain (22

yards) which used to be standard units of measurement in the

UK A square mile contains 640 acres In the metric system a

*hectare is approximately 0.4 acre

action limits The outer limits set on a *control chart in a

production process If the observed value falls outside theselimits, then action will be taken, often resetting the machine

For the means of samples of sizes n in a process with standard

deviation σ with target mean μ, the action limits will be set at

active constraint An inequality such as y + 2x ≥ 13 is said to

be active at a point on the boundary, i.e where equality holds,for example (6, 1) and (0, 13)

activity networks (edges as activities) The networks used in

*critical path analysis where the edges (arcs) representactivities to be performed Paths in the network represent theprecedence relations between the activities, and *dummyactivities are required to link paths where common activitiesappear, but the paths are at least partly independent Whilethis is a complication, each activity appears on only one edgeand the sequence of activities needed is easier to follow thanwhen the vertices are used to represent the activities Once theactivity network has been constructed from the precedencetable, the *critical path algorithm (edges as activities) can beapplied

activity networks (vertices as activities) The networks used

in *critical path analysis where the vertices (nodes) representactivities to be performed The edges (arcs) coming out from

any vertex X join X to any vertex Y whose activity cannot start until X has been completed, and the edge is labelled with the time taken for activity X Note that there will often be more

than one such activity, in which case each edge will carry thesame time The construction of an activity network requires alisting of the activities, their duration, and the precedencerelations which identify which activities are dependent on theprior completion of other activities With this structure,

*dummy activities are not needed, but the same activity will

be represented by more than one edge when more than oneother activity depends on its prior completion, and thesequence of

activities is less easy to follow than the alternative structure

*activity networks (edges as activities)

acute angle An angle that is less than a *right angle An

*acute-angled triangle is one all of whose angles are acute

adders (in combinatorial circuits) The half-adder and full

adders are sections of circuits which use a system of logicgates to add binary digits, by using a combination for whichthe truth table output is identical to the output required by thebinary addition

• An article demonstrating how a simple adder works

addition (of complex numbers) Let the complex numbers z1

and z2, where z1 = a + bi and z2 = c + di, be represented by the points P1and P2in the *complex plane Then z1 + z2= (a + c) + (b + d)i, and z1+ z2is represented in the complex plane

by the point Q such that OP1QP2 is a parallelogram; that is,such that Thus, if the complex number

z is associated with the *directed line-segment , where P represents z, then the addition of complex numbers

corresponds exactly to the addition of the directedline-segments

addition (of directed line-segments) See ADDITION (ofvectors)

addition (of matrices) Let A and B be m × n matrices, with A

= [a ij ] and B = [b ij] The operation of addition is defined bytaking the SUM A + B to be the m × n matrix C, where C =

[c ij ] and c ij = a ij + b ij The sum A + B is not defined if A and

B are not of the same order This operation + of addition on

the set of all m × n matrices is *associative and

*commutative

addition (of vectors) Given vectors a and b, let and

be *directed line-segments that represent a and b, with the

same initial point O The sum of and is the directed

line-segment , where OACB is a parallelogram, and the

SUM a + b is defined to be the vector c represented by This is called the parallelogram law Alternatively, the sum of

vectors a and b can be defined by representing a by a directed

line-segment and b by where the final point of thefirst

directed line-segment is the initial point of the second Then a + b is the vector represented by This is called thetriangle law Addition of vectors has the following properties,

which hold for all a, b and c:

(i) a + b = b + a, the commutative law.

(ii) a + (b + c) = (a + b) + c, the associative law.

(iii) a + 0 = 0 + a = a, where 0 is the zero vector.

(iv) a + (−a) = (−a) + a = 0, where −a is the negative of a.

addition formula SeeCOMPOUND ANGLE FORMULA.

addition law If A and B are two events then the addition law

states that the Pr(A or B) = Pr(A) + Pr(B) − Pr(A and B), or using set notation P(A∪B) = P(A) + P(B) − P(A ∩ B) In the

special case where A and B are *mutually exclusive events

this reduces to P(A∪B) = P(A) + P(B).

addition modulo n See MODULO N, ADDITION andMULTIPLICATION.

additive function A function for which f(x + y) = f(x) + f(y).

For example, f(x) = 3x is an additive function since f(x + y) = 3(x + y) = f(x) + f(y) but is not additive since

is not in general equal to

additive group A *group with the operation +, called

addition, may be called an additive group The operation in agroup is normally denoted by addition only if it is

*commutative, so an additive group is usually *abelian

additive identity The identity element under an operation of

addition, usually denoted by 0, so a + 0 = 0 + a = a.

additive inverse SeeINVERSE ELEMENT.

adherent point A point of the *closure of a set.

ad infinitum Repeating infinitely many times.

adj Abbreviation for *adjoint.

adjacency matrix For a *simple graph G, with n vertices v1,

v2, …, v n , the adjacency matrix A is the n × n matrix [a ij]

with a ij = 1, if vi is joined to v j , and a ij = 0, otherwise The

matrix A is *symmetric and the diagonal entries are zero The

number of ones in any row (or column) is equal to the

*degree of the corresponding vertex An example of a graph

and its adjacency matrix A is shown below.

adjacent angles A pair of angles on a straight line formed by

a line or half-line meeting it Adjacent angles will add to180°

adjacent edges A pair of edges in a graph joined by a

common vertex

adjacent side The side of a right-angled triangle between the

right angle and the given angle

adjacent vertices A pair of vertices in a *graph joined by a

common edge

adjoint The adjoint of a square matrix A, denoted by adj A, is

the transpose of the matrix of cofactors of A For A = [a ij], let

A ij denote the *cofactor of the entry a ij Then the matrix of

cofactors is the matrix [A ij ] and adj A = [A ij]T For example, a

3 × 3 matrix A and its adjoint can be written

In the 2 × 2 case, a matrix A and its adjoint have the form

The adjoint is important because it can be used to find the

*inverse of a matrix From the properties of cofactors, it can

be shown that A adj A = (det A)I It follows that, when det A

≠ 0, the inverse of A is (1/det A) adj A.

adjugate =ADJOINT.

aerodynamic drag A body moving through the air, such as

an aeroplane flying in the Earth’s atmosphere, experiences aforce due to the flow of air over the surface of the body Theforce is the sum of the aerodynamic drag, which is tangential

to the flight path, and the LIFT, which is normal to the flightpath

affine geometry A geometry in which some properties are

preserved by *parallel projection from one plane to another

However, others are not, and in particular *Euclid’s third andfourth axioms do not hold.

affine transformation A *transformation which preserves

*collinearity and therefore the straightness and parallel nature

of lines, and the ratios of distances

aggregate Census returns are used to construct aggregate

statistics concerning a wide range of characteristics such aseconomic, social, health, education, often grouped bygeographical region, gender, age, etc The name derives fromthe way they are compiled as counts from a large number ofindividual census returns Other examples of aggregatestatistics include indices such as the *retail price index

agree If f(x) and g(x) are defined on a set S, and f(x) = g(x) for

all x∈S, then f and g agree on the set S.

air resistance The resistance to motion experienced by an

object moving through the air caused by the flow of air overthe surface of the object It is a force that affects, for example,the speed of a drop of rain or of a parachutist falling towardsthe Earth’s surface As well as depending on the nature of theobject, air resistance depends on the speed of the object.Possible *mathematical models are to assume that themagnitude of the air resistance is proportional to the speed or

to the square of the speed

Aitken’s method (in numerical methods) If an iterative

formula x r+1 = f(x r) is to be used to solve an equation,Aitken’s method of accelerating convergence uses the initialvalue and the first two values obtained by the formula tocalculate a better approximation than the iterative formula

would produce This can then be used as a new starting pointfrom which to repeat the process until the required accuracyhas been reached While this is computationally intensive, it

is the sort of process which spreadsheets handle very easily

If x0, x1, x2are the initial value and the first two iterations and

Δx r = x r+1 − x r, Δ2x r = Δx r+1 − Δx r are the forwarddifferences then

More generally this will be expressed as

aleph Any infinite *cardinal number, usually denoted by the

Hebrew letter א See alsoTRANSFINITE NUMBER.

aleph-null The smallest infinite cardinal number The

cardinality of any set which can be put in one-to-onecorrespondence with the set of natural numbers Such sets aresaid to be *countable or *denumerable One of the apparentparadoxes in number theory is that the set of rational numbersbetween 0 and 1, the set of all rational numbers, and the set ofnatural numbers all have the same cardinality The symbol א0

is used

algebra The area of mathematics related to the general

properties of arithmetic Relationships can be summarized by

using variables, usually denoted by letters x, y, n, … to stand

for unknown quantities, whose value(s) may be determined

by solving the resulting equations See also ABSTRACT ALGEBRAandLINEAR ALGEBRA.

Algebra, Fundamental Theorem of See FUNDAMENTAL

THEOREM OFALGEBRA

algebra of sets The set of all subsets of a *universal set E is

closed under the binary operations ∪ (*union) and ∩(*intersection) and the unary operation ′ (*complementation).The following are some of the properties, or laws, that hold

for subsets A, B and C of E:

(i) A∪(B∪C) = (A∪B)∪C and A ∩ (B ∩ C) = (A ∩ B) ∩

C, the associative properties.

(ii) A ∪ B = B ∪ A and A ∩ B = B ∩ A, the commutative

∪B) ∩ (A∪C), the distributive properties.

(vii) A∪A′ = E and A ∩ A′ = Ø.

(viii) E′ = Ø and Ø′ = E.

(ix) (A′)′ = A.

(x) (A∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪B′, De Morgan’s

laws

The application of these laws to subsets of E is known as the

algebra of sets Despite some similarities with the algebra ofnumbers, there are important and striking differences

algebraic closure The extension of a given set to include all

the roots of polynomials with coefficients in the given set

The smallest algebraically closed set of numbers is C, the set

of complex numbers, since the very simple equation x2 + 1 =

0 has a complex solution

algebraic geometry The area of mathematics related to the

study of *geometry by algebraic methods

algebraic number A real number that is the root of a

*polynomial equation with integer coefficients All *rational

numbers are algebraic, since a/b is the root of the equation bx

− a = 0 Some *irrational numbers are algebraic; for example,

is the root of the equation x2 − 2 = 0 An irrationalnumber that is not algebraic (such as π) is called a

*transcendental number

algebraic structure The term used to describe an abstract

concept defined as consisting of certain elements withoperations satisfying given axioms Thus, a *group or a *ring

or a *field is an algebraic structure The purpose of thedefinition is to recognize similarities that appear in differentcontexts within mathematics and to encapsulate these bymeans of a set of axioms

algebraic system A set together with the *operations and

*relations defined on that set

algorithm A precisely described routine procedure that can

be applied and systematically followed through to aconclusion

algorithmic complexity The number of steps required in an

algorithm depends on the process and on the number of itemsbeing operated on An algorithm where the number of stepsincreases in direct proportion to the number of items is said tohave linear complexity, so if there are three times as manyitems the algorithm will have three times as many steps If thealgorithm was of quadratic complexity then there would havebeen nine times as many steps

aliquant part A number or expression which is not an exact

divisor of a given number or expression For example, 2 is an

aliquant part of any odd number, and x + 1 is an aliquant part

of x2+ 1

aliquot part A number or expression which is an exact

divisor of a given number or expression, and is usuallyrequired to be a proper divisor For example, 2 is an aliquot

part of any even number larger than 2, and x + 1 is an aliquot part of x2− 1

almost all (almost everywhere) Holding for all values except

on a set of *zero measure The most striking example of a set

of zero measure is

the set of rational numbers, so if f(x) = a when x is rational, and f(x) = b when x is irrational, and g(x) = b everywhere, then f and g agree almost everywhere.

almost surely = ALMOST EVERYWHERE in a probabilitymeasure

al-Khwārizmī See under KHWĀRIZMĪ, MUHAMMAD IBN

MŪSĀ AL-

alternate angles SeeTRANSVERSAL.

alternative hypothesis SeeHYPOTHESIS TESTING.

alternating group The *subgroup of the *symmetric group,

S n , containing all the *even permutations of n objects It has order n!/2 For n > 4, it is the only proper, *normal subgroup

of S n apart from the *empty set The alternating group is a

*simple group

alternating series A series in which the sign alternates

between positive and negative So any series in the form a n =(−1)np nor (−1)n−1p n where all p n> 0 is alternating

altitude A line through one vertex of a triangle and

perpendicular to the opposite side The three altitudes of atriangle are concurrent at the *orthocentre

ambiguous case The case where two sides of a triangle are

known, and an acute angle which is not the angle between theknown sides There can often be two possible triangles whichsatisfy all the given information, hence the name

If you know the length of AB, and of BC, and the acute angle

at A, you can construct the triangle as follows Draw AB From A draw a line making the required angle to AB Place the point of a compass at B and draw an arc of a circle with radius equal to the required length of BC Where the line from

A and the arc intersect is the position of C Unless the angle at

C is

exactly 90° or the given information is inconsistent, there will

be two different points of intersection and hence two possibletriangles

• An animation showing the construction of the two possiblesolutions

amicable numbers A pair of numbers with the property that

each is equal to the sum of the positive divisors of the other.(For the purposes of this definition, a number is not included

as one of its own divisors.) For example, 220 and 284 areamicable numbers because the positive divisors of 220 are 1,

2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, whose sum is 284, and