COIN , n. coin-match ing game. See game. CO'IN-CIDE ', v. To be coincident.
CO-IN'CI-DENT , adj. coincident configura tions. Two configurations which are such lhal any point of either one lies on the other. Two lines (or curves or surfaces) which have the same equation are coincident.
CO-LAT'I-TUDE, n. colatitude of a point on t he earth . Ninety degrees minus the latitude; the complementary angle of the latitude.
COL-LAT'E-RAL, adj. eollaleral security. Assets deposited to guarantee the fulfillment of some contract to pay, and to be returned upon the fulfillment of the contract, collateral trust bonds. Sec BOND.
COL-LECT'ING, p. collecting terms . Group- ing terms in a parenthesis or adding like terms. E.g., to collect terms in 
we write it in the form 
; to collect terms in 
we write it in the form 
COL-LIN'E-AR , adj. collinear planes. Planes having a common line. Syn. coaxial planes. Three planes arc cither collinear or parallel if the equation of any one of them is a linear combination of the equations of the others. See consistency —consistency of linear equations.
collinear points . Points lying on the same line. Two points in the plane are collinear with the origin if and only if their corresponding rectangular Cartesian coordinates arc pro portional, or the determinant, whose first row is composed of the Cartesian coordinates of one of the points and the second of those of the other, is zero; i.e., 
where the points are 
and 
Two points in space are collinear with the origin if, and only if, their corresponding Cartesian coordinates are proportional, i.e., the matrix 
whose columns are composed of the co ordinates of the points, is of rank one. Three points in a plane are collinear if, and only if, the third-order determinant whose rows are 
and 
where the
and 
are the coordinates of the three points, is zero . Three points in space are collinear if, and only if, lines through different pairs of the points have their direction numbers proportional, or if, and only if, the coordinates of any one of them can be ex pressed as a linear combination of the other two, in which the constants of the linear com bination have their sum equal to unity.
COL-LIN-E-A'TION , n. A transformation of the plane or space which carries points into points, lines into lines, and planes into planes. See transformation —collineatory transformation.
COL-LIN'E,-A-TO-RY , adj. collineatory transformation. See TRANSFORMATION—collinea- tory transformation.
CO-LOG'A-RITHM , n. The colorarithm of a number is the logarithm of the reciprocal of the number [i.e., the negative of the logarithm), expressed with the decimal part positive. Used in computations to avoid subtracting mantissas and the confusion of dealing wilh the negatives of mantissas. E.g., to evaluate 
by use of logarithms, write 
. 
where colog 
"COLONEL BLOTTO" GAME. See game.
COL'UMN , n . A vertical array of terms, used in addition and subtraction and in deter minants and matrices,
column in a determinant. See determinant.
COMBESCURE, Jean Joseph Antoine Eduard (1824-1884]. Combescure transformation of a curve. A one-to-one continuous mapping of the points of one space curve on another in such a way that the tangents at corresponding points arc parallel. It follows that the principal normals and the binomials, respectively, at corresponding points must also be parallel.
Combescure transformation of a triply orthogonal system of surfaces. A one-to-one continuous mapping of the points of three- dimensional Euclidean space on itself, such thai the normals lo the members of one triply orthogonal system of surfaces arc parallel to t he normals to the members of another system al points corresponding under the trans formation.
COM'BI-NA'TION, n. A combination of a set of objects is any subset without regard to order. The number of combinations of n things, r at a time, is the number of sets that can be made up from the n things, each set containing r different things and no two sets containing exactly the same r things. This is equal to the number of permutations of the n things, taken r at a lime, divided by the number of permutations of r things taken r at a time; that is, 
which is denoted by 
, or 
.- E.g., the combinations of a, b, c, two at a time, are 
is also the coefficient in the 
term of the binomial expansion, 
with the 
all equal. That is, the coefficient of 
is 
times the number of combinations of the n different 
at a lime. See BINOMIAL binomial coefficients. The numher of com binations of n things, r at a time, when repetitions arc allowed is the number of sets which can be made up of r things chosen from the given n, each being used as many limes as desired. The number of such combinations is the same as the number of combinations of 
different things taken rat a time, repetitions not allowed; i.e..
The combinations of a, b, c two at a time, repetilions allowed, arc: 
The total numher of combinations of n different things (repetilions not allowed) is the sum of the number of combinations taker 
at a time, i.e., 
which is the sum of the binomial coefficients in 
linear combination. See linear.
COM'BI-NA-TO'RI-AL , adj. combinatorial topology. See topology.
COM-MAND', n. An instruction, in machine language, for a computing machine to per form a certain operation.
COM-MEN'SU-RA-BLE, adj. commensurable quantities. Two quantities which have a common measure; i.e., there is a measure which is contained an integral number of times in each of them. A rule, a yard long, is commensurable with a rod, for they each contain, for instance, 6 inches an integral number of times. Two real numbers are commensurable if and only if their ratio is a rational number.
COM-MER'CIAL, adj. commercial bank . A bank that carries checking accounts.
commercial draft. See draft.
commercial paper. Negotiable paper used in transacting business, such as drafts, negotiable notes, endorsed checks, etc.
commercial year . See year.
COM-MIS'SION, n. A fee charged for transacting business for another person.
commission man or merchant. See BROKER.
COM'MON , adj. common denominator . For two or more fractions, a common denominator is a common multiple of the denominators. For example, a common denominator for the fraclions 
and 
is 12, or any multiple of 12. The l east common denominator (L.C.D.) is the smallest of all common denominators, i.e., the least common multiple of the de nominators. In the example, 12 is the least common denominator.
common difference in an arithmetic pro gression. The difference between any term and the preceding term, usually denoted by d.
common divisor or common factor . See divisor —common divisor.
common fraction . See fraction.
common logarithms. Logarithms having 10 for their base. See logarithm.
common multiple . See multiple —common multiple.
common stock . Stock upon which the dividends paid are determined by the net profits of the corporation after all other costs, including dividends on preferred stock, have been paid.
common tangent of two circles. A line which is tangent to each of the circles. If each circle is in the exterior of the other, there are four common tangents. The two tangents that separate the circles are internal tangents.
The other two tangents are external tangents ; both circles are on the same side of each external tangent.
COM-MU-TA'TION , adj. commutation sym bols in life insurance. Symbols denoting the nature of the numbers in the columns of a commutation table. For instance. 
and 
See below, commutation tables. commutation tables (columns). Tables from which the values of certain types of insurance can be quickly computed. E.g., suppose that one has a commutation table with the values of 
and 
for all ages appearing in the mortality tables, where 
is the product of the number of persons who attain the age x in any year and the present value of a sum of money x years hence, at some given rate, and 
is the sum of the series ( 
to end of table). The value of an immediate annuity of $1.00 at age x is the quotient 
and that of an annuity due is 
Sometimes (following Davies) 
is defined as the sum of the series ( 
]. With this definition, the annuity values must be 
and 
respectively. Commutation tables based on the latter definition of 
are called the terminal form, while those based on the former definition of 
are called the initial form .
COM-MU'TA-TIVE, adj. A method of com bining objects two at a time is commutative if the result of the combination of two objects docs not depend on the order in which the objects are given. For example, the com mutative Ian of addition states that the order of addition does not affect the sum: 
for any numbers a and b {e.g., 
The commutative law of multiplication slates that the order of multiplication does not affect the product: 
for any numbers a and b {e.g., 
'). There are many mathematical systems that satisfy commutative laws, but many others that do not. For example, (vector) multiplication of vectors and multiplication of matrices are not commutative. See group, RING, commutative group. See group.
COM'MU-TA-TOR, n. commutator of ele ments of a group. T ne commutator of two elements a and b of a group is the element 
or the clement c such that 
The group of all elements of the form 
where each 
is the commutator of some pair of elements, is called the commulator subgroup. The commutator subgroup of an Abelian group contains only the identity element. A group is said to be perfect if it is identical with its commutator subgroup. A commutator subgroup is an invariant subgroup and the factor group formed with it is Abelian.
COM-MUT'ING , p. commuting obligations.
Exchanging one set of obligations to pay a certain sum (or sums) at various times for another to pay according to some other plan. The common date of comparison at which the two sets are equivalent (equal in value at that time) is the focal date .
COM'PACT, adj. A topological space E being compact (originally, bicontpact ) means that it has the property that, for any union of open sets that contains E, there is a finite number of these open sets whose union con tains E. A space E is compact if and only if each collection of closed sets contained in E has a nonempty intersection if it has the finite intersection property. All closed subsets of a compact space arc compact; all compact subsets of a Hausdorff topological space are closed, A locally compact space is a space E with the property that each point of E has a neighborhood that is contained in a compact subset of E. The set 
is compact. The set R of real numbers is not compact, since R is contained in the union of all open intervals of unit length, but R is not contained in any finite union of such intervals (see heine —Heine-Borel theorem). A count- ably compac space is a space E with the properly that, lor any union of a countable number of open sets that contains E, there is a finite number of these open sets whose union contains E. A space E is countably compact if and only if each sequence in E has an accumu lation point in E. A sequentially compact (originally, compact ) space is a space E with the property that each sequence in E contains a subsequence that converges to a point in E. For a space to have the Bolzano-Weierstrass property means that each infinite subset has at least one accumulation point (see bolzano — Bolzano-Weierstrass theorem). For a Lindelof space (and therefore for a metric space), com pactness, countable compactness, sequential compactness, and the Bolzano-Weierstrass property are equivalent. All compact spaces are countably compact, and all countably compact spaces have the Bolzano-Weierstrass property. All 
-spaces with the Bolzano- Weierstrass property arc countably compact. Ail sequential compact spaces arc countably compact, and all countably compact spaces satisfying the first countability axiom are sequentially compact.
COM-PACT'I-FI-CA'TION , n. A compacti fication of a topological space T is a compact topological space W which contains T (or is such that T is homeomorphic with a subset of W). The complex plane (or sphere) is the compactification of the Euclidean plane ob tained by adjoining a single point (usually designated by the symbol 
) and defining the neighborhoods of 
to be sets containing 
and the complement of a bounded, closed (i.e., compact) subset of the plane. Likewise, a locally compact Hausdorff space H has a one-point compactification (also a Hausdorff space) obtained by adjoining a single point, which can be designated by the symbol 
whose neighborhoods are sets containing 
and the complement of a compact subset of II. The Stone-Cech compactiflcation of a Tychonoff space T is the closure of the image of T in the space 
, where 
i.s the Cartesian product of the closed unit interval I (taken 
times) and 
is the cardinal number of the family F of all continuous functions from T to I (the image of a point x of T in 
is the member of 
whose f "component"' is f ( x ) f or each member /of F). The Stonc-Ceeh compaclification is (in a certain real sense) a maximal compactifi- cation. The entire space 
is compact, a consequence of the Tychonoff theorem (see product —Cartesian product).
COM-PAC'TUM , n. A topological space which is compact and metrizable. Examples of compacta are closed intervals, closed spheres (with or without their interiors), and closed polyhedra. See compact —compact set.
COM'PA-RA-BLE , adj. comparable functions . Functions f and g which have real-number values, which have a common domain of definition D, and which are such that cither 
for all x in for
all x in D. 
COM-PAR'I-SON , adj., n. comparison date .
See equation-- equation of payments.
comparison property of the real numbers. The property that exactly one of the state ments 
and 
is true for any numbers x and y. See Trichotomy.
comparison test for convergence of an infinite series, if, after some chosen term of a series, the absolute value of each term is equal to of less than the value of the corresponding term of some convergent scries of positive terms, the series converges (and converges absolute ly); if each term is equal to or greater than the corresponding term of some divergent series oi positive terms, the series diverges.
COM'PASS , it. An instrument for describ ing circles or for measuring distances between two points. Usually used in the plural, as compasses .
mariner's compass . A magnetic needle that rotates about an axis perpendicular to a card (see figure) on which the directions are indi cated. The needle always indicates the direction of the magnetic meridian.
COM-PAT'I-BIL'l-TY , adj., n. Same as CONSISTEINCY.
compatibility equations [Elasticity). The differential equations connecting Ihe com ponents of the strain tensor which guarantee that the state of strain be possible in a continuous body.
COM'PLE-MENT, n, complement of an
angle. The complement of an angle A is the angle 
Sec complimentary —complementary angles.
complement of a set. The set of all objects that do not belong to the given set 
but belong to a given whole space (or set) that contains 
The complement of the set of
positive numbers with respect to the space of all real numbers is the set containing all negative numbers and zero. See lattice.
COM'PLE-MEN'TA-RY , adj. complementary acceleration . See acceleration —acceleration of Coriolis.
complementary angles . Two angles whose sum is a right angle. The two acute angles in a right triangle are always complementary. See trigonometric —trigonometric cofunc-tions,
complementary function . See differential —linear differential equations.
complementary minor. See minor —minor of an element in a determinant.
complementary trigonometric functions. Same as trigonometric cofunctions. See TRIGONOMETRIC.
surface complementary to a given surface. Given a surface S, there is an infinitude of parallel surfaces such that S is a surface of center relative to each of them. See surface —surfaces of center relative to a given surface, PARALLEL-parallel surfaces. The other common surface of center of the family of parallel surfaces is said to be complemen tary to S.
COM-PLETE' , adj. complete annuity . See ANNUITY.
complete field . See FIELD—ordered field.
complete induction . Sec induction- — mathematical induction.
complete lattice. See lattice.
complete scale . See scale- number scale.
complete space . A complete metric space is a metric space such that every Cauehy sequence converges to a point of the space (see SE QUENCE—Cauchy sequence). The space of all real numbers (or all complex numbers) is complete. The space of all continuous functions defined on the interval 
is not complete if the distance between / and g is defined as 
since the sequence 
does not then converge to a con tinuous function if 
and 
A topologically complete topological space is a topo logical space that is homeomorphic to some complete metric space. A subset of a com plete metric space is topologically complete if and only if it is a 
subset (see borel -Borel set). A complete linear topological space is a linear topological space for which each Cauehy net converges to some point in the space. A Cauehy net is a net 
for which 
converges to 7ero, where 
is a member of the space for each 
in the given directed set [see moore (e. h.) —Moore-Smith convergence].
complete system of functions. See orthog onal— orthogonal functions.
complete system of representations for a group. See representation —representation of a group.
weakly complete space . See weak— weak completeness.
COM-PLET'ING , n. completing the square .
A process used in solving quadratic equations. It consists of transposing all terms to the left side of the equation, dividing through by the coefficient of the square term, then adding to the constant (and to the right side) a number that will make the left member a perfect trinomial square. This method is sometimes modified by first multiplying through by a number chosen to make the coefficient of the square of the variable a perfect square, then adding a constant to both sides of the equation, as before, to make the left side a perfect trinomial square. E.g., to complete the square in 
divide both mem-bers of the equation by 2, obtaining 
Now add 3 to both sides, obtaining 
Frequently, completing the square refers to writing any polynomial of the form 
in the form 
a procedure used, for example, in reducing the equations of conies to their standard form.
COM'PLEX , adj., n. As a noun, a complex may mean simply a set (a!so see below, simplicial complex).
absolute value of a complex number . See MODULUS—modulus of a complex number.
amplitude, or argument, of a complex number. Sec polar polar form of a com plex number.
complex conjugate of a matrix . See matrix —complex conjugate of a matrix.
complex domain (field). The set of all complex numbers. Sec FIELD.
complex fraction. See FRACTION.
complex integration . Sec contour —con tour integral.
complex measure . See MEASURE—mea sure of a set.
complex number . Any number, real or imaginary, of the form 
where a and b arc real numbers and 
Called imaginary numbers when 
and pure imaginary when 
and (although 
complex numbers are not imaginary in the usual sense). Two complex numbers are defined to be equal if and only if they are identical. I.e., 
means 
and 
A complex number 
can be represented in the plane by the vector with components a- and y, or by the point 
(see the figure below, and argand - Argand diagram). Thus two complex numbers are
equal if and only if they are represented by the same vectors, or by the same points, in the above figure, 
and 
Therefore 
which is the polar form of 
(see polar).
The sum of complex numbers is obtained by adding the real parts and the coefficients of i separately; e.g.. 
Geometrically, this is the same as the addition of the corresponding vectors in the plane. In the figure below, 
The product of complex numbers is computed by treating the numbers as polynomials in i with the special property 
Thus:
If the complex numbers are in the form 
and 
their product is 
; i.e., to multiply two complex numbers, multiply their moduli and add their amplitudes (see DE moivre— De Moivre's theorem). Similarly, the quotient of two complex numbers is the complex number whose modulus is the quo tient of the modulus of the dividend by that of the divisor and whose amplitude is the amplitude of the dividend minus that of the divisor; that is,
When the numbers are not in polar form, the quotient can be computed by multiplying dividend and divisor by the conjugate of the divisor, as illustrated in the following example:
Tech. The system of complex numbers is the set of ordered pairs 
of real numbers in which two pairs are considered equal if, and only if, they are identical 
if, and only if, 
and in which addition and multiplication are defined by
The system satisfies most of the fundamental algebraic laws, such as the associative and commutative laws for addition and multipli cation. It is a field, but not an ordered field. A remarkable consequence of these definitions is:
That is, the number 
has the two square roots (0, 1) and 
See fundamental fundamental theorem of algebra.
complex plane . The plane (of complex numbers) with a single point at infinity whose neighborhoods are exteriors of circles with center at 0. The complex plane is topologic- ally (and conformally) equivalent to a sphere. See projection —stereographic projection.
complex roots of a quadratic equation 
Used in contrasting roots of the form 
with real roots, although the latter are special cases of the former for which 
See discriminant —discriminant of a quadratic equation in one variable.
complex sphere . A unit sphere on which the complex plane is represented by a stereo- graphic projection. The complex plane is usually the equatorial plane relative to the pole of projection, or the tangent plane at the point diametrically opposite the pole of projection.
conjugate complex numbers. Two numbers of type 
and 
where a and b are real numbers. Frequently called conjugate imaginaries. If 
denotes the conjugate of z, then 
if 
and x and y are real, and 
is a root of any polynomial equation with real coefficients which has z as a root.
modulus of a complex number . Sec modulus.
real and imaginary parts of a complex number. See REAL, IMACINARY.
root of a complex number . See ROOT.
simplicial complex . A set which consists of a finite number of simplexcs (not necessarily all of the same dimension) with the property that the intersection of any two of the sim- plexes either is empty or is a face of each of them. This definition is sometimes modified in various ways, e.g., by requiring that each simplex be oriented. A simplicial complex is sometimes called a complex, but a complex is sometimes defined with fewer restrictions (e.g., it may be a countable set of simplexcs such that the intersection of any two of thcsimplexes e ither is empty or is a face of each of them, and no vertex of a simplex belongs to more than a finite number of the simplcxes). The dimension of a simplicial complex is the largest of the dimensions of the simplexes making up the simplicial complex. The class of all simplexes which belong to a simplicial complex K and have dimension less than that of K is called the skeleton of K. A finite set K of elements 
is called an abstract simplicial complex (or an abstract complex or a skeleton complex ), and the elements 
are called vertices , if cerlain nonempty subsets of K, which are called abstract simplexes (or skeletons ) are such that each subset (called a face) of an abstract simplex is an abstract simplex and each of the vertices is an abstract simplex. The dimension of an abstract simplex of 
point? is r, and the dimension of an abstract complex is the largest of the dimensions of its abstract simplexes. An abstract complex of dimension n can always be represented by a simplicial complex imbedded in the Euclidean space of dimension 
A simplicial complex is sometimes called a geometric complex (or simply a complex ), or a triangula- tion. The set of all those points which belong to simplexes of a simplicial complex is called a polyhedron . A topological space is said to be triangulable , and is sometimes called a polyhedron or a topological simplicial complex , if it is homeomorphic to the set of points belonging to simplexes of a simplicial complex K; the homeomorphism together with the complex K is a triangulation of the polyhedron, A simplicial complex is oriented if each of its simplexes is oriented. See chain— chain of a complex, manifold, simplex, surface, TRIANGULATION.
unit complex number. A complex number whose modulus is 1 ; a complex number of the form 
Unit complex numbers are represented by points on the unit circle in the plane. The products and quotients of unit complex numbers are unit complex numbers.
COM-PO'NENT, n . component of an acceleration, force, or velocity. See below, component of a vector. component of an algebraic plane curve. See curve —algebraic plane curve.
component of a computing machine . Any physical mechanism or abstract concept having a distinct role in automatic com putation. See headings below and under ARITHMETIC, CONTROL, INPUT, OUTPUT, STORAGE.
component of a set of points . A subset which is connected and is not contained in any other connected subset of the given set of points, A component is necessarily a closed subset relative to the set.
component of the stress tensor . In linear theory of elasticity, a set of six functions determining the state of stress at any point of the substance.
component of a vector. See vector.
direction components. See direction — direction numbers.
elementary potential digital computing com ponent. In a computing machine, any com ponent that can assume any one of a fixed discrete set of stable states, and that can influence and/or be influenced by other com ponents of the machine. See circuit —flip- flop circuit.