AD'JU-GATE , n. See adjoint —adjoint of a matrix.
AD-MIS'SI-BLE , adj. admissible hypothesis.
See HYPOTHESIS.
AF-FINE' , adj. afflne transformation. (I) A transformation of the form
(2) A transformation of the form given in (1) except that the determinant of the coefficients may ot may not be zero (it is singu lar or nonsingular according as this determi nant is zero or nonzero). The determinant of the coefficients is denoted by 
The fol lowing are important special cases of the affine transformation, 
(a) translations 
; (b) rotations 
;(c) stretchings and stir in kings 
i, called transformations of similitude or homothetic transformations; (d) reflections in the x-axis and j>-axis, respectively 
; (e) simple elongations and compressions 
or 
; (f) simple shear transfor- mations 
Ad isogonal affline transformation is an affline transformation which does not change the size of angles. It has the form
The affine transformation carries parallel lines into parallel lines, finite points into finite points, and leaves the line at infinity fixed. An affine transformation can always be factored into the product of transformations belonging to the above special cases. An homogeneous affine trans formation is an affine transformation in which the constant terms are zero; an affine transformation which does not contain a translation as a factor. Its form is where cither 
and 
or 
and 
AGE, n. (Life Insurance) The age at issue is the age of the insured at his birthday nearest the policy date. The age year is a year in the lives of a group of people of a certain age. The age year 
refers to the year from x to x+l, the year during which the group is x years old.
AG'GRE-GA'TION , n. signs of aggregation:
Parenthesis, (); bracket, []; brace, {}; and vinculum or bar, ——. Each means that the terms enclosed are to be treated as a single term. E.g., 3(2-1+4) means 3 times 5, or !5. See various headings under distributive. AGNESI, Maria Gaetana (1718-1799). Distinguished Italian mathematician. See WITCH.
AHLFORS, Lars Valerian (1907- ). Finnish-Am eric an mathematician and Fields Medal recipient (1936). Research in complex-variable theory and theory of quasicon- formal mappings, with important contribu tions to Riemann surfaces and meromorphic functions.
AHMES (RHYND or RHIND) PAPYRUS. Probably the oldest mathematical book known, written 2000 to 1800 B.C. and copied by the Egyptian scribe Ahmes about 1650 B.C. See rhind.
ALBERT, Abraham Adrian (1905-1972). American algebraist. Made fundamental contributions to the theory of Riemannian matrices and to the structure theory of associative and nonassociative algebras, Jor dan algebras, quasigroups, and division rings.
ALBERTI, Leone Battista (1404-1472). Italian mathematician and architect. Wrote on art, discussing perspective and raising questions that pointed toward the development of projective geometry.
A'LEPH , n . The first letter of the Hebrew alphabet, written 
aleph-null or aleph-zero . The cardinal num ber of countably infinite sets, written 
See cardinal —cardinal number.
ALEXANDER, James Waddell (1888- 1971). American algebraic topologist who did research in complex-variable theory, homology and ring theory, fixed points, and the theory of knots.
Alexander's subbase theorem . A topologi- cal space is compact if and only if there is a subbase S for its topology which has the property that, whenever the union of a col lection of members of S contains X, then X is contained in the union of a finite number of members of this collection.
AL'GE-BRA , n. (1) A generalization of arithmetic. E.g., the arithmetic facts that 
etc., are all special cases of the (general) algebraic statement that 
where x is any number. Letters denoting any number, or any one of a certain set of numbers, such as all real numbers, are related by laws that hold for any numbers in the set; e.g., 
for all x (all numbers). On the other hand, con ditions may be imposed upon a letter, repre senting any one of a set, so that it can take on but one value, as in the study of equations; e.g., if 
then x is restricted to 4.
Equations are met in arithmetic, although not so named. For instance, in percentage one has to find one of the unknowns in the equation, interest = principal x rate, or 
when the other two arc given. (2) A system of logic expressed in algebraic symbols, or a Boolean algebra (see boolean). (3) See below, algebra over a field.
algebra over a field . An algebra (or linear algebra ) over a field F is a ring R that is also a vector space with members of F as scalars and satisfies 
for all scalars a and b and ail members x and y of R. The dimension of the vector space is the order of R. The algebra is a commutative algebra, or an algebra with unit element, according as the ring is a commutative ring, or a ring with unit element. A division algebra is an algebra that is also a division ring. A simple algebra is an algebra that is a simple ring. The set of real numbers is a commutative division algebra over the field of rational numbers; for iny positive integer n, the set of all square matrices of order n with complex numbers (or real numbers) as elements is an algebra (non-commutative) over the field of real numbers. Any algebra consisting of all n by n matrices with elements in a given field is a simple algebra. An algebra of order n with a unit element is isomorphic to an algebra of n by n square matrices.
algebra of propositions . See BOOLE—Bool-e an algebra.
algebra of subsets . An algebra of subsets of a set X is a class of subsets of X which contains the complement of each of its members and the union of any two of its members (or the inter section of any two of its members). It is called a 
if it also contains the union of any sequence of its members. An algebra of subsets is a Boolean algebra relative to the operations of union and intersection. A ring of subsets of a set X is an algebra of subsets of X if and only if it contains X as a member. For any class C of subsets of a set X, the intersection of all algebras (or 
) which contain C is the smallest algebra fa- algebra) which contains C and is said to be the algebra ( 
) generated by C. For the real line (or /(-dimensional space) examples of 
are the system of all measurable sets, the system of all Borel sets, and the system of all sets having the property of Baire. See ring —ring of sets.
Banach algebra . An algebra over the field of real numbers (or complex numbers) which is also a real (or complex) Banach space for which 
for all x and y. It is called a real or a complex Banach algebra according as the field is the real or the complex number field. The set of all functions which are continuous on the closed interval [0, 1] is a Banach algebra over the field of real numbers if 
is defined to be the largest value of 
for 
. Syn. normed vector ring.
Boolean algebra . See boole.
fundamental theorem of algebra . See fundamental —fundamental theorem of algebra.
measure algebrra . See measure. —measure ring and measure algebra.
AL'GE-BRAIC , adj. algebraic adder. See ADDER.
algebraic addition . See sum algebraic sum, sum of real numbers.
algebraic curve. See curve.
algebraic deviation . See deviation.
algebraic expression, equation, function, operation, etc. An expression, etc., containing or using only algebraic symbols and opera tions, such as 
Algebraic operations arc the operations of addition, subtraction, multiplication, divi sion, extraction of roots, and raising to integral or fractional powers. A rational algebraic expression is an expression that can he written as a quotient of polynomials. An irrational algebraic expression is one that is not rational, as 
See function —algebraic function and various headings under rational.
algebraic extension of a field . See extension.
algebraic hypersurface . See hypersurface.
algebraic multiplication. See multipli cation.
algebraic number . (1) Any ordinary positive or negative number; any real directed num ber. (2) Any number which is a root of a polynomial equation with rational coefficients; the degree of the polynomial is said to be the degree of the algebraic number 
and the equation is the minimal equation of 
if 
is not a root of such an equation of lower degree. An algebraic integer is an algebraic number which satisfies some manic equation, 
with integers as coefficients. The minimal equation of an algebraic integer is also monic. A rational number is an algebraic integer if and only if it is an ordinary integer. The set of all algebraic numbers is an integral domain (sec domain integral domain). (3) Let F* be a field and F a subfield of F*. A member c of F* is algebraic with respect to Fif c is a zero of a polynomial with coefficients in F; otherwise, c is transcendental with respect to F.
algebraic operations. Addition, subtraction, multiplication, division, evolution and involu tion (extracting roots and raising to powers).
algebraic proofs and solutions. Proofs and solutions which use algebraic symbols and no operations other than those which arc alge braic. Sec above, algebraic operations.
algebraic subtraction . See SUBTRACTION
algebraic symbols . Letters representing numbers, and the various operational symbols indicating algebraic operations. See mathe matical symbols in the appendix.
algebraic variety . Let 
be an n -dimen- sional vector space with F the field of scalars. An algebraic variety is a subset of 
that is the sef of all points 
which satisfy a finite set of polynomial equations 
irrational 
algebraic surface. See irra tional.
AL-GE-BRA'IC-AL-LY , a. algebraically complete field. A field F which has the property that every polynomial equation with coefficients in F has a root in F. The field of algebraic numbers and the field of complex numbers are algebraically complete. Every field has an extension that is algebraically complete. Syn. algebraically closed field.
AL'GO-RITHM , n. Some special process of solving a certain type of problem, particularly a method that continually repeats some basic process.
division algorithm. Sec division.
Euclid's algorithm. A method of finding the greatest common divisor (G.C.D.) of two numbers—one number is divided by the other, then the second by the remainder, the first remainder by the second remainder, the second by the third, etc. When exact division is finally reached, the last divisor is the greatest common divisor of the given numbers (integers). In algebra, the same process can be applied to polynomials. E.g., to find the greatest com mon divisor of 12 and 20, we have 
Is 1 with remainder 8; 
is 1 with remainder 4; and 
hence 4 is the G.C.D.
AL'IEN-A'TION , n. coefficient of alienation.
See correlation normal correlation.
A-LIGN'MENT , adj. alignment chart. Same as NOMOGRAM.
AL'I-QUOT PART. Any exact divisor of a quantity; any factor of a quantity; used almost entirely when dealing with integers. E.g., 2 and 3 are aliquot parts of 6.
AL MOST, adj. almost all and almost every where. See MEASURE—measure zero, almost periodic. See PERIODIC.
AL'PHA, n. The first letter in the Greek alphabet: lower ease, a; capital, A.
AL-TER'NANT, n. A determinant for which there are n functions 
{if the determinant is oforder n) and n quantities 
for which Ihe element in the ith column and 
row is 
for each / and j (this determinant with rows and columns inter changed is also called an alternant). The Vandermonde determinant is an alternant (see DF.TtRMlNANT).
AL'TFR-NATE, adj. Two alternate angles
are angles on opposite sides of a transversal cutting two lines, each having one of Ihe lines for one of its sides. They are alternate exterior angles if neither lies between the two lines cut by the transversal. They are alternate interior angles if both lie between the two lines. See angle —angles made by a transversal.
AL-TER-NAT'ING, adj. alternating function.
A function 
of more than one variable for which 
changes sign if two of the variables are interchanged. If 
is an alternating multilinear function of n vectors in an rt-dimensional vector space and 
is a basis for V, then 
where
and det 
is the determinantwith 
in the 
row and 
column.
alternating group. See group —aliernating group.
alternating series. A series whose terms arc alternately positive and negative, as 
An alternating series converges if each term is numerically equal to or less than the pre ceding and if the nth term approaches zero as n increases without limit. This is a sufficient, but not a necessary set of conditions -the term-by-term sum of any two convergent series converges and, if one scries has all negative terms and the other all positive terms, their indicated sum may be a convergent alternating series and not have its terms monotonically decreasing. The series 
is such a scries. See necessary —necessary condition for convergence.
AL'TER-NA'TION, n . (1) In logic, the same as disjunction. (2) See proporjion.
AL-TER'NA-TIVE, adj. alternative hypoth esis. See HYPOTHttSIS-test of a hypothesis.
AL'TI-TUDE, n. A line segment indicating the heighi of a figure in some sense (or the length of such a line segment). See cone, cylinder, parabolic —parabolic segment, PARALLELOGRAM, PARAI.LI.LEPIPED, PRISM, PYRAMID, RLCTANGLE, SMJMENT—Spherical segment, frapezoid, triangle, zone.
altitude of a celestial point. Its angular distance above, or below, the observer's horizon, measured along a great celestial circle (vertical circle) passing through the point, the zenith, and the nadir. The altitude is taken positive when the object is above the horizon and negative when below. Sec figure under hour —hour-angle and hour-circle.
AM-BIG'U-OUS, adj. Not uniquely dctcr- minablc.
ambiguous case in the solution of triangles. For a plane triangle, the ease in which two sides and the angle opposite one of them is given. One of the other angles is then found by use of the law of sines; but there are always Iwo angles less than 
corresponding to any given value of Ihe sine (unless the sine be unity, in which case Ihe angle is 
and the triangle is a right triangle). When the sine gives two distinct values of the angle, two triangles result if the side opposite is less than the side adjacent to the given angle (assuming the data are not such that there is no triangle possible, a situation that may arise in any case, ambiguous or nonamhiguous). In the figure, angle A and sides a and b are given 
triangles 
and 
are both solutions. If a = b sin A, the right triangle ABC is the unique solution.
For a spherical triangle, the ambiguous case is the ease in which a side and the opposite angle are given (the given parts may then be either two sides and an angle opposite one side, or two angles and the side opposite one angle).
A-MER'I-CAN, adj. American experience table of mortality. See mortality.
AM'I-CA-BLE , adj. amicable numbers. Two numbers, each of which is equal lo the sum of all the exact divisors of the other except ihe number itself. E.g., 220 and 284 are amicable numbers, for 220 has the exact divisors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, whose sum is 284; and 284 has the exact divisors 1, 2, 4, 71, and 142, whose sum is 220.
A-MOR-TT-ZA'TION , n. amortization of a debt. The discharge of the debt, including interest, by periodic payments, usually equal, which continue until the debt is paid without any renewal of the contract. The mathe matical principles are the same as those used for annuities.
amortization equation . An equation re lating the amount of an obligation to be amortized, the interest rate, and the amount of the period payments. See AMOrtizf.
amortization of a premium on a bond. Writing down (decreasing) the book value of the bond on each dividend date by an amount equal to the difference between the dividend and the interest on the investment (interest on the book value at the yield rale). See value — book value.
amortization schedule . A table giving the annual payment, the amount applied to principal, the amount applied to interest, and the balance of principal due. See amortize.
A-MOUNT' , n. amount of a sum of money at a given date. The sum of the principal and interest (simple or compound) to the dale; designated as amount at simple interest or amount at compound interest (or compound amount), according as interest is simple or compound. In practice, the word amount without any qualification usually refers to amount at compound interest.
amount of an annuity . See accumulated — accumulated value of an annuity at a given date.
compound amount . See compound.
AM'PERE, n. A unit of measure of electric current. The legal standard of current since 1950 is the absolute ampere, which is the current in each of two long parallel wires which carry equal currents and for which there is a force of 
newton per meter acting on
each wire. The legal standard of current before 1950 was the international ampere, which is the current which when passed through a standard solution of silver nitrate deposits silver at the rate of .001118 gram per sec. One international ampere equals 0.999835 absolute ampere. Sec coulomb, ohm.
AM'PLI-TUDE , n. amplitude of a complex number. The angle that the vector represent ing the complex number makes with the positive horizontal axis. E.g., the amplitude of 
See polar polar form of a complex number.
amplitude of a curve . Half the difference between the greatest and the least values of the ordinatcs of a periodic curve. The amplitude of y — sin x is 1; of y = 2 sin x is 2.
amplitude of a point. See polar —polar coordinates in the plane.
amplitude of simple harmonic motion. See harmonic —simple harmonic motion,
AN'A-LOG , adj. analog computer. See computer.
A-NAL'O-GY, n. A form of inference some times used in mathematics to set up new theorems. It is reasoned that, if two or more things agree in some respects, they will prob ably agree in others. Exact proofs must, of course, be made to determine the validity of any theorems set up by this method.
A-NAL'Y-SIS, n. [pi. analyses]. That part of mathematics which uses, for the most part, algebraic and calculus methods—as dis tinguished from such subjects as synthetic geometry, number theory, and group theory.
analysis of a problem . The exposition of the principles involved; a listing, in mathe matical language, of the data given in the statement of the problem, other related data, the end sought, and the steps to be taken.
analysis of variance . Sec variance.
analysis situs . The field of mathematics now called topology.
diophantine analysis . Sec diophanius.
proof by analysis. Proceeding from the thing to be proved lo some known truth, as opposed to synthesis which proceeds from the true to that which is to be proved. The most common method of proof by analysis is, in fact, by analysis and synthesis, in that the steps in the analysis are required to be reversible.
unitary analysis . A system of analysis that proceeds from a given number of units to a unit, then to the required number of units. Consider the problem of finding the cost of 7 tons of hay if 
tons cost S25.O0. Analysis: If 
tons cost 125.00, 1 ton costs $10,000. Hence 7 tons cost $70.00.
AN-A-LYT'IC, adj. analytic continuation. If 
is given to be a single-valued analytic function in a domain D, then possibly there is a function Fanalytic in a domain of which D is a proper subdomain, and such that 
in D.
If so, the function F is necessarily unique.
The process of obtaining F from 
is called analytic continuation. E.g., the function 
defined by 
is thereby defined only for 
the radius of convergence of the series being 1 and !he circle of convergence having center at 0. The series represents the function 
but if this function is given a new representation, say by
where the coefficients are determined from the original series, the new circle of converg ence extends outside the old one (see the figure). The given function 
usually given as a power series (but not necessarily so), is called a function-element of F . The analytic continuation might well lead to a many- sheeted Riemann surface of definition of F. See monogenic- monogenic analytic function, analytic curve. A curve in n-dimensional Euclidean space which, in the neighborhood of each of its points, admits a representation of the form 
where the 
are real analytic functions of the real variable I. If in addition we have the curve is said to be a regular analytic 
curve and the parameter / is a regular parameter for the curve. For three-dimensional space, an analytic curve is a curve which has a para metric representation x = x{t), y = y(t), z =z ( t ). for which each of these functions is an analytic function of the real variable ;; it is a regular analytic curve if dx/dt, dy/dt and dz/dt do not vanish simultaneously. See POINT—ordinary point of a curve.
analytic function of a complex variable. A single-valued function, 
or a multiple-valued function considered as a single-valued function on its Riemann surface, which is differentiable at each point of its domain (a nonnull connected open set) of definition D is analytic in D. It can be shown that an analytic function/of a complex variable has continuous derivatives of all orders and can be expanded as a Taylor scries in a neighborhood of each point 
of D:
A function is sometimes said to be analytic in D if it is continuous in D and has a deriva tive at all except at most a finite number of points of D. If 
is differentiable at all points of D, it is a regular function, or a regular analytic function, or a holomOrphic function, in D. See cauchy— Cauchy-Riemann partial differential equations, MONOGENIC.
analytic function of a real variable. A function / is analytic at h if it can be repre sented by a Taylor's series in powers of (x — h) whose sum is equal to the value of the function at each x in some neighborhood of h. The function is said to be analytic in the interval (a, b) if the above is true for every h in the interval (a, b). See fayiok —Taylor's theorem.
analytic geometry . See geometry —analytic geometry.
analytic at a point . A single-valued function 
of the complex variable z is analytic at the point 
if there is a neighborhood N of 
such that 
is different iable at every point of N. I.e., 
is analytic at 
if it is analytic in a neighborhood of 
Syn. holomorphic, regular, or monogenic at a point. See above, analytic function of a complex variable,
analytic proof or solution. A proof or solution which depends upon that sort of procedure in mathematics called analysis; a proof which consists, essentially, of algebraic (rather than geometric) methods and/or of methods based on limiting processes (such as the methods of differential and integral calculus).
analytic structure for a space. A covering of a locally Euclidean space of dimension n by a set of open sets each of which is homeo- morphic to an open set in n-dimcnsional Euclidean space 
and which are such that whenever any two of these open sets overlap, the coordinate transformation (in both direc tions) is given by analytic functions (i.e., functions which can be expanded in power scries in some neighborhood of any point). If neighborhoods U and V overlap and P is in their intersection, then the homeomorphisms of U and V with open sets of n-dimensional Euclidean space define coordinates 
and 
for P, and the functions 
and 
are the functions required to be analytic. The analytic structure is real or complex according as the coordinates of points in 
are taken as real or complex numbers. See EUCLIDEAN- locally Euclid ean space, manifold.
of an analytic function. An a -point of the analytic function 
of the complex variable z is a zero point of the analytic function 
The order of an a -point is the order of the zero of 
at the point.
See zero zero point of an analytic function of a complex variable.
normal family of analytic functions . See NORMAL.
quasi-analytic function . For a sequence of positive numbers 
and a closed interval 
the class of quasi-analytic
functions is the set of all functions which possess derivatives of all orders on / and which arc such that for each function 
there is a constant k such that 
provided this set of functions has the property that if/is a member of the set and 
for 
and 
then 
on /.
If 
or 
then the corresponding class of functions is precisely the class of all analytic functions on /. Every function which possesses derivatives of all orders on / 
is the sum of two functions each of which belongs to a quasi- analytic class. Even if the class defined by 
... and / is not quasi-analytic, certain subclasses are sometimes said to be quasi- analytic if they do not contain a nonzero function 
for which 
and 
Quasi-analyticity is one of the most important properties of analytic functions, but there exist classes of quasi-analytic functions which contain nonanalytic functions, singular point of an analytic function. See singular.