ARC CO'SINE, n. The arc cosine of a number x is an angle (or number) whose cosine is x, written 
x or arc cos x. E.g.,
arc 
is 
', or in general 
See TRIGONOMETRIC -inverse trigonometric function. Syn. inverse cosine, anticosine. The figure shows the graph of 
(y in radians).
ARC CO-TANGENT , n. The arc cotangent of a number x is an angle (or number) whose cotangent is x , written 
x or
arc cot x. E.g., arc cot 1 is 
, or in
general 
. See trigonometric —
inverse trigonometric function. Syn. inverse cotangent, anticotangent. (In the above figure, y is in radians.)
ARCHIMEDES (c. 287-212 B.C.). Greek geometer, analyst and physicist. He used the limiting processes of both differential and integral calculus. One of the greatest mathematicians of all time.
Archimedean property. The property of real numbers that for any positive numbers a and b there is a positive integer n such that 
method of exhaustion. See exhaustion .
spiral of Archimedes. See SPIRAL.
ARC-HY-PER-BOL'IC, adj. arc-hyperbolic sine, cosine, etc. See HYPERBOLIC-inverse hyperbolic functions.
ARC SE-CANT' , n. The arc secant of a number x is an angle (or number) whose secant is x, written 
or arc sec x. An angle whose secant is x. E.g., arc sec 2 is 
, 
or in general, 
. See trigonometric inverse trigonometric function. Syn. inverse secant, anti-secant. (In the following figure, y is in radians.)
ARC SINE, n. The arc sine of a number x is an angle (or number) whose sine is x, written 
x or arc sin x. E.g., 
is 
, or in general 
. See trigonometric —inverse trigonometric func tion. Syn. inverse sine, antisine. The figure above shows the graph of ( y in radians). 
ARC TAN-CENT", n. The arc tangent of a number x is an angle (or number) whose tangent is x, written 
x or arc tan x.
E.g., arc tan 1 is 
or in genera 
See TRiGONOMETRIC- inverse trigonometric function. Syn. inverse tangent, antitangent. The figure above shows the graph of 
ARE , n. A unit of measure of area in the metric system, equal to 100 square meters or 119.6 square yards. See hectare.
A'RE-A , n. area of a lune. See lune.
area of a plane set. A rectangle with adja cent edges of length a and b has area ab. The area of any bounded plane set is the least upper bound 
of the sum of the areas of a finite collection of nonoverlapping rectangles contained in the set, or the greatest lower bound 
of the sum of the areas of a finite collection of rectangles which together completely cover the set, provided 
(if 
the set has area and the area is zero; if 
" the set does not have area). An unbounded set with area is an unbounded set S Tor which there is a number m such that 
has area not greater than m whenever R is a rectangle; the area of S is then the least upper bound of the areas of 
for rect angles R. This definition can be used to prove the usual formulas for area (see the specific configuration: circle, triangle, etc.) Calculus is very useful for computing area (see integral —definite integral). The "method of exhaustion" is related to the methods of calculus (see exhaustion). Syn. two-dimen sional content. See content, dido, mea surable -measurable set, PAPPUS.
area of a surface. See SURFACE—surface area.
differential (or element) of area. See ELE MENT—element of integration, SURFACE- surface area, surface of revolution.
lateral area of a cone, cylinder, parallele piped, etc. See the specific configuration.
relations between areas of similar surfaces. Areas of similar surfaces have the same ratio (vary as) the squares of corresponding lines. E.g., (1) the areas of two circles are in the same ratio as the squares of their radii, (2) the areas of two similar triangles arc in the same ratio as the squares of corresponding side? or altitudes.
ARGAND, Jean Robert (1768-1822). Swiss mathematician. One of the first to publish (1806) an account of the graphical representation of complex numbers. See gauss -Gauss plane, wallis, wessel.
Argand diagram. Two perpendicular axes on one of which real numbers are represented and on the other pure imaginaries, thus providing a frame of reference for
graphing complex numbers. These axes are called the real axis and the imaginary axis or the axis of reals and the axis of imaginaries.
AR'GU-MF.NT , n. argument of a complex number. Same as amplitude. See ampli- tude- amplitude of a complex number.
argument of a function. Same as the INDEPENDENT VARIABLE See FUNCTION.
arguments in a table of values of a function.
The values in the domain of the function lor which the values in the range arc tabulated. The arguments in a trigonometric table are the angles for which the functions are tabu lated ; in a table of logarithms, the numbers for which the logarithms arc tabulated.
A-RITH'ME-TIC, n. The study of the positive integers 1, 2, 3, 4, 5, ... under the operations of addition, subtraction, multiplica tion, and division, and the use of the results of these studies in everyday life.
arithmetic modulo n. See congruence.
fundamental operations of arithmetic. Addi tion, subtraction, multiplication, and division.
AR-ITH-MET'IC or AR-ITH-MET'I-CAL,
adj. F.mploying the principles and symbols of arithmetic.
arithmetic component. In a computing machine, any component that is used in performing arithmetic, logical, or other similar operations.
arithmetic mean (or average). See MEAN .
arithmetic means between two numbers. The other terms of an arithmetic sequence of which the given numbers are the first and last terms; a single mean between two num bers x and y is their mean, 
See mean , and below, arithmetic sequence.
arithmetic number. See number -arith metic numbers.
arithmetic progression. Same as ARITH METIC SEQUENCE.
arithmetic sequence. A sequence, each term of which is equal to the sum of the preceding term and a constant; written:
where a
is the first term, d is the common differ ence, or simply the difference, and 
is the last, or 
term. The positive integers, 
, form an arithmetic sequence. Syn. arithmetic pro gression. See below, arithmetic series.
arithmetic series. The indicated sum of the terms of an arithmetic sequence. The sum of the arithmetic sequence described above is equal to
AR-ITH-MOM'E-TER, n. A computing machine.
ARM , n. arm of an angle. A side of the angle.
AR-RANGE'MENT, n. Same as PERMUTA TION (1).
AR-RAY , n. A display of objects in some regular arrangement, as a rectangular array or matrix in which numbers are displayed in rows and columns, or an arrangement of statistical data in order of increasing (or decreasing) magnitude,
ARTIN, Emil (1898-1962). German alge braist and group theorist who spent many years in the U.S.
Artinian ring. See CHAIN—chain conditions on rings.
AS-CENDING, adj. ascending powers of a variable in a polynomial. Powers of the variable that increase as the terms are counted from left to right, as in the polynomial 
ascending chain condition on rings. See CHAIN—chain conditions on rings.
ASCOLI, Giulio (1843-1896). Italian analyst.
Ascoli's theorem. Let A be an infinite set of functions all of whose domains are the same closed bounded set D in a finite- dimensional Euclidean space (e.g., a closed bounded interval) and whose ranges are sets of real numbers. If these functions arc equlcontinuous and there is a number M such that 
for all 
in A and all x
in D, then there is a sequence 
of distinct members of A that converges uni formly to a continuous function. The fol lowing stronger theorem also is true: Let , A be a set of functions whose domains arc the same separable metric space X and whose ranges are in a metric space Y. If these functions arc equicontinuous and for each x in a dense subset of X the set 
is compact, then there is a sequence 
of distinct members of which converges pointwise to a continuous function 
and the convergence is uniform on each compact subset of X.
AS-SESSED' , adj. assessed value. A value set upon properly for the purpose of taxation.
AS-SES'SOR , n . One who estimates the value of (evaluates) property as a basis for taxation.
AS'SETS , n, assets of an individual or firm.
All of his (or its) goods, money, collectable accounts, etc., which have value; the opposite of liabilities.
fixed assets. Assets represented by equip ment for use but not for sale such as factories, buildings, machinery, and tools.
wasting assets. See depreciation.
AS-SO'CI-ATE, adj., n. In a commutative semigroup or ring, associates are members a and b for which there exist members x and y such that a = bx and b = ay.
associate matrix. See filrmiiian -Hcrmi- tian conjugate of a matrix.
AS-SO'CI-AT'ED , adj. associated radius of convergence. If the power series 
converges for and diverges for 
where 
is positive, then the set 
is called a set of associated radii of convergence for the series. E.g., for the series 
associated radii are any positive numbers 
with 
AS-SO'CI-A'TIVE , adj. A method of com bining objects two at a time is associative if the result of the combination of three objects (order being preserved) does not depend on the way in which the objects are grouped. If the operation is denoted by 
and the result of combining x and y by 
then 
for any x, y and z for which the "products" are defined. For ordinary addition of num bers, the associative law states that (a + b) + c = a + (b + c) for any numbers a, b, c. This law can he extended to slate that in any sum of several terms any method of grouping may be used {i.e., at any stage of the addition one may add two adjacent terms). The associative law for multiplication stales that 
for any numbers a, b, c. This law can be extended to state that in any product of several factors any method of grouping may he used (i.e., at any stage of the multiplication one may multiply any two adjacent factors). See group. For a type of non-associative multiplication, see cayley -Cayley algebra.
AS-SUMP'TION, n. See axiom, and below, fundamental assumptions of a subject.
empirical assumption. See empirical — empirical formula, assumption, or rule.
fundamental assumptions of a subject. A set of assumptions upon which the subject is built. For instance, in algebra the commutative and associative laws arc fundamental assumptions. Sets of fundamental assumptions for the same subject vary more or less with different writers.
AS-SUR'ANCE , n. Same as insurance.
AS'TROID, n . The hypocycoid of four cusps.
A'SYM-MET'RIC, adj. asymmetric relation. See SYMMETRIC—symmetric relation.
AS'YMP'TOTE , n. For a plane curve, an asymptote is a line which has the property that the distance from a point P on the curve to the line approaches zero as the distance from P to the origin increases without bound and P is on a suitable piece of the curve. Often it is required that the curve not oscillate about the line. See below, asymptote to the hyperbola.
asymptote to the hyperbola. When the equation of the hyperbola is in the standard form 
the lines 
and 
are its asymptotes. This can be sensed by writing the above equation in the form
and noting that 
approaches zero as x increases without limit. Tech. The numerical difference between the corresponding ordinates of the lines and the hyperbola is
which approaches zero as x increases, and the distances from the hyperbola to the lines are the product of this infinitesimal by the cosines of the angles the lines make with the x -axis; hence the distances between the lines and the hyperbola each approach zero as x increases. See above, asymptote, and the figure under HYPERBOLA.
AS'YMP-TOT'IC , adj. asymptotic directions on a surface at a point. Directions al a point P on a surface S for which 
See fundamental —funda mental coefficients of a surface. Asymptotic directions at P on S are the directions at P on S in which the tangent plane at P has contact of at least the third order. See distance — distance from a surface to a tangent plane. Asymptotic directions arc also the directions in which the normal curvature vanishes. At a planar point, all directions are asymptotic directions; otherwise there arc exactly two asymptotic directions, which are real and distincl, real and coincident, or conjugate imaginary, according as the point on the real surface S is hyperbolic, parabolic, or elliptic. asymptotic cone of an hypcrboloid. If either of the hyperboloids
and
is cut by the plane y = mx, hyperbolas are formed whose asymptotes pass through the origin. The cone described by these lines as m varies is the asymptotic cone of the hyper- boloid under consideration, asymptotic distribution. If a distribution 
of a random variable x is a function of a parameter n (e.g., n may be the size of a sample and x the mean), the limit of 
as 
is the asymptotic distribution funclion of x. In particular, if two quantities u and 
can be obtained so that the distribution function of
will be, in the limit as 
equal to
then F(x) is asymptotically normally distri buted. This means that x is asymptotically normally distributed in the sense that the limit, as 
, of the probability of 
is given by the normal distribution regardless of whether or not 
has a mean and variance of 
and 
Whatever the distribution of 
the probability of the variable 
is given in the limit by the normal distribution, if x can be so transformed as to be asymptotically normal.
B.T.U . See BRITISH—British thermal unit.