# Algebraic Sentence Examples

For an

**algebraic**solution the invariants must fulfil certain conditions.It was then proposed to arrange a detector so that it was affected by the

**algebraic**sum of the two oscillations, and by swivelling round the double receiving antennae to locate the direction of the sending station by finding out when the detector gave the best signal.Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of

**Algebraic**Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.Evolution and involution are usually regarded as operations of ordinary algebra; this leads to a notation for powers and roots, and a theory of irrational

**algebraic**quantities analogous to that of irrational numbers.The name l'arte magiore, the greater art, is designed to distinguish it from l'arte minore, the lesser art, a term which he applied to the modern arithmetic. His second variant, la regula de la cosa, the rule of the thing or unknown quantity, appears to have been in common use in Italy, and the word cosa was preserved for several centuries in the forms toss or algebra, cossic or

**algebraic**, cossist or algebraist, &c. Other Italian writers termed it the Regula rei et census, the rule of the thing and the product, or the root and the square.It is important to begin the study of graphics with concrete cases rather than with tracing values of an

**algebraic**function.From A Merely Formal Point Of View, We Have In The Barycentric Calculus A Set Of " Special Symbols Of Quantity " Or " Extraordinaries " A, B, C, &C., Which Combine With Each Other By Means Of Operations And Which Obey The Ordinary Rules, And With Ordinary

**Algebraic**Quantities By Operations X And =, Also According To The Ordinary Rules, Except That Division By An Extraordinary Is Not Used.It was formerly the custom to assign the invention of algebra to the Greeks, but since the decipherment of the Rhind papyrus by Eisenlohr this view has changed, for in this work there are distinct signs of an

**algebraic**analysis.Notwithstanding the prolixity of writers and the number of the writings, all attempts at extracting an

**algebraic**analysis from their geometrical theorems and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had little or no affinity to algebra.One of the most recent developments of algebra is the

**algebraic**theory of number, which is devised with the view of removing these difficulties.AdvertisementThis insertion of irrational numbers (with corresponding negative numbers) requires for its exact treatment certain special methods, which form part of the

**algebraic**theory of number, and are dealt with under Number.These works possess considerable originality, and contain many new improvements in

**algebraic**notation; the unknown (res) is denoted by a small circle, in which he places an integer corresponding to the power.Thus, 1 - x would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 - x) (1 - y) would give us all things neither horned nor sheep. By the use of such symbols propositions could be reduced to the form of equations, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary

**algebraic**rules.Energyequations, such as the above, may be operated with precisely as if they were

**algebraic**equations, a property which is of great advantage in calculation.Such a number is a "one-many" relation which relates n signed real numbers (or n

**algebraic**complex numbers when they are already defined by this procedure) to the n cardinal numbers I, 2..AdvertisementWhile, therefore, the logical development of

**algebraic**reasoning must depend on certain fundamental relations, it is important that in the early study of the subject these relations should be introduced gradually, and not until there is some empirical acquaintance with the phenomena with which they are concerned.Thus every quaternion may be written in the form q = Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary

**algebraic**quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of quaternions.This equation is generally true for any series of transformations, provided that we regard H and W as representing the

**algebraic**sums of all the quantities of heat supplied to, and of work done by the body, heat taken from the body or work done on the body being reckoned negative in the summation.The three subjects to which Smith's writings relate are theory of numbers, elliptic functions and modern geometry; but in all that he wrote an "arithmetical" made of thought is apparent, his methods and processes being arithmetical as distinguished from

**algebraic**. He had the most intense admiration of Gauss.We will confine ourselves here to

**algebraic**complex numbers - that is, to complex numbers of the second order taken in connexion with that definition of multiplication which leads to ordinary algebra.AdvertisementThis fact is of extreme importance in the theory of

**algebraic**forms, and is easily representable whatever be the number of the systems of quantities.The progress of analytical geometry led to a geometrical interpretation both of negative and also of imaginary quantities; and when a " meaning " or, more properly, an interpretation, had thus been found for the symbols in question, a reconsideration of the old

**algebraic**problem became inevitable, and the true solution, now so obvious, was eventually obtained.This is also possible if the lenses have the same

**algebraic**sign.He also published several papers on

**algebraic**forms and projective geometry.Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "

**Algebraic**Functions and their Integrals," with the topics**algebraic**functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.AdvertisementIn his theory of graphs, or geometrical representations of

**algebraic**functions, there are valuable suggestions which have been worked out by others.The magnitudes of the maximum shearing stresses are indicated by the

**algebraic**differences of the thicknesses of the lines of principal stress.It is remarkable that although the seven integrals were found almost from the beginning of the investigation, no others have since been added; and indeed it has recently been shown that no others exist that can be expressed in an

**algebraic**form.Knowing these variations it becomes possible to represent by integration the value of the elements as

**algebraic**expressions containing the time, and the elements with which we started.When the varying elements are known these are computed by the equations (2) because, from the nature of the

**algebraic**relations, the slowly varying elements are continuously determined by the equations (4), which express the same relations between the elements and the variables as do the equations (2) and (3).The computations are, as a general rule, simpler, and the

**algebraic**expressions less complex, than when the computations of the longitude itself are calculated.By the second general method the moon's co-ordinates are obtained in terms of the time by the direct integration of the differential equations of motion, retaining as

**algebraic**symbols the values of the various elements.The third method seeks to avoid the difficulty by using the numerical values of the elements instead of their

**algebraic**symbols.Seebeck found that the metals could be arranged in a Thermoelectric Series, in the order of their power when combined with any one metal, such that the power of any thermocouple p, composed of the metals A and B, was equal to the

**algebraic**difference (p'-p") of their powers when combined with the standard metal C. The order of the metals in this series was found to be different from that in the corresponding Volta series, and to be considerably affected by variations in purity, hardness and other physical conditions.The components for any other combination of two are found by taking the

**algebraic**difference of the values with respect to lead.In analytical geometry the conic is represented by an

**algebraic**equation of the second degree, and the species of conic is solely determined by means of certain relations between the coefficients.Although the Arabs were in full possession of the store of knowledge of the geometry of conics which the Greeks had accumulated, they did little to increase it; the only advance made consisted in the application of describing intersecting conics so as to solve

**algebraic**equations.While Desargues and Pascal were founding modern synthetic geometry, Rene Descartes was developing the

**algebraic**representation of geometric relations.The subject of analytical geometry which he virtually created enabled him to view the conic sections as

**algebraic**equations of the second degree, the form of the section depending solely on the coefficients.This method rivals in elegance all other methods; problems are investigated by purely

**algebraic**means, and generalizations discovered which elevate the method to a position of paramount importance.Another approach is to construct algebras directly from the

**algebraic**group itself.In the

**algebraic**theory of data, continuous data types are modeled by topological algebras.He sent his results on making the differential calculus into a purely

**algebraic**theory to the Royal Society.In many ways the heart of his argument is found not in the text but in the almost

**algebraic**appendices.The system contains only

**algebraic**equations and o.d.e.s with a single independent variable (normally time ).Will QCA's revised criteria improve students ' mastery of basic skills and particularly

**algebraic**techniques?It is called

**algebraic**Curves over a Finite Field and is currently 644 pages.To consolidate and further develop

**algebraic**, geometric and trigonometric techniques.Her research interests are representation theory,

**algebraic**combinatorics and computational algebra.This usually requires large

**algebraic**computations due to the geometrical quantities entering the field equations and equations of motion.Based on our

**algebraic**theory we describe a category of models for the pi-calculus, and show that they all preserve bisimulation congruence.On the other hand, in agent-based simulations g t easily grows enormous, hindering any attempt at

**algebraic**manipulation.Expression If the subset has been defined by an

**algebraic**expression If the subset has been defined by an**algebraic**expression, this will be here.There are also geometric and

**algebraic**tips and tricks.Professor Griffiths is well known for his work in

**algebraic**geometry.His treatment has the great merit of being completely

**algebraic**in character and of meeting every difficulty without an appeal to geometric intuition.The wonder of it all is the theory of

**algebraic**invariants was successful far beyond the hopes of its originators.ViÃ¨te introduced the first systematic

**algebraic**notation in his book In artem analyticam isagoge published at Tours in 1591.Children are known to often invent idiosyncratic notation to describe their mathematical findings, or to use

**algebraic**notation in unusual ways.In 1657 he became the first to find the arc length of an

**algebraic**curve when he rectified the cubical parabola.A-Level Mathematics requires strict self-discipline to practice a large number of

**algebraic**techniques to achieve a high standard of proficiency.Simple

**algebraic**operations are then used to intersect two spheres, or a line and a sphere.The goal of

**algebraic**semantics is to capture the semantics of behavior by a set of axioms with purely syntactic properties.This culminated in the great results of

**algebraic**topology in the middle of the previous century.Research interests My main research interest is in the application of

**algebraic**topology to differential topology.An obituary notice by his friend Auguste Chevalier appeared in the Revue encyclopedique (1832); and his collected works are published, Journal de Liouville (1846), pp. 381-444, about fifty of these pages being occupied by researches on the resolubility of

**algebraic**equations by radicals.For the subjects of this general heading see the articles ALGEBRA;

**ALGEBRAIC**FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.**Algebraic**treatment consists in replacing either of the terms A or B by an expression which we know from the laws of arithmetic to be equivalent to it.He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n 2 differential coefficients of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations (see

**Algebraic**Forms).Generally, we may say that

**algebraic**reasoning in reference to equations consists in the alteration of the form of a statement rather than in the deduction of a new statement; i.e.For in such a construction every point of the figure is obtained by the intersection of two straight lines, a straight line and a circle, or two circles; and as this implies that, when a unit of length is introduced, numbers employed, and the problem transformed into one of

**algebraic**geometry, the equations to be solved can only be of the first or second degree, it follows that the equation to which we must be finally led is a rational equation of even degree.This condition is represented in the

**algebraic**theory when we have one more unknown quantity than the number of equations; i.e.For instance, 237578 w was printed @ 5070 8 3D; and the fact that Stevinus meant those encircled numerals to denote mere exponents is evident from his employing the very same sign for powers of

**algebraic**quantities, e.g.The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the

**algebraic**imaginary.He has given by means of it a simple proof of the existence of n roots, and no more, in every rational

**algebraic**equation of the nth order with real coefficients.That we have here a perfectly real and intelligible interpretation of the ordinary

**algebraic**imaginary is easily seen by an illustration, even if it be a somewhat extravagant one.In his famous Geometria (1637), which is really a treatise on the

**algebraic**representation of geometric theorems, he founded the modern theory of analytical geometry (see Geometry), and at the same time he rendered signal service to algebra, more especially in the theory of equations.If we define the positive direction along a tube of electric force as the direction in which a small body charged with positive electricity would tend to move, we can summarize the above facts in a simple form by saying that, if we have any closed surface described in any manner in an electric field, the excess of the number of unit tubes which leave the surface over those which enter it is equal to 47r-times the

**algebraic**sum of all the electricity included within the surface.Existing theories of abstract syntax are inadequate; we will develop improved

**algebraic**theories, covering variable binding and structured types.How in the world will their child be able to ever get a job unless he memorizes those math facts and, later,

**algebraic**equations?Graphs reinforce multiplication while bar models reinforce

**algebraic**concepts.They are linked to inverse operations, many

**algebraic**functions, and proportions.The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any

**algebraic**formula we can at once write down an operation formula.Although Hippocrates could not determine the proportionals, his statement of the problem in this form was a great advance, for it was perceived that the problem of trisecting an angle was reducible to a similar form which, in the language of

**algebraic**geometry, is to solve geometrically a cubic equation.Another of his works, Recensio canonica effectionum geometricarum, bears a stamp not less modern, being what we now call an

**algebraic**geometry - in other words, a collection of precepts how to construct**algebraic**expressions with the use of rule and compass only.The work of Wallis had evidently an important influence on the next notable personality in the history of the subject, James Gregory, who lived during the period when the higher

**algebraic**analysis was coming into power, and whose genius helped materially to develop it.