# Algebraic Sentence Examples

**ALGEBRAIC**FORMS. The subject-matter of**algebraic**forms is to a large extent connected with the linear transformation of**algebraical**polynomials which involve two or more variables.- For an
**algebraic**solution the invariants must fulfil certain conditions. - 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. - 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. - 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.. - 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. - Which is satisfied by every symmetric fraction whose partition contains no unit (called by Cayley non-unitary symmetric functions), is of particular importance in
**algebraic**theories. - 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. - Of two or more binary forms there are also complete systems containing a finite number of forms. There are also
**algebraic**systems, as above mentioned, involving fewer covariants which are such that all other covariants are rationally expressible in terms of them; but these smaller systems do not possess the same mathematical interest as those first mentioned. **Algebraic**Forms; Binomial; Combinatorial Analysis; Determin Ants; Equation; Continued Fraction; Function; Theory of groups; Logarithm; Number; Probability; Series.- While, 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. - One of the most recent developments of algebra is the
**algebraic**theory of number, which is devised with the view of removing these difficulties. - 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. - It is important to begin the study of graphics with concrete cases rather than with tracing values of an
**algebraic**function. - 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. - 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. - A 3 ordinary
**algebraic**quantities, which ma S' g q ? - 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. - 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 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. - 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. - If therefore we choose a quantity e such that log e I o X X= I, log i oe = X, which gives (by more accurate calculation) e=2.71828..., we shall have lim(loge(I+0))}/0=I, and conversely 'lim' {ex+0 - e x } 143= The deduction of the expansions log e (' +x) = x - Zx 2 + 3x 3 - ..., e x = I +.x+x2/2!+x3/3!-}-..., is then more simply obtained by the differential calculus than by ordinary
**algebraic**methods. - This 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. - 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. **Algebraic**Forms >>- 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. - 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. - 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. - 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. - This is also possible if the lenses have the same
**algebraic**sign. - 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. - 1890, p. 490) that exp(mldl +m2d2+m3d3+...) = exp (Midi +M2d2+M3d3+...), where now the multiplications on the dexter denote successive operations, provided that pp t exp(MiE+M2 2+M3E3+...) +mlH+m2V+m3S3+..., being an undetermined
**algebraic**quantity. - This fact is of extreme importance in the theory of
**algebraic**forms, and is easily representable whatever be the number of the systems of quantities. - Therefore 4) 1 and 42 must have different
**algebraic**signs, or the system must be composed of a collective and a dispersive lens. **ALGEBRAIC**FORMS. The subject-matter of**algebraic**forms is to a large extent connected with the linear transformation of**algebraical**polynomials which involve two or more variables.- Laws of
**Algebraic**Form. - 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. - The composition of two such lines by the
**algebraic**1 Theory of Conjugate Functions, or**Algebraic**Couples, with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time, read in 1833 and 1835, and published in Trans. - The composition of two such lines by the
**algebraic**1 Theory of Conjugate Functions, or**Algebraic**Couples, with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time, read in 1833 and 1835, and published in Trans. - As an
**algebraic**1 It appears from Joly's and Macfarlane's references that J. - In the third-order complex the centre locus becomes a finite closed quartic surface, with three (one always real) intersecting nodal axes, every plane section of which is a trinodal quartic. The chief defect of the geometrical properties of these bi-quaternions is that the ordinary
**algebraic**scalar finds no place among them, and in consequence Q:1 is meaningless.