# Arithmetical Sentence Examples

- The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between
**arithmetical**and geometrical measurement. - Even when the formal evolution of the science was fairly complete, it was taken for granted that its symbols of quantity invariably stood for numbers, and that its symbols of operation were restricted to their ordinary
**arithmetical**meanings. - When algebra had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked; but it is one of the surprises in the history of science that logarithms were invented as an
**arithmetical**improvement years before their connexion with exponents was known. - 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. - Na-nun, one; nar, two; and ne', three, or variants of these; all higher
**arithmetical**ideas being expressed by the word kerpn, which means " many." - Hence if all the energy supplied to the train is utilized at one axle there is the fundamental relation RV (I) Continuing the above
**arithmetical**illustration, if the wheels to the axle of which the torque is applied are 4 ft. - He was thus led to conclude that chemistry is a branch of applied mathematics and to endeavour to trace a law according to which the quantities of different bases required to saturate a given acid formed an
**arithmetical**, and the quantities of acids saturating a given base a geometrical, progression. - He took a passionate delight in the pursuit of knowledge from his very infancy, and is reported to have worked out long
**arithmetical**sums by means of pebbles and biscuit crumbs before he knew the figures. - The most famous outcome of his inquiries is the law known as Weber's or Fechner's law which may be expressed as follows:- "In order that the intensity of a sensation may increase in
**arithmetical**progression, the stimulus must increase in geometrical progression." - In mathematics, he was the first to draw up a methodical treatment of mechanics with the aid of geometry; he first distinguished harmonic progression from
**arithmetical**and geometrical progressions. **Arithmetical**groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry.- The harmony between
**arithmetical**and geometrical measurement, which was disturbed by the Greek geometers on the discovery of irrational numbers, is restored by an unlimited supply of the causes of disturbance. - They are preceded by two sections dealing with the introduction to algebra from4the
**arithmetical**and the graphical sides, and are followed by a section dealing briefly with the developments mentioned in ï¿½ï¿½ 9 and 1 o above. **Arithmetical**Introduction to Algebra.- Order of
**Arithmetical**Operations. - - It is important, before beginning the study of algebra, to have a clear idea as to the meanings of the symbols used to denote
**arithmetical**operations. - - The equation exists, without being shown as an equation, in all those elementary
**arithmetical**processes which come under the head of inverse operations; i.e. - (ii.) In the above case the two different kinds of statement lead to
**arithmetical**formulae of the same kind. - In the case of division we get two kinds of
**arithmetical**formula, which, however, may be regarded as requiring a single kind of numerical process in order to determine the final result. - So far as the nature of
**arithmetical**operations is concerned, we launched out on the unknown. - Expressed Equations.-The simplest forms of
**arithmetical**equation arise out of abbreviated solutions of particular problems. In accordance with ï¿½ 15, it is desirable that our statements should be statements of equality of quantities rather than of numbers; and it is convenient in the early stages to have a distinctive notation, e.g. - (2) The first step towards
**arithmetical**reasoning in such a case is the introduction of the sign of equality. **Arithmetical**and Algebraical Treatment of Equations.-The following will illustrate the passage from**arithmetical**to algebraical reasoning.**Arithmetical**and Algebraical Treatment of Equations.-The following will illustrate the passage from**arithmetical**to algebraical reasoning.- In the same way, the transition from (x 2 +4x+4) - 4= 21 to x 2 +4x+4 = 25, or from (5+2) 2 =25 to x+2= 1 /25, is
**arithmetical**; but the transition from 5 2 + 45+4= 25 to (5+2) 2 = 25 is algebraical, since it involves a change of the number we are thinking about. - In addition to these, there are cases in which letters can usefully be employed for general
**arithmetical**reasoning. - (v.) Permutations and Combinations may be regarded as
**arithmetical**recreations; they become important algebraically in reference to the binomial theroem (ï¿½ï¿½ 41, 44)ï¿½ (vi.) Surds and Approximate Logarithms. - From the**arithmetical**point of view, surds present a greater difficulty than negative quantities and fractional numbers. - "), that the graphic method leads without
**arithmetical**reasoning to the properties of negative values. - It must, of course, be remembered (ï¿½ 23) that this is a statement of
**arithmetical**equality; we call the statement an " identity," but we do not mean that the expressions are the same, but that, whatever the numerical values of a, b and c may be, the expressions give the same numerical result. - (iv.) In algebra we have a theory of highest common factor and lowest common multiple, but it is different from the
**arithmetical**theory of greatest common divisor and least common multiple. - (ii.) The solution of equations is effected by transformation, which may be either
**arithmetical**or algebraical. - The principles of
**arithmetical**transformation follow from those stated in ï¿½ï¿½ 15-18 by replacing X, A, B, m, M, x, n, a and p by any expressions involving or not involving the unknown quantity or number and representing positive numbers or (in the case of X, A, B and M) positive quantities. - These numbers constitute an
**arithmetical**progression of the rth order. - In algebraical transformations, however, such as (x-a)2 = x 2 - 2ax+a 2, the
**arithmetical**rule of signs enables us to combine the sign-with a number and to treat the result as a whole, subject to its own laws of operation. - The divisions in ï¿½ï¿½ 50-52 have been supposed to be performed by a process similar to the process of
**arithmetical**division, viz. - If, moreover, we examine the process of algebraical division as illustrated in ï¿½ 50, we shall find that, just as
**arithmetical**division is really the solution of an equation (ï¿½ 14), and involves the tacit use of a symbol to denote an unknown quantity or number, so algebraical division by a multinomial really implies the use of undetermined coefficients (ï¿½ 42). - (v.) The further extension to fractional values (positive or negative) of n depends in the first instance on the establishment of a method of algebraical evolution which bears the same relation to
**arithmetical**evolution (calculation of a surd) that algebraical division bears to**arithmetical**division. - It is often convenient, as in ï¿½ 56 (ii.) and (vi.), to consider the mode of development of such a series, without regard to
**arithmetical**calculation; i.e. - The idea of continuity must in the first instance be introduced from the graphical point of view;
**arithmetical**continuity being impossible without a considerable extension of the idea of number (ï¿½ 65). - The word " sequence," as defined in ï¿½ 58 (i.), includes progressions such as the
**arithmetical**and geometrical progressions, and, generally, the succession of terms of a series. - The development is based on the necessity of being able to represent geometrical magnitude by
**arithmetical**magnitude; and it may be regarded as consisting of three stages. - Ordinary algebra developed very gradually as a kind of shorthand, devised to abbreviate the discussion of
**arithmetical**problems and the statement of**arithmetical**facts. - It could not escape notice that one and the same symbol, such as -1 (a - b), or even (a - b), sometimes did and sometimes did not admit of
**arithmetical**interpretation, according to the values attributed to the letters involved. - It was at last realized that the laws of algebra do not depend for their validity upon any particular interpretation, whether
**arithmetical**, geometrical or other; the only question is whether these laws do or do not involve any logical contradiction. - The only known type of algebra which does not contain
**arithmetical**elements is substantially due to George Boole. - The Romans, who succeeded the Greeks as the chief civilized power in Europe, failed to set store on their literary and scientific treasures; mathematics was all but neglected; and beyond a few improvements in
**arithmetical**computations, there are no material advances to be recorded. - Investigation of the writings of Indian mathematicians has exhibited a fundamental distinction between the Greek and Indian mind, the former being pre-eminently geometrical and speculative, the latter
**arithmetical**and mainly practical. - Although this transition from the discontinuous to continuous is not truly scientific, yet it materially augmented the development of algebra, and Hankel affirms that if we define algebra as the application of
**arithmetical**operations to both rational and irrational numbers or magnitudes, then the Brahmans are the real inventors of algebra. - His travels and mercantile experience had led E t u eopre him to conclude that the Hindu methods of computing were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both algebra and arithmetic. In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for numbers, and solves many problems, both
**arithmetical**and algebraical. - It includes the properties of numbers; extraction of roots of
**arithmetical**and algebraical quantities, solutions of simple and quadratic equations, and a fairly complete account of surds. - In order to save
**arithmetical**labour it is convenient to be provided with conversion factors for reducing variously expressed results to the standard form. - The distinction between algebraical and
**arithmetical**reasoning then lies mainly in the fact that the former is in a more condensed form than the latter; an unknown quantity being represented by a special symbol, and other symbols being used as a kind of shorthand for verbal expressions.