## Arithmetical Sentence Examples

- The nature of logarithms is explained by reference to the motion of points in a straight line, and the principle upon which they are based is that of the correspondence of a geometrical and an
**arithmetical**series of numbers. - 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. - Nothing shows more clearly the rude state of
**arithmetical**knowledge at the beginning of the 17th century than the universal satisfaction with which Napier's invention was welcomed by all classes and regarded as a real aid to calculation. - This single instance of the use of the decimal point in the midst of an
**arithmetical**process, if it stood alone, would not suffice to establish a claim for its introduction, as the real introducer of the decimal point is the person who first saw that a point or line as separator was all that was required to distinguish between the integers and fractions, and used it as a permanent notation and not merely in the course of performing an**arithmetical**operation. - 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.- In order to save
**arithmetical**labour it is convenient to be provided with conversion factors for reducing variously expressed results to the standard form. - It may be regarded as based on arithmetic, or as dealing in the first instance with formal results of the laws of
**arithmetical**number; and in this sense Sir Isaac Newton gave the title Universal Arithmetic to a work on algebra. - 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. - The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between
**arithmetical**and geometrical measurement. - The association of algebra with arithmetic on the one hand, and with geometry on the other, presents difficulties, in that geometrical measurement is based essentially on the idea of continuity, while
**arithmetical**measurement is based essentially on the idea of discontinuity; both ideas being equally matters of intuition. - The difficulty first arises in elementary mensuration, where it is partly met by associating
**arithmetical**and geometrical measurement with the cardinal and the ordinal aspects of number respectively (see Arithmetic). - 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. **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.- 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. - General
**Arithmetical**Theorems. (i.) The fundamental laws of arithmetic should be constantly borne in mind, though not necessarily stated. - (i.)
**Arithmetical**Progressions such as 2, 5, 8, ... - (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. - - (i.) An expression such as a.2.a.a.b.c.3.a.a.c, denoting that a series of multiplications is to be performed, is called a monomial; the numbers (
**arithmetical**or algebraical) which are multiplied together being its factors. - 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. - - (i.) Special
**arithmetical**results may often be used to lead up to algebraical formulae. - (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. - (i.) The sum of the first n terms of an ordinary
**arithmetical**progression (a+b), (a+2b), ... - 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 use of negative coefficients leads to a difference between
**arithmetical**division and algebraical division (by a multinomial), in that the latter may give rise to a quotient containing subtractive terms. The most important case is division by a binomial, as illustrated by the following examples: - 2.10+1) 6.100+5.10+ 1(3.10+I 2.10+I) 6.100+I.10 - I (3.10 - I 6.100+3.10 6.100+3.10 2.10+ I - 2.10 - I 2.10 +I - 2.10 - I In (1) the division is both**arithmetical**and algebraical, while in (2) it is algebraical, the quotient for**arithmetical**division being 2.10+9. - - (i.) The results of the addition, subtraction and multiplication of multinomials (including monomials as a particular case) are subject to certain laws which correspond with the laws of arithmetic (ï¿½ 26 (i.)) but differ from them in relating, not to
**arithmetical**value, but to algebraic form. - 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. - 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.