Arithmetic progression

“ Moreover, the general laws of numerical combination appear much more clearly if the signs are kept, since the result is immediately seen to be composed of the participating members. Thus, algebra, that is, calculation with abstract or general quantities, has developed as an extensive and important field of general mathematics.
By the theory of numbers we understand the most general part of arithmetic which treats of the properties of the "numerical bodies" formed in some regular way. ”

“ Logarithms were originally invented for the sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and division may be replaced by those of addition and subtraction, and the operations of raising to powers and extraction of roots by those of multiplication and division. For the purpose of thus simplifying the operations of arithmetic, the base is taken equal to 10, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated. ”

“ All such careful arithmetical computations add clearness and definiteness to historical and geographical ideas. The natural sciences have been so little systematically taught in our common schools, that we are scarcely able to realize what connection may be made between them and arithmetic. ”

“ As to the main point, however, there is no doubt, that modern arithmetic comes out of ancient counting on the columns of the abacus, improved by writing a dot or a round 0 to show the empty column, and by this means young children now work calculations which would have been serious labour to the arithmeticians of the ancient world. ”

“ While the putting of arithmetic after geometry and algebra may excite the greater surprise, it is practically the most defensible part of the whole inversion. The most important processes of advanced arithmetic are only explainable on algebraic or geometric grounds. ”

“ And, indeed, it is the distinctive peculiarity of the calculus in general, that it proceeds on the hypothesis of that absolute equality of its abstract units, which no real units possess; and that the exactness of its results holds only in virtue of this hypothesis. The first ideas of number must necessarily then have been derived from like or equal magnitudes as seen chiefly in organic objects; and as the like magnitudes most frequently observed were magnitudes of extension, it follows that geometry and arithmetic had a simultaneous origin. ”

“ Why not instead adopt the easier course of simply trying to translate each of the applications of mathematics?" In order to give the empirical/realist account we need not say that every statement of arithmetic is induced from observations of real world quantities. Nor need we say that the system of arithmetic is open to falsification, but is not in fact never falsified by the observation of real world quantities. ”

“ The principle is obvious enough and would probably be admitted in theory by any intelligent teacher, but in practice we are still wedded to conventional usages which arose long before the psychology of arithmetic was studied. For example, we inherit the convention of studying addition of integers thoroughly, and then subtraction, and then multiplication, and then division, and many of us follow it though nobody has ever given a proof that this is the best order for arithmetical learning. ”

“ The transformation of functions belongs strictly to algebra. Teachers who love arithmetic, and who have themselves success in working out the so-called numerical conundrums, defend with much earnestness the current practice which uses so much time for arithmetic. They see in it a valuable training for ingenuity and logical analysis, and believe that the industry which discovers arithmetical ways of transforming the functions given in such problems into plain numerical operations of adding, subtracting, multiplying, or dividing is well bestowed. ”

“ It is a useful exercise to multiply and divide numbers represented in other scales of notation than the common or decimal one. The rules are in all respects the same for all systems, the number carried being always the radix of the system. Thus, in the quinary system we carry fives instead of tens. I now give an example of multiplication and division: [Pg 170]
Another way of turning a number from one scale into another is as follows: Multiply the first digit by the old radix in the new scale, and add the next digit ”

“ Results led to the conclusion that arithmetical ability tends to be represented in two main divisions: (1) the power to compute with ease and readiness, and (2) the power to solve problems by arithmetic, which involves the application of a higher degree of ability than is required in computation.
Arithmetical tests of various kinds correlate more closely than do arithmetical tests with non-arithmetical tests. “Hence we are compelled to interpret this relationship as evidence distinctly in favor of Burt’s suggestion, that there is an essential unity in arithmetical ability.” ”

“ At the time when this work was first published, the importance of establishing arithmetic in the young mind upon reason and demonstration, was not admitted by many. The case is now altered: [Pg vi] schools exist in which rational arithmetic is taught, and mere rules are made to do no more than their proper duty. There is no necessity to advocate a change which is actually in progress, as the works which are published every day sufficiently shew. ”

“ And what is the nature of the mental process by which numbers are found capable of having their relations expressed algebraically? It is just the same. It is the formation of certain abstract conceptions of numerical functions which are the same whatever be the magnitudes of the numbers. It is the invention of general symbols serving to express the relations between numbers, as numbers express the relations between things. And transcendental analysis stands to algebra in the same position that algebra stands in to arithmetic. ”

“ In seeking to determine with precision in what determinations of values properly consist, we easily recognize that they are nothing else but veritable transformations of the functions to be valued; transformations which, in spite of their special end, are none the less essentially of the same nature as all those taught by analysis. In this point of view, the calculus of values might be simply conceived as an appendix, and a particular application of the calculus of functions, so that arithmetic would disappear, so to say, as a distinct section in the whole body of abstract mathematics. ”

“ The £1 is termed the unit. A numerical quantity, therefore, represents a certain unit, taken a certain number of times. If we take £3 twice, we get £6; and if we take 3s. twice, we get 6s., i.e. 6 times 1s. Thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout. If we retain the unit, the arithmetic is concrete; if we ignore it, the arithmetic is abstract. But in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced. ”

“ The celebrated interminable fraction 3.14159..., which the mathematician calls , is the ratio of the circumference to the diameter. But it is thousands of things besides. It is constantly turning up in mathematics: and if arithmetic and algebra had been studied without geometry, must have come in somehow, though at what stage or under what name must have depended upon the casualties of algebraical invention. ”

“ IN this age the knowledge of arithmetic is so widespread that it is difficult to conceive that it is comparatively young. The three R's, Reading, 'Riting and 'Rithmetic, seems to our mind almost axiomatic and fundamental in regard to education. It is, therefore, very interesting to read in histories of arithmetic that the earliest known book which contains a systematic exposition of the decimal system of numeration was written in the seventh century, and that our familiar arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, can not be traced to an earlier century than the seventh. ”

“ In the ordinary arithmetic the numbers or figures employed, taken separately, have always the same value, and the result, when, sometimes by a tedious process, obtained, is applicable only to the particular question proposed, but in solving the problem by algebra, since we employ letters to which any values may be attributed at pleasure, the result obtained is largely applicable to all questions of a particular class. ”

“ In this point of view the numerical solution of questions which we cannot resolve algebraically, and even the calculation of "Definite Integrals," whose general integrals we do not know, really make a part, in spite of all appearances, of the domain of arithmetic, in which we must necessarily comprise all that which has for its object the determination of the values of functions. ”

“ Vital problems as nuclei around which to organize arithmetical learning are of prime importance. It is even safe probably to insist that some genuine problem-situation requiring a new process, such as addition with carrying, multiplication by two-place numbers, or division with decimals, be provided in every case as a part of the introduction to that process. The sacrifice should not be too great, however; the search for vital problems that fit an economical order of subject matter is as much needed as the amendment of that order to fit known interests ”

“ But in the series, every term is only the half of the preceding; consequently no number of terms, however great, can be made as great as 2 by adding one more. The sum, therefore, of 1, 1⁄2, 1⁄4, 1⁄8 &c. continually approaches to 2, diminishing its distance from 2 at every step, but never reaching it. Hence, 2 is celled the limit of 1 + 1⁄2 + 1⁄4 + &c. We are not, therefore, to conclude that every series of decreasing terms has a limit. ”

“ What we have obtained is merely the concept of number. The individual members may be chosen quite arbitrarily. They need in no way be equal. Each number represents a quantity type; and it is the sphere of arithmetic to examine these different types in respect to subdivision and combination. If this be done without considering the amount of the number, we call the corresponding science algebra. On the other hand, the extension of formal rules beyond their original application has led to one development of numbers after the other. ”

“ It is possible, as one knows from experience, for a boy to have mastered all the arithmetical rules and tables with which his master has supplied him, and to have all his measures and weights at his fingers' ends, and yet to be so destitute of the arithmetical sense as to give without a moment's misgiving an entirely nonsensical answer to a simple arithmetical problem,—to say, for example, as I have known half a class of boys say, that a room is five shillings and sixpence wide. Such a boy, though his head may be stuffed with arithmetical information, has no knowledge of arithmetic. ”

“ Efforts to arrive at a much more accurate definition of the term arithmetic are apt to meet with disappointment. On the one hand, we meet with contradictory classifications among works of the highest authority. The great mathematical encyclopedia which is being published almost simultaneously in German and French includes determinants under arithmetic, while the International catalogue of scientific literature places determinants under algebra. ”

“ Although only those gifted with high mathematical powers could have originated the minute and abstruse symbolical reasoning at the source of the methods whereby the inter-relationship of these phenomena have been measured and calculated, yet the application of these methods, and of the formulæ which have now been provided, are open to any intelligent worker who has knowledge of arithmetic and of simple mathematics, and the computer's zeal for precision and accuracy. ”

“ They should not include any elaborate practice with very large numbers, or decimals beyond thousandths, or the addition and subtraction of fractions which not one person in a hundred has to add or subtract oftener than once a year.
When we have an adequate sociology of arithmetic, stating accurately just who should use each arithmetical ability and how often, we shall be able to define the task of the elementary school in this respect. ”

“ When infinite numbers are first introduced to people, they are apt to refuse the name of numbers to them, because their behaviour is so different from that of finite numbers that it seems a wilful misuse of terms to call them numbers at all. In order to meet this feeling, we must now turn to the logical basis of arithmetic, and consider the logical definition of numbers. ”

“ All we need to say is that in any system for the calculation of real world quantities that employs logically true and non-falsifiable statements of arithmetic these statements can be replaced by a statement or statements that are not non-falsifiable statements of arithmetic. ”

“ On the contrary, the premisses of arithmetic can be put in other forms, and, furthermore, an indefinite number of propositions of arithmetic can be proved directly from logical principles without mentioning them. Thus, while arithmetic may be defined as that branch of deductive reasoning concerning classes and relations which is concerned with the establishment of propositions concerning cardinal numbers, it must be added that the introduction of cardinal numbers makes no great break in this general science. It is no more than an interesting subdivision in a general theory. ”