Integers

“ Now, since a, B, b, A, are integers, so also is P; and thence Q; and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1 and +1, it follows that the unlimited succession of integers P, Q, R, are each less in numerical value than the preceding. ”

“ As a matter of fact, so long as we are content with the arithmetic of finite integers, and do not introduce either infinite integers or infinite classes or series of finite integers or ratios, it is possible to obtain all desired results without the axiom of infinity. ”

“ Among the even figures we include 0.
[12] Including both ciphers and others.
[13] For shortness, I abbreviate the words greatest common measure into their initial letters, g. c. m.
[14] Numbers which contain an exact number of units, such as 5, 7, 100, &c., are called whole numbers or integers, when we wish to distinguish them from fractions.
[15] A factor of a number is a number which divides it without remainder: thus, 4, 6, 8, are factors of 24, and 6 × 4, 8 × 3, 2 × 2 × 2 × 3, are several ways of decomposing 24 into factors. ”

“ All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry the child's mind along with them in learning arithmetic; all who wish to teach numbers, and not mere ciphers—now teach it through the evidence of the senses, in the manner we have described.
We may, if we please, call the proposition, "Three is two and one," a definition of the number three, and assert that arithmetic, as it has been asserted that geometry, is a science founded on definitions. ”

“ For we often do, and might always, verify the simplest numerical propositions, by reckoning up the numbers. The operations of addition, subtraction, multiplication, division, and extraction of roots, with all the most complex ones relating to algebraic quantities, considered as the exponents of numbers, are no more than methods of producing this coincidence of words, founded upon and rising above one another. ”

“ Arithmetic is the prose of Mathematics where everything is stated and nothing left to the imagination, but Algebra is very different. 'Known Numbers are usually represented by the first letters of the alphabet, as a, b, c. Unknown Numbers, or those whose values are to be determined, are usually represented by the last letters of the alphabet, as x, y, z.' The unknown numbers,—the mysterious numbers,—for what is unknown is in a measure mysterious and what is mysterious is romantic or poetic. ”

“ But the grand agent for transforming experimental into deductive sciences, is the science of number. The properties of numbers, alone among all known phenomena, are, in the most rigorous sense, properties of all things whatever. All things are not coloured, or ponderable, or even extended; but all things are numerable. And if we consider this science in its whole extent, from common arithmetic up to the calculus of variations, the truths already ascertained seem all but infinite, and admit of indefinite extension. ”

“ 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 the case of complex numbers, if they are defined as ordered couples of real numbers, we secure at once some of the properties required, namely, that two real numbers are required to determine a complex number, and that among these we can distinguish a first and a second, and that two complex numbers are only identical when the first real number involved in the one is equal to the first involved in the other, and the second to the second. What is needed further can be secured by defining the rules of addition and multiplication. ”

“ Number is a common and easily distinguishable property of all bodies and of all phenomena. By the process of grouping and counting we derive the fundamental laws concerning collections of bodies, such as the associative law, the distributive law, the permutative law, etc. Having secured its fundamental data from the physical world, mathematics withdraws, so to speak, into the realm of thought and, aided by reason alone, weaves its material by successive steps into those remarkable systems which we call arithmetic, algebra, the infinitesimal calculus, the theory of functions, etc. ”

“ In the third place, the real desideratum about such a definition as that of number is not that it should represent as nearly as possible the ideas of those who have not gone through the analysis required in order to reach a definition, but that it should give us objects having the requisite properties. Numbers, in fact, must satisfy the formulae of arithmetic ”

“ 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. ”

“ 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. ”

“ I shall now teach you how to multiply the unknown numbers, that is to say, the roots, one by the other, if they stand alone, or if numbers are added to them, or if numbers are subtracted from them, or if they are subtracted from numbers; also how to add them one to the other, or how to subtract one from the other.
Whenever one number is to be multiplied by another, the one must be repeated as many times as the other contains units. ”

“ 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. ”

“ From this last arrangement seems to have come the fundamental law of the decimal notation in which its superior utility consists, and upon which quite recently has been based the metric system of weights and measures. By placing any of the digits in the place of the zero to make the numbers between ten and twenty, we have the law established. The science of arithmetic, like all other sciences, was very limited and imperfect at the beginning, and the successive steps by which it has reached its present extension and perfection have been taken at long intervals, and among different nations. ”

“ While considering the extent of the savage's arithmetical knowledge, of his ability to count and to grasp the meaning of number, it may not be amiss to ask ourselves the question, what is the extent of the development of our own number sense? To what limit can we absorb the idea of number, with a complete appreciation of the idea of the number of units involved in any written or spoken quantity? Our perfect system of numeration enables us to express without difficulty any desired number, no matter how great or how small it be. ”

“ If we consider that we are dealing with numerical quantities, we must recognize the fact that, while addition and subtraction might in the first instance be limited to such quantities, multiplication and division necessarily introduce the idea of pure number. (ii) If on the other hand we regard ourselves as dealing with pure number throughout, then, as multiplication is continued addition, we ought to include in our classification involution as continued multiplication. ”

“ 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. ”

“ 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. ”

“ When we enter into the spirit of the methods of arithmetic, we perceive that they all flow clearly and simply from the very principles of numeration, from some precise definitions, and from certain ideas of relations between numbers, which all minds easily perceive, and which they even possessed in advance, before their teacher made them recognize them and taught them to class them in a methodical and fruitful order. ”

“ To have a really correct apprehension of the principle of decimal notation, for instance, to understand that it is purely arbitrary, and that we might in the same way take any other number than ten as the base of a numerical scale,—that we might increase for instance by fives, or eights, or nines, or twelves, just as well as by tens—all this requires considerable maturity of intellect, and some subtlety of reasoning. ”

“ For, in the first place, mastery by deductive reasoning of such matters as 'carrying' in addition, 'borrowing' in subtraction, the value of the digits in the partial products in multiplication, the manipulation of the figures in division, the placing of the decimal point after multiplication or division with decimals, or the manipulation of the figures in the multiplication and division of [Pg 71] fractions, is impossible or extremely unlikely in the case of children of the ages and experience in question. ”

“ How far every boy ought to be made to do mathematics (beyond simple arithmetic) I cannot say. I know that I am extremely grateful for the small amount of mathematics forced into me. I am even thankful for a very mechanical side of the subject, namely, the use of mathematical tables in general, and for being compelled to work out innumerable sums by logarithms, which we had to do in a “neat tabular form” to quote our precise master’s words. ”

“ 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. ”

“ Indeed I doubt whether many of the pretentious sciolists, who insist so much on young children giving the rationale of everything, have themselves ever yet made an ultimate analysis of the first step in arithmetical notation. Many of them would open their eyes were you to tell them, for instance, that the number of fingers on your two hands may be just as correctly expressed by the figures 11, 12, 13, 14, or 15, as by the figures 10,—a truism perfectly familiar to every one acquainted with the generalizations of higher arithmetic. ”

“ A logical reformulation The apples example can be reformulated as an experiment. Take a basket that contains a pair of apples and nothing else. Take a bucket that contains a pair of apples and nothing else. Empty the entire contents of the basket into the bucket taking care to make sure that everything that is in the basket goes into the bucket. Now how can we determine how many apples are in the bucket?
One way is to use the calculus of arithmetic. We can take the contents of the basket as an instantiation of the arithmetical notion "2". ”

“ He laughed at difficulties, whether in the science of magnitudes, that is algebra, or in the science of numbers, that is arithmetic; and it was a treat to see him handle the symbols, the conventional signs which form the algebraic notation, whether letters of the alphabet, representing quantities or magnitudes, or lines coupled or crossed, which indicate the relation between the quantities and the operations to which they are submitted. ”

“ Again, let none be alarmed at vast numbers and fractions, for in calculation it is as easy to set down or to reflect upon a thou­sand as a unit, or the thousandth part of an integer as an integer itself. ”

“ Any other origin is difficult to reconcile with observed facts, and still more difficult to reconcile with any rational theory of number system development. Dismissing from consideration the quinary scale, let us briefly examine once more the natural process of evolution through which the decimal and the vigesimal scales come into being. After the completion of one count of the fingers the savage announces his result in some form which definitely states to his mind the fact that the end of a well-marked series has been reached. ”