Multiplication

“ 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. ”

“ Properly speaking, this sum cannot be done.
Multiplication is merely a contracted form of addition: it means taking a number or quantity a certain number of times. Every multiplication can be proved by addition. All numbers are abstract or concrete—3 is abstract, £3 is concrete. ”

“ To do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects. If we want to give, to 5 boys, 4 apples each, we are said to multiply 4 apples by 5. We cannot multiply 4 apples by 5 boys, for then we should get 20 “boy-apples,” an expression which has no meaning. Or, again, to distribute 20 apples amongst 5 boys, we are not regarded as dividing 20 apples by 5 boys, but as dividing 20 apples by the number 5. The multiplication or division here involves the omission of the unit “boy,” and the operation is incomplete. ”

“ In the last analysis, division is only the obverse side of multiplication, and any one who has not learned his multiplication table is reduced to some such expedient as that of the Egyptian. Indeed, whenever we pass beyond the range of our memorized multiplication table-which for most of us ends with the twelves—the experimental character of the trial multiplication through which division is finally effected does not so greatly differ from the experimental efforts which the Egyptian was obliged to apply to smaller numbers. ”

“ You subtract thing from ten; there remain ten less thing. Multiply the one by the other, it is ten things less a square. This is the product of the multiplication of one of the two parts by the other. At present you divide this by the difference between the two parts, which is ten less two things. The quotient of this division is, according to the statement, five and a fourth. If, therefore, you muliply five and one-fourth by ten less two things, the product must be equal to the above amount, obtained by multiplication, namely, ten things less one square. ”

“ In teaching mental arithmetic, too, for I would certainly teach it to some extent, I would suggest the expediency of teaching children, in performing these mental operations, to think in figures, in other words, to form conceptions of the arithmetical figures and signs, which are visible objects, rather than of quantities and relations, which are mere abstractions. Multiplication is a mere metaphysical entity. The sign of multiplication is a simple, visible symbol, addressed to the eye, and capable of being conceived by the mind with unmistakable clearness and precision. ”

“ But we soon perceive that, while all multiplication implies the possibility of treating any number whatever as a provisional unit which can be added to itself, inversely the units in their turn are true numbers which are as big as we like, but are regarded as provisionally indivisible for the purpose of compounding them with one another. ”

“ To Prove a Subtraction.—Add from bottom to top the difference and the subtrahend; the sum must equal the minuend.
Multiplication.
Multiplication.—An operation which consists in repeating a number called multiplicand (M) as many times as there are units in another column called multiplier (m) ; the result is called the product (p) of the numbers, and the numbers themselves are called factors of the product. This definition may be extended to the case where the factors are not whole numbers. ”

“ Our present method was known specifically as Italian division, and was not introduced until the beginning of the last century.
One writer on arithmetic, a pious monk, furnishes a good illustration of mediæval logic. He is embarrassed by the usage and meaning of the term "multiplication" in the case of fractions in which the product is less than the multiplicand, and he proposes the question, "Whether the multiplication of fractions is an increasing process?" ”

“ And though in some things, as in numbers, besides adding and subtracting, men name other operations, as multiplying and dividing, yet they are the same; for multiplication, is but adding together of things equal ”

“ After thousandths, come ten thousandths, hundred thousandths, mil- Konths, &c. but for ordinary uses we seldom come down to so small a fraction. Multiplication is performed as if there were no comma ; but in the sum total, as many figures must be cut off at the right hand of the comma, as are cut off in both the multiplier and sum multiplied. ”

“ 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. ”

“ For elementary work the multiplicand may come immediately after the multiplier, as in D; the last figure of each partial product then comes immediately under the corresponding figure of the multiplier. A better method, which leads up to the multiplication of decimals and of approximate values of numbers, is to place the first figure of the multiplier under the first figure of the multiplicand, as in E; the first figure of each partial product will then come under the corresponding figure of the multiplier. ”

“ Within them is embraced the whole of computation; but because every number, every aggregation of units, is also a new unit capable of being added or subtracted, there are also the operations of multiplication and division, which consists in one case of the addition of several equal numbers together, and in the other, of the subtraction of several equal numbers from a greater until that is exhausted. ”

“ The development of these various operations followed logically on the practical exercise in the store, where multiplication proved to be a product of sums, division, a process of successive subtractions.
In our classes we have arithmetic lessons every day. ”

“ Division is the opposite of multiplication. In multiplication you have a number of heaps, with the same number of pebbles in each, and you want to know how many pebbles there are in all. In division [Pg 35] you know how many there are in all, and how many there are to be in each heap, and you want to know how many heaps there are. ”

“ Multiplication.We should next proceed to continued addition or multiplication. Many children come to school not knowing that multiplication is continued addition, and still fewer have any idea that division is continued subtraction. ”

“ I answer, that multiplication is the efficient cause of agriculture, though I allow, that, in the infancy of society, the spontaneous fruits of the earth, which are free to all, are the efficient cause of a multiplication, which may rise to the exact proportion of them ”

“ So that any mere multiplication by a constant reappears as a mere multiplication when the thing is differentiated. And, what is true about multiplication is equally true about division: for if, in the example above, we had taken as the constant instead of , we should have had the same come out in the result after differentiation. ”

“ Division is defined as the operation inverse to multiplication; but begin by an example taken from the familiar notion of partition and show on this example that multiplication reproduces the dividend.
There still remain the operations on fractions. The only difficulty is for multiplication. It is best to expound first the theory of proportion; from it alone can come a logical definition ”

“ Just as the addition combinations function mainly in the higher-decade modifications of them, so the multiplication combinations function chiefly in the cases where the bond has to operate while the added tasks of keeping one's place, adding what has been carried, writing down the right figure in the right place, and holding the right number for later addition, are also taken care of. It seems best to introduce such short multiplication as soon as the × 5s, × 2s, × 3s, and × 4s are learned and to put the × 6s, × 7s, and the rest to work in such short multiplication as soon as each is learned. ”

“ It makes no difference what the character of the man was who made the first multiplication table. It is absolutely true, and whenever you find an absolute fact, it makes no difference who discovered it. ”

“ I had been to school most all the time and could spell and read and write just a little, and could say the multiplication table up to six times seven is thirty-five, and I don’t reckon I could ever get any further than that if I was to live forever. I don’t take no stock in mathematics, anyway. ”

“ Propose to a person of the company to multiply, by whatever number he pleases, one of the three sums which you will give him on a piece of paper; desire him to strike out whatever figure he pleases of the product of his multiplication, let him change and invert the order of the remaining figures after the defalcation he has chosen. ”

“ The propriety of the name consists in the fact that the product is of less degree than either of the factors. For the contrary reason, the ordinary external or combinatorial multiplication is sometimes called by Grassmann progressive. Regressive multiplication is associative and exhibits a very remarkable analogy with the progressive. This analogy I have not time here to develop, but will only remark that in this analogy lies in its most general form that celebrated principle of duality, which appears in various forms in geometry and certain branches of analysis. ”

“ What I have said will suffice, I hope, as an answer to Balbus, who holds that (a) and (c) are the only possible varieties of the problem, and that to say "We cannot use addition, therefore we must be intended to use multiplication," is "no more illogical than, from knowledge that one was not born in the night, to infer that he was born in the daytime"; and also to Fifee, who says "I think a little more consideration will show you that our 'error of adding the proportional numbers together for each candidate instead of multiplying' is no error at all." ”

“ You will find it quite impossible to get any smaller result.
Now, my puzzle is to find the largest possible result. Divide the blocks into any two groups of five that you like, and arrange them to form two multiplication sums that shall produce the same product and the largest amount possible. That is all, and yet it is a nut that requires some cracking. Of course, fractions are not allowed, nor any tricks whatever. The puzzle is quite interesting enough in the simple form in which I have given it. Perhaps it should be added that the multipliers may contain two figures. ”

“ It is evident that by the combination of the operations of indeterminate, algebraic, and combinatorial multiplication we obtain multiple quantities of a more complicated nature than by the use of only one of these kinds of multiplication. The indeterminate product of combinatorial products we have already mentioned. The combinatorial product of algebraic products, and the indeterminate product of algebraic products, are also of great importance, especially in the theory of quantics. ”

“ Counting is fundamental, and counting that is spontaneous, free from sensible observation, and from the strain of reason. A study of these original methods shows that multiplication was developed out of counting, and not from addition as nearly all textbooks treat it. Multiplication is counting. When children count by 4's, etc., they accent the same as counting gymnastics or music. When a child now counts on its fingers it simply reproduces a stage in the growth of the civilization of all nations. ”

“ Section 43 If the first term is multiplied by the fourth term, it will be equal to the number returned by multiplying the second term by the third term, like so: 2:3 = 4:6 has extremes whose multiplication produce 12 as well as means whose multiplication produce 12. ”

“ Similarly where multiplication has arisen out of the subdivision of a unit into smaller units, we can again subdivide these smaller units. Thus we get successive multiplication; but it represents quite different operations according as it is due to repetition, in the sense of § 34, or to subdivision, and these operations will be exhibited by different diagrams. Of the two diagrams below, A exhibits the successive multiplication of £3 by 20, 12 and 4, and B the successive reduction of £3 to shillings, pence and farthings. ”

“ If any two quantities be made of the mutual multiplication of many quantities, which have proportion to one another, and the efficient quantities on both sides be equal in number, the proportion of the products will be compounded of the several proportions, which the efficient quantities have to one another. ”

“ It is, that we may sometimes know a general proposition in cases where we do not know a single instance of it. Take such a case as the following: We know that any two numbers can be multiplied together, and will give a third called their product. We know that all pairs of integers the product of which is less than 100 have been actually multiplied together, and the value of the product recorded in the multiplication table. ”

“ Multiplication teaches a short way of adding one number together any number of times. Its sign is a cat o'-nine-tails; its symbol a whipping-post. Since the wonderful powers of the number nine have been publicly discussed, we have had no more shooting at her Majesty, (Heaven preserve her!) which shows the transcendant powers of arithmetical argument. The Egyptian plague of frogs and flies exemplifies this rule. In Modern Rome we have multiplication of fleas. In Modern Babylon we have multiplication of bugs, particularly humbugs. ”