MULTIPLE, SUBMULTIPLE, MULTIPLICATION. Any num ber of equal magnitudes added together give a multiple of any ono among them. Thus 4 + 4 + 4, or 12, is ,a multiple of 4. And sub multiple is the inverse term to multiple : thus 12 being a multiple of 4, 4 is a sub-multiple of 12. The term submultiple is equivalent to ALIQUOT PART.
The derivation of the word is from multiplex, manifold, and multi plication is the process of forming a multiple. Thus to multiply 184 yards by 279 is to repeat 184 yards 279 times, and to add all tho results, together. And this is the first and fundamental meaning of multiplication. Its usual symbol is x : thus 4 x 3 is 12.
If we look at the primary rides of arithmetic, we shall see that multiplication is the only one which cannot be entirely performed upon concrete quantities. To or from 100 yards 50 yards can be added or subtracted, and 100 yards can be divided by 50 yards; but 100 yards cannot be multiplied by 50 yards. The very definition of multiplica tion requires that every question should contain a number of taws which another number, abstract or concrete, is to be repeated; and this number of times or repetitions cannot be a number of anything else. Thus, to talk of multiplying 10 feet by 7 feet is a contradiction in terms; if it mean that 10 feet is to be multiplied by 7, or that 7 repetitions of 10 feet are to be made, 10 feet is multiplied seven times, not seven feet times. But if it be meant that 10 feet is to be repeated as often as 7 feet contains one foot, the question bas three data, and belongs to a class which will be considered in PROPORTION : it is in fact a question of multiplication in which the number of repetitions is not given, but is to be extracted from the result of a question in division. On this subject see also RECTANGLE.
it being now distinctly understood that a number of times or repetitions is an essential element of every question of multiplication, the extension is obvious by which a fraction of a time, or a fraction of a repetition, is allowed to enter. Thus 12+12 +12+ 6 is 12 repeated three times and half a time, or 12 multiplied by 31 is 42. Similarly 21 + 21 +21 +1I, or Ai, is 21 taken 31 times. Up to this point there is no violation of etymology; the multiplicand (multiplicandunn, number to be multiplied) is taken manifold times. But [Nosreea] by the same sort of extension of language by which 1, and even 0, are called numbers, the mere exhibition of a multiplicand is called multiplying it by one : thus 7 is 7 taken once, or 7 multiplied by 1, though, cally, multiplication does not take place. Again, when the half of a number is taken, or when it is taken half a time, it is said to be multi plied by # ; and so on for any other fraction. The advantage of such
extension in practice more than counterbalances its obvious defect, namely, that the beginner must, without great care, be confused by the application of a word in a sense diametrically opposed to its literal meaning.
The abbreviated process of multiplication rests upon the following principles. (1.) if the parts of a number be multiplied, and the results added together, the whole is multiplied; thus 18, composed of 13 and 5, is taken 7 times by taking 13 and 5 each 7 times, and adding the results. (2.) Multiplication by the parts of any number, and addition of the results, is equivalent to multiplication by the whole : thus 13 taken 7 times and 8 times gives two products, the sum of which is 13 taken 7 + 8 or 15 times. c3.) Successive multiplication by two numbers is equivalent to one multiplication by the product of these two num bers: thus 7 taken 3 times, and the result taken 4 times, is 7 taken as many times as there are units in 4 times 3, or 12 times. (4.) If one number be multiplied by another, the result is the same if the multi plicand and multiplier be changed : thus 7 times 8 is the same thing as 8 times 7. (5.) In the decimal system, the annexing of one cipher multiplies by 10, of two ciphers by 100, The application of these principles requires that, in the decimal system of notation, the products of all simple digits up to 9 times 9 should be remembered : this is usually done by learning what is called the multiplication table; and this table, which is only absolutely neces sary up to 9 times 9, is usually committed to memory up to 12 times 12. This being supposed to be done, we shall now show the process of multiplying 1234 by 5073. By (2.) we must take 1234, 5000 times, 70 times, and 3 times, and add the results. To take 1234, 3 times, we subdivide it into 1000, 200, 30, and 4, each of which taken 3 times, and the results added together, gives 3000 600 90 12 3702 from which process the rule for multiplying by a single figure may easily be derived. The next step is to take 1234, 70 times, that is, first 7 times, and the result 10 times. The full process is 7000 1400 210 28 86380 Similarly 1234 taken 5000 times, gives 6,170,000. Now put the three results together, and add them; which gives the first column follow ing :I- 3702 1234 86380 5073 6170000 3702 6260082 86386170 6260082 The second column shows the usual manner of performing the operation, which we suppose the reader to know. We have given the preceding detail that he may do what many have never done, namely, compare the common process with the deduction of the result from first principles.