MULTIPLICATION.
In multiplication two numbers are given, and it is required to find how much the first amounts to, when reckon ed as many times as there are units in the second. Thus 8 multiplied by 5, or 5 times 8, is 40. The given numbers (8 and 5) are called factors ; the first (8) the multiplicand; the second (5) the multi plier ; and the amount (40) the product. This operation is nothing else than addi tion of the same number several times re peated. If we mark 8 five times under each other, and add them, the sum is 40 : but as this kind of addition is of frequent and extensive use, in order to shorten the operation, we mark down the number only once, and conceive it to be repeated as often as there are units in the multipli er. For this purpose, the learner must be thoroughly acquainted with the follow ing multiplication table, which is com posed by adding each digit 12 times.
In this table the multiplicand figures are in the upper horizontal row ; the multipliers are in the left hand column, and the products will be found under the multiplicand, and in the same row with the multiplier ; thus 9 times 11 are 99; and 99 will be found in the column under the 11, and in the same horizontal row with the 9, among the multipliers.
If both factors be under 12, the table exhibits the product at once. If the mul tiplier only be under 12, we begin at the unit place, and multiply the figures in their order, carrying the tens to the higher place, as in addition.
Example.
76859 multiplied by 4 4 Ans. 307436 or, 76859 added 4 times.
76859 76859 76859 Ans. 307436 the same as before.
If the multiplier be 10, we annex a cy pher to the multiplicand. if the multi plier be 100, we annex two cyphers ; and so on. The reason is obvious, from the use of cyphers in notation. if the multi plier be any digit, with one or more cy phers on the right hand, we multiply by the figure, and annex an equal number of cyphers to the product.
Thus, if it be required to multiply by 60, we first multiply by 6, and then annex a cypher. It is the same thing as to add the multiplicand 60 times ; and this might be clone by writing the account at large, dividing the column into 10 parts of 6 lines, finding the sum of each part, and adding these ten sums together. If the multiplier consist of several significant figures, we multiply separately by each, and add the products. It is the same as if we divided a long account of Addition into parts corresponding to the figures of the multiplier.
Example.
To multiply 7329 by 365 7329 7329 7329 5 60 30036645 439740 2,98700 36645 = 5 times. 439740 = 60 times. 2198700 = 300 times. 2675085 = 365 times.
It is obvious that 5 times the multipli cand added to 60 times, and to 300 times, the same must amount to the product re. quired. In practice, we place the pro. ducts at once under each other; and as the cyphers arising from the higher places of the multiplier are lost in the addition, we omit them. Hence may be inferred the following Rule. Place the multiplier under the
multiplicand, and multiply the latter suc cessively by the significant figures of the former by placing the right-hand figure of each product under the figure of the multiplier from which it arises ; then add the product.
Example.
7329 93956 365 870436645 875824 43974 657692 21987 751648 2675085 817793024 A number, which cannot be produced by the multiplication of two others, is called a prime number; as 3, 5, 7, 11, and many others. A number, which may be produced by the multiplication of two or more smaller ones, is called a compo site number. For example, 27, which arises from the multiplication of 9 by 3 ; and these numbers (9 and 3) are called the component parts of 27.
1. If the multiplier be a composite number, we may multiply successively by the component parts.
Example.
7638 by 45, or 5 times 9, 7638 45 9 38190 68742 30552 5 343710 343710 Because the second product is equal to five times the first, and the first is equal to nine times the multiplicand, it is obvi, ous that the second product must be five times nine, or forty-five times as great a: the multiplicand.
2. If the multiplier be 5, which is the half of 10, we may annex a cypher, itri divide by 2. If it be 25, which is the fourth part of 100, we may annex twc cyphers, and divide by 4. Other con tractions of the like kind will readily oc cur to the learner.
3. To multiply by 9, which is one les. than 10, we may annex a cypher, and sub tract the multiplicand from the numbe. it composes. To multiply by 99,999, any number of 9's, annex as many cy phers, and subtract the multiplicand The reason is obvious ; and a like rub may be found, though the unit place be different from 9. Multiplication is prove( by repeating the operation, using the multiplier for the multiplicand, and the multiplicand for the multiplier. It may also be proved by division, or by casting out the 9's ; of which afterwards: and an account, wrought by any contraction, may be proved by performing the opera tion at large, or by a different contraction.
The following examples will serve to exercise a learner in this rule.
1. Multiply 87945 by 2 2. 947321 by 3. . . . 7914735 by 4 4. . . . 49782147. by 5 5. . . . 5721321 by 6 6. . . . 4794321 by 7 7. . . . 7654319 by 8 8. . . . 3721478 by 9 9. . . . 4783219 by 11 10. . . . 4733218 by 12 11. . . . 4783882 by 21 12. . . . 2179929 by 32 13. . . . 921577394 by 84 14. . . . 217431473 by 132 15. . . . 47314796 by 144 16 . . . 217932173 by 96 17. . . . 314731271 by 121 18. . . . 4796427 by 238 19. . . . 470621472 by 432 20. . . . 479621473 by 453 21. . . . 479632179 by 473 22. . . 473457896 by 963 23. . . . 943446788 by 987 24. . . 49416739 by 298 25. . . . 479327 by 403 26. . .. . 493 2149 by 3028 27. . . 4731214 by 3008 28. . . . 49496 by 4009 29. . . . 97213217 by 904 30. . . . 49521729 by 706 31. . . . 4932920 by 720 32. . . . 493679310 by 970 33. . . . 7893470 by 760 34. . . . 479379340 by 900