There are several abbreviations of multiplication which are very valuable, but which are not commonly taught.
1. Five times is half of ten times : to multiply by 5 annex a cipher and divide by 2 : thus 76783 x 5 is most easily done as follows : 2) 767830 383915 2. Nine times is one less than ten times, so that 76783 x 9 can be found as fellows: 767830 76783 691047 This may be best done by subtracting every figure of the multipli cand from the preceding, carrying and borinwing where necessary, in the usual way, on the supposition that the first figure is to be sub tracted from ten. Thus the process of multiplying 27293 by 9 is as follows : 27293 245637 3 from 10, 7, carry I ; 1 and 9 is 10, 10 from 13, 3, carry I ; 1 and 2 is 3, 3 from 9, 6; 7 from 12, 5, carry 1 ; 1 and 2 is 3, 3 from 7, 4 ; 0 from 2, 2.
3. Eleven times is one more than ten times ,-• so that the addition corresponding to the preceding subtraction must be made. Thus to multiply 62781 by 11, proceed as follows : 62781 690591 Let 1 remain ; 1 and 8 is 9 ; 8 and 7 is 15, carry 1; 1 and 7 is 8 and 2 are 10, carry 1 ; and 2 is 3 and 6 are 9 ; 6 and 0 is 6.
4. To multiply by any number from 12 to 19 inclusive, multiply by the last figure, i and to the carrying figure add the figure of the multi plicand which s just done with. Thus 2734 17 46478 7 times 4 is 28, carry 2, adding 4, or carry 6 ; 7 x 3 is 21, and (3 is 27, carry (2 + 3 or) 5 ; 7 x 7 is 49 and 5 is 54, carry (5+7 or) 12 ; 7 x 2 is 14 and 12 is 26, carry (2 + 2 or) 4.
5. To multiply by 25, annex two ciphers and divide by 4 : to multiply by 125 annex three ciphers and divide by 8.
6. in multiplying by a number of two figures, ending with 7 or 8, as 68, it may be advisable to take the multiplicand 70 times, and subtract it twice, in preference to taking it 60 times, and adding it 8 times. The following rules are taken from the Itisala Hisab.' (Taylor's
Liliwati; Introduction, p. 17.) The first at least can easily be done without paper.
1. To multiply two numbers together, each of which is between 11 and 19; to the whole of one number add the units of the other ; ten times this, together with the product of the units' places, is the product required. Thus, 17 times 14 is 21 times 10 and 28, or 238.
2. To multiply two numbers together, each of which has only two places : to the whole of one factor, multiplied by the tens of the other, add the tens of that factor multiplied by the units of the other ; ten times the result, together with the product of the units, is the product required. Thus 76 x 38 is done as follows: 76 x 3 is 228, which in creased by 7 x 8, or 56, is 284, and 2840 increased by 48 is 2888, the answer required.
The multiplication of sums of money is facilitated by a process known by the name of Prtaceice.
The multiplication of fractions offers no difficulty when the extension of the word multiplication, already described, is understood and admitted. For instance, when we have to multiply ; by or to take / 4-elevenths of a time, we see that / being [Faacetosrs], one-eleventh of this is A, and 4-elevenths is A : whence the rule commonly given, namely, multiply the numerators together for a numerator, and the denominators fora denominator. In the multiplication of one decimal fraction by another, as 1.23 by .018, the multiplication of the nume rators gives 123 x 18, or 2214, and that of the denominators 100 x 1000, or 100,000. But a decimal fraction which has 100,000 for its denomi nator, has as many places as there are in both of the others together, whose denominators are 100 and 1000. From this consideration the common rule immediately follows.
For a mechanical contrivance for expediting multiplication, see NAPIER'S RODS.