100. Special Cases.—The following are some special rules:—
(i) To multiply by 5, multiply by 10 and divide by 2. (And conversely, to divide by 5, we multiply by 2 and divide by 10.)
(ii) In multiplying by 2, from the left, add 1 if the next figure of the multiplicand is 5, 6, 7, 8 or 9.
(iii) In multiplying by 3, from the left, add 1 when the next figures are not less than 33 ... 334 and not greater than 66 ... 666, and 2 when they are 66 ... 667 and upwards.
(iv) To multiply by 7, 8, 9, 11 or 12, treat the multiplier as 10 − 3, 10 − 2, 10 − 1, 10 + 1 or 10 + 2; and similarly for 13, 17, 18, 19, &c.
(v) To multiply by 4 or 6, we can either multiply from the left by 2 and then by 2 or 3, or multiply from the right by 4 or 6; or we can treat the multiplier as 5 − 1 or 5 + 1.
101. Multiplication by a Large Number.—When both the numbers are large, we split up one of them, preferably the multiplier, into separate portions. Thus 231·4273 = (200 + 30 + 1)·4273 = 200·4273 + 30·4273 + 1·4273. This gives the partial products, the sum of which is the complete products. The process is shown fully in A below,—
and more concisely in B. To multiply 4273 by 200, we use the commutative law, which gives 200·4273 = 2 × 100 × 4273 = 2 × 4273 × 100 = 8546 × 100 = 854600; and similarly for 30·4273. In B the terminal 0’s of the partial products are omitted. It is usually convenient to make out a preliminary table of multiples up to 10 times; the table being checked at 5 times (§ 100) and at 10 times.
The main difficulty is in the correct placing of the curtailed partial products. The first step is to regard the product of two numbers as containing as many digits as the two numbers put together. The table of multiples will them be as in C. The next step is to arrange the multiplier and the multiplicand above the partial products. 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 multipler 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.