13 × 10 = 130 13 × 100 = 1300 13 × 1000 = 13000 13 × 10000 = 130000 142 × 1000 = 142000 23700 × 10 = 237000 3040 × 1000 = 3040000 10000 × 100000 = 1000000000 58. I now shew how to multiply by one of the numbers, 2, 3, 4, 5, 6, 7, 8, or 9. I do not include 1, because multiplying by 1, or taking the number once, is what is meant by simply writing down the number. I want to multiply 1368 by 8. Write the first number at full length, which is 1 thousand, 3 hundreds, 6 tens, and 8 units. To multiply this by 8, multiply each of these parts by 8 (50) and (52), which will give 8 thousands, 24 hundreds, 48 tens, and 64 units. Now 64 units are written thus 64 48 tens 480 24 hundreds 2400 8 thousands 8000 Add these together, which gives 10944 as the product of 1368 and 8, or 1368 × 8 = 10944. By working a few examples in this way you will see for following rule. 59. I. Multiply the first figure of the multiplicand by the multiplier, write down the units’ figure, and reserve the tens. II. Do the same with the second figure of the multiplicand, and add to the product the number of tens from the first; put down the units’ figure of this, and reserve the tens. III. Proceed in this way till you come to the last figure, and then write down the whole number obtained from that figure. IV. If there be a cipher in the multiplicand, treat it as if it were a number, observing that 0 × 1 = 0, 0 × 2 = 0, &c. 60. In a similar way a number can be multiplied by a figure which is accompanied by ciphers, as, for example, 8000. For 8000 is 8 × 1000, and therefore (55) you must first multiply by 8 and then by 1000, which last operation (57) is done by placing 3 ciphers on the right. Hence the rule in this case is, multiply by the simple number, and place the number of ciphers which follow it at the right of the product. EXAMPLE. Multiply 1679423800872 by 60000 ------------------ 100765428052320000 61. EXERCISES. What is 1007360 × 7? _Answer_, 7051520. 123456789 × 9 + 10 and 123 × 9 + 4?--_Ans._ 1111111111 and 1111. What is 136 × 3 + 129 × 4 + 147 × 8 + 27 × 3000?--_Ans._ 83100. An army is made up of 33 regiments of infantry, each containing 800 men; 14 of cavalry, each containing 600 men; and 2 of artillery, each containing 300 men. The enemy has 6 more regiments of infantry, each containing 100 more men; 3 more regiments of cavalry, each containing 100 men less; and 4 corps of artillery of the same magnitude as those of the first: two regiments of cavalry and one of infantry desert from the former to the latter. How many men has the second army more than the first?--_Answer_, 13400.