The sign, placed between two numbers, denotes that they are to be multiplied together. It is called, the sign of multiplication. MULTIPLICATION TABLE. 8 5 times 5 times 2 times. 4 are 2 times 5 are 10 2 times 6 are 12 2 times 7 are 14 2 times 8 are 16 2 times 9 are 18 2 times 10 are 20 2 times 11 are 22 2 times 12 are 24 3 times 0 are 3 times 1 are 3 times 2 are 3 times 3 are 3 times 4 are 12 3 times 5 are 15 3 times 6 are 18 3 times 7 are 21 3 times 8 are 24 times 9 are 27 es 10 are 30 a 11 are 33 12 are 36 0 3 6 9 8 times 6 are 48 8 times 7 are 56 8 are 64 72 0/11 times 0 are 0112 times 0 are 1 are 11 12 times 1 are 12 2 are 22 12 times 2 are 24 3 are 33 12 times 3 are 36 4 are 44 12 times 4 are 48 5 are 55 12 times 5 are 60 10 times 0 are 10 times 1 are 10 times 2 are 10 times 3 are 10 times 4 are 10 times 5 are 10 times 6 are 10 times 7 are 10 times 8 are 10 times 9 are 10 times 10 are 100 11 times 10 are 110 12 times 10 are 120 10 times 11 are 110 11 times 11 are 121 12 times 11 are 132 10 times 12 are 120 11 times 12 are 132 12 times 12 are 144 Ex. 1. Let it be required to multiply 4 by 2. Here 4 is the multiplicand, and 2 is the multiplier, and it is required to find the product, which is the number arising from repeating 4 two times. The product of 4 by 2 is found by multiplication, or by adding two 4's together. Multiplicand. 4 4× 28 Product. Ex. 2. Let it be required to multiply 4 by 3, and also to multiply 5 by 3.. From these examples we see, that the product of 4 multiplied by 3 is 12, the number which from adding three 4's together; and that the of 5 by 3 is equal to 15, the number which arises from adding three 5's together. 5 If 3 be multiplied by 5 the product is still 15, as before. 1 1 1 1 1 For, when 5 is the multipli1 1 1 1 1 cand and 3 the multiplier, the 1 1 1 1 1 product is equal to the 5 units in the upper horizontal line taken 3 times; that is, equal to 15, all the units in the 3 lines. And when 3 is the multiplicand and 5 the multiplier, the product is equal to all the units in the 5 vertical rows, which is 15. 30. Therefore, either of the factors may be used as the multiplier without altering the product. We see from the above examples, that any product may be found by setting down the multiplicand as many times as there are units in the multiplier, and adding all the numbers together. § 31. Multiplication is therefore a short method of addition. 236 4 24 units. 12 tens. 8 hundreds. Ex. 3. Multiply 236 by 4. First set down the 236, then place the 4 under the units place 6, and draw a line beneath it. Then multiply the 6 by 4: the product is 24 units; set them down. Next multiply the 3 tens by 4: the product is 12 tens; set down the 2 under the tens of the 24, leaving the I to the left, which is the place of hundreds. Next multiply the 2 by 4: the product is 8, which being hundreds, is set down under the 1. The sum of these numbers, 944, is the entire product. 944 The product can also be found, thus: say 4 times 6 are 24: set down the 4, and then ay, 4 times 3 are 12 and 2 to carry are 14: 4, and then say, 4 times 2 are 236 4 944 are 9. Set down the 9, and the before. Ex. 4. Multiply 627 by 84. Multiply by the 4 units, as in the last example. Then multiply by the 8 tens. The first product 56 is 56 tens; the 6, therefore, must be set down under the 0, which is the place of tens, and the 5 carried to the product of the 2 by 8. Then multi, ply the 6 by 8, carrying the duct, and set down the result 50. numbers 52668, is the required product. Ex. 5. Multiply 506 by 302. 627 84 2508 5016 52668 2 from the last proThe sum of the 506 302 1012 000 1518 In this example, we say, 2 times 6 are 12: then set down the 2, and say, 2 times 0 are 0 and 1 to carry make 1. Set down the 1, and say, 2 times 5 are 10: set down the 10. Then beginning with the 0, we say, 0 times 6 is 0: set down the 0. Then say, 0 times 0 is 0; set down the 0, and 152812 then say, 0 times 5 is 0. Then multiply by the 3 hundreds and set down the first figure 8 in the place of hundreds, and place the other figures to the left. 32. When an 0 appears in the multiplier, we need not multiply by it, since each of the products is 0; but when we multiply by the next figure to the left, we must observe to set the first figure of the product directly under its multiplier. Thus, we have placed 8 directly under the multiplier 3. CASE I. 33. When the multiplier does not exceed 12. RULE. I. Set down the multiplicand and under it set the multiplier, so that units shall fall under units, an draw a line beneath them. II. Multiply every figure of the multiplicand by the multiplier, setting down and carrying as in ad I. Set down the multiplier under the multiplicand. so that units shall fall under units, tens under tens, &c. and draw a line beneath them. II. Begin with the right hand figure and multiply all the figures of the multiplicand by each figure of the multiplier, and when any of the products exceeds 9, set down and carry to the next product as in addition; observing to write the first figure of each product directly under its multiplier. III. Add up the several products and their sum will be the product sought. PROOF OF MULTIPLICATION. $35. Write the multiplicand in the place of the and find the product as before: if the two e the same, the work is supposed right. |