36. When the multiplier is 1 and any number of ciphers after it, as 10, 100, 1000, &c. Placing a cipher on the right of a number changes the units place into tens, the tens into hundreds, the hundreds into thousands, &c. and therefore increases the number ten times. Thus, 55 is increased ten times by making it 550. So the addition of two ciphers increases a number one hundred times; the addition of three ciphers, a thousand times, &c. Thus, 36 is increased a hundred times, by making it 3600, and 25 is increased a thousand times, by making it 25000. Hence, we have the following RULE. Place as many ciphers as there are in the multiplier, on the right of the multiplicand, and the number so formed will be the required product. EXAMPLES. 1. Multiply 254 by 10. Ans. 2540. Ans. 64800. Ans. 7987000. Ans. 98400000. CASE IV. § 37. When there are ciphers on the right hand of one or both of the factors. RULE. Neglect the ciphers and multiply the significant figures: then place as many ciphers to the right hand of the product, as there are in both of the factors. § 38. When the multiplier is a composite number. A composite number is produced by the multiplication of two or more numbers, which are called the components or factors. Thus, 2×3=6. Here 6 is the composite number, and 2 and 3 are the factors, or components. The number 16=8×2: here 16 is a composite number, and 8 and 2 are the factors; and since 4×4=16, we may also regard 4 and 4 as factors or components of 16. Ex. 1. Let it be required to multiply 8 composite number 6, in which the factors are 2 and 3. If we write 6 horizontal lines with 8 units in each, it is evident that the product of 8×6=48, the number of units in all the lines. But let us first connect the lines in sets of 2 each, as on the right; there will then be in each set 8×2 =16; or 16 units in each set. But there are 3 sets; hence 16x3=48, the number of units in all the sets. If we divide the lines into sets of 3 each, as on the left, the number of units in each set will be equal to 8×3=24, and there being 2 sets, 24×2=48, the whole number of units. As the same may be shown for all numbers we have the following RULE. When the multiplier is a composite number, multiply by each of the factors in succession, and the last product will be the entire product sought. EXAMPLES. 1. Multiply 327 by 12. The factors of 12 are 2 and 6, or they are 3 and or they are 3, 2 and 2, for, 2×6=12, 3×4=12, 3×2×2=12. 2. Multiply 5709 by 48; the factors being 8 and 6, or 16 and 3. § 27. If a number be added to itself, how many times greater will the sumn be than the number? How may a number be repeated? 28. What is multiplication? What is the number called which is to be repeated? What is the multiplier? What is the product? §29. What are the multiplier and multiplicand called? How do you denote that two numbers are to be multiplied together? What is the sign called? Repeat the table. §30. If the multiplicand be made the multiplier will the product be altered? § 31. What may multiplication be considered? 32. When an 0 appears in the multiplier, what do you do? § 33. When the multiplier does not exceed 12, how do you set it down? How do you multiply by it? § 34. When the multiplier exceeds 12, how do you set it down? How do you multiply by it? How do you add up? § 35. How do you prove multiplication? §36. When the multiplier is 1, 10, 100, &c. what is the rule? Why? § 37. When there are ciphers on the right of both the factors, what do you do? §38. What is a composite number? What are the parts called? How do you multiply by it? |