the sum multiplied by ten, is equal to the square of eight diminished by four ?" Ans. (64+2)×10=82-4. MULTIPLICATION OF COMPOUND EXPRESSIONS. 4. In multiplication, the multiplier must always be regarded as an abstract number. The multiplicand may be a quantity or denominate number of any kind. Thus, we may repeat $7 once, twice, thrice, or any number of times, but we could not multiply dollars by yards, pounds, hours, or by any other denominate number. It is sometimes proposed to multiply money by money, as 2s. 6d. by 2 s. 6d., but this is not philosophically correct. We may repeat 2 s. 6 d. once, twice, &c., but we cannot repeat it 2 s. 6d. times. In estimating the cost of 20 bushels of apples at 25 cents per bushel, we do not repeat 25 cents 20 bushels of times, neither do we repeat 20 bushels 25 cents times; but, having fixed upon a quantity of apples equal to one bushel as our unit, we find that the quantity whose cost is to be estimated is 20 bushels, or twenty times the unit; now as one bushel is worth 25 cents, 20 bushels, being 20 times as many apples, must be worth 20 times as many cents as one bushel; we therefore repeat 25 cents twenty times, not 20 bushels of times, and thus obtain 500 cents, or 5 dollars, for the whole cost. And, in a similar way, we might show that, in all operations of multiplication, the multiplier is an abstract number, denoting how many times the multiplier is to be repeated. The product must evidently be of the same name as the MULTIPLICATION OF COMPOUND EXPRESSIONS. 15 multiplicand, since repeating a quantity once, twice, thrice, or any number of times, cannot change its denomination. Sometimes the multiplicand is also an abstract number. It would be equally absurd to suppose the multiplier to be a negative quantity; for we could not repeat a quantity a minus number of times. If, then, in the course of an operation, we have for factors a positive and a negative quantity, we must regard the positive factor as the multiplier, and the negative quantity as the multiplicand; as, for example, if we wish the product of 4 and -7, we must repeat -7 four times, and the result will still be negative, we shall thus obtain - 28. Or we might have multiplied the 4 by 7, and changed the sign of the product. From which we see that when a minus quantity occurs in the multiplier, we are to multiply by it considered as positive, and then to change the sign of the product. Applying this principle to the case when both factors are negative, as, for example, - 5 multiplied by -7. In this case, it will be necessary to repeat -5 seven times, and then to change the sign of the product, we thus obtain for our result, 35. From what has been said and done, we have for the multiplication of compound expressions the following RULE. Multiply each term of one of the factors by each term of the other factor, observing that like signs produce plus, and unlike signs produce minus. Let it be required to multiply 3+2 by 4+5. We must repeat 3+2 as many times as there are units in 4+5. First, repeating 3+2 as many times as there are units in 4, we get (3+2)×4=12+8, for first partial product: Secondly, repeating 3+2 as many times as there are units in 5, we get (3+2)×5=15+10, for second partial product: Hence, 3+2, repeated as many times as there are units in 4+5, becomes (3+2)×(4+5)=12+8+15+ 10. Again, let it be required to multiply 7-3 by 4+2: Proceeding as in the last example, we find (7-3)× (4+2)=28-12+14-6. In a similar way, we find that 4-3, multiplied by 3-2, gives (4-3)×(3-2)=12-9-8+6. EXAMPLES. 1. What is the product of 8+3 by 6+4? Ans. 48+18+32+12. Ans. 24-8+18-6. 2. What is the product of 6-2 by 4+3? 3. What is the product of 11-3 by 13-7? Ans. 143-39-77+21. 4. What is the product of 3+2-1 by 4-1+5? Ans. 12+8-4-3-2+1+15+10-5. 5. What is the product of 1+2-3 by 4-5+6? Ans. 4+8-12-5-10+15+6+12-18. 6. What is the product of 7-9 by 5-11? Ans. 35-45-77+99. 7. What is the product of 21-3 by 9-2? Ans. 189-27-42+6. 8. What is the product of 1+7+5 by 2+3? Ans. 2+14+10+3+21+15. SINGULAR PROPERTY OF THE FIGURE 9. 5. Every number will divide by 9, when the sum of its digits is divisible by 9. For, take any number, as 78534; this number is, by the nature of decimal arithmetic, the same as 70000+ 8000+500+30+4. .. 78534-9999×7+999×8+99×5+9×3+(7+8+ 5+3+4.) Now, since each expression, 9999 × 7, 999 × 8, 99 × 5, and 9×3, is divisible by 9, it follows that the first number, 78534, will be divisible by 9 when the sum of its digits (7+8+5+3+4) is. Hence, it follows that any number being diminished by the sum of its digits, will become divisible by 9. Also, any number divided by 9, will leave the same remainder as the sum of its digits when divided by 9. The above properties belong to the digit 3, as well as to that of 9, since 3 is a divisor of 9. No other digit has such properties. NOTE. These singular properties of the digit 9, have been made use of by many authors for proving the work of the four fundamental rules of arithmetic. PRIME NUMBERS. 6. No even number can, with the single exception of the number 2, be a prime, since all even numbers are divisible by 2. It is also evident that there are many odd numbers which are not primes. If we write in order the natural series of odd numbers, we discover that every third term, counting from 3, is divisible by 3; every fifth term, counting from 5, is divisible by 5; every seventh term, counting from 7, is divisible by 7, and so on. 39 Commencing at 3, under every third term, I have placed a small figure, to denote that the term under which it is placed is divisible by 3. Under every fifth term, counting from 5, I have, in like manner, placed a small, indicating that the corresponding term is divisible by 5. I have proceeded in the same way for the higher primes. Now it is evident that all the terms, under which there are no small figures found, are primes. We may also remark, that the numbers expressed by the small figures are the different prime factors of the numbers under which they are placed. 3-139 1, 3, 5, 7, 93, 11, 13, 153.5, 17, 19, 21 3.7, 23, 255, 273, 29, 31, 333.11, 355.7, 37, 39 41, 43, 453.5, 47, 497, 513-179 53, 555.119 573.19, 59, 61, 633.7, 655-13) 67, 69 3.2 31 93.23, 71, 73, 75 3.5) 777.11, 79, 813, 83, 855-175 87 3.29, 89, 917.13, 933.31) 955-19, 97, 99 3.119 &c. In the above operation, we have found the primes only which are less than 100; but this process may be extended as far as we wish. This method of finding the successive primes was employed by Eratosthenes, who inscribed the series of odd numbers upon parchment, then cutting out such numbers as he found to be composite, his parchment with its holes resembled somewhat a sieve; hence, this method is called Eratosthenes' Sieve. The number 2, although an even number, must be regarded as coming under our definition of a prime, since the only number which will divide it is itself. |