rows of a dots each, we shall arrive by similar reasoning at the result, that for all positive integral values of a and b ax b = by a. This is called the Law of Commutation for Multiplication. It may be shown that the principle applies whenever there are three or more factors. 15. Continued Product. The product of three or more numbers, a, b, c, and d, written ax bx c x d, or abcd, is defined to be the number obtained by multiplying a by b, this result by c, and finally the last result by d. If we represent any three arithmetic whole numbers by a, b, and c, we may indicate the continued product of three positive numbers by writing (a)(+b)(+c). (The following proof may be omitted when the chapter is read for the first time.) By the definition of a continued product we are to understand (a)(+b)(*c) as meaning that we are to first multiply +a by +b, and this product by +c. But (+a)(+b)(+c) = [(+a) (+b)](+c). By definition of multiplication. (a)(+b)+(ab). Hence (a)(b)(+c) = [+(ab)](+c). Regarding +(ab) as a single number, and multiplying by +c, = +(abc). By assuming some, all, or none of the three factors of the product to be positive numbers, we may extend the principle to include such combinations of positive and negative numbers as the following: (+a)(+b)(+c) = +(ab)(+c) = +(abc). (−a)(+b)(+c) =−(ab)(+c) = −(abc). (−a)(−b)(+c) = +(ab)(+c) = *(abc). and so on for more factors. 16. The essential thing to be observed in the identities above is that a continued product is positive if it contains no negative factors or if it contains an even number of negative factors, and it is negative if it contains an odd number of negative factors. E. g. 1. (+3)(+2)(+4) (+5) = +120. 17. The Associative Law for Multiplication The value of a product remains unaltered if, in the process of multiplying several numbers, two successive factors are associated or grouped together to form a single product. E. g. 2 × 3 × 4 x 5 = 2 × 3 × (4 × 5) = (2 × 3) × 20 = 6 × 20 = 120. (The following proof may be omitted when the chapter is read for the first time.) For any three arithmetic whole numbers, a, b, c, We have abc = a(bc). abc (ab)c. By definition of a product. Considering the product (ab) as one number, by the Commutative Law for two factors we have: Or, = c(ab), = (ca)b, by definition of a product. = (bc)a, by definition of a product. abc (ab)ca(bc). 18. By repeated applications of the Commutative and Associative Laws for multiplication, it may be shown that both laws hold for three or more factors. That is : Also, abcacb= bac = bca = cab = cba. abcda (bcd) = a (bc) d = b (acd) = etc., and so on for any number of factors. 19. From the above, it appears that we may arrange the factors of a product in any order, and group them together in any convenient way, without altering the value of the result. 20. Both the multiplicand and multiplier receive the name of factor, since they may be interchanged without altering the value of the product. E. g. a and b are factors of the product a × b. Similarly, each number of a continued product, abcdef called a factor of that product. E. g. 5, a, b, and c are all factors of the continued product 5 abc. 21. The Distributive Law for Multiplication is Up to this point we have considered products in which both multiplicand and multiplier consisted of single numbers. In case either or both are sums or differences, we are led to consider the third Fundamental Law of Algebra, namely, the Distributive Law. In particular, we will show that the product of 3 multiplied by the sum of 4 and 5 is the same as the product of 3 multiplied by 4, increased by the product of 3 multiplied by 5. Let a series of dots be arranged as below, forming a set of three rows, each containing 9 dots. The dots may be counted in either of two ways: first, as a single group consisting of three rows containing 9 dots each, that is, 27 dots in all; second, as consisting of one group of three rows containing 4 dots each, and a second group consisting of three rows containing 5 dots each, — the two groups being separated as shown. Hence we may write 3(4+ 5) = (3 × 4) + (3 × 5) = 27. 22. The process may be applied to any three whole numbers, a, b, c, and we may assert as a general principle that The product of an algebraic sum multiplied by a single number may be obtained by multiplying each term of the sum by the given number, and finding the algebraic sum of the results obtained. This is called the Distributive Law for Multiplication, and it may be shown to hold when the multiplier consists of any number of terms, which may be positive or negative, integral or fractional. 23. Zero as a Factor. It follows directly from the Fundamental Laws that a product is zero if one of its factors is zero. That is a0=0, (Multiplier 0). 0a0. (Multiplicand 0). (The following proof may be omitted when the chapter is read for the first time.) Also, 0⋅ a = (n−n)a (Multiplicand 0.) = na - na = 0. 24. Since the proof of an identity establishes at the same time the truth of its converse, it follows that, if a product is zero, at least one of its factors must be zero, then either a is 0, or b is 0, or both a and b are 0. 25. The product obtained by using the same factor repeatedly is called a power of that factor. E. g. 3 x 3 is called the second power of 3, or 3 raised to the second power, since 3 occurs twice as a factor. Also 2 x 2 x 2 x 2 is called the fourth power of 2; etc. 26. The number of times a factor appears in a product may be indicated by writing a small number called the exponent or the index of the power at the right of and immediately above the factor. E. g. We may write 52 instead of 5 × 5; 48 instead of 4 x 4 x 4; etc. 27. The number which is used repeatedly as a factor to obtain a power is called the base of the power. 28. The definition of an exponent as given is that it indicates the number of times a factor appears in a product. This definition requires that the exponent should be a positive whole number. In a later chapter this notion of an exponent will be somewhat extended. 29. In arithmetic a number is defined as being even or odd according as it is or is not divisible by 2. E. g. 2, 4, 6, 10, 16, etc., are even numbers. 3, 5, 7, 11, 17, etc., are odd numbers. 30. A power is defined as being even or odd according as its exponent is even or odd. E. g. while (+4)2, (+6)1, (−3), (−7)3, etc., are even powers, 31. An odd power of a negative base contains an odd number of negative factors, and accordingly, by the rule of signs for continued products, it is of negative quality. E. g. The power (-2)3 is of negative quality. For, (-2)=(-2)(2)(-2) = −8. 32. Whenever we speak of a positive integral power we have reference to the exponent rather than to the value of the base, which may itself be fractional or negative. E. g. The following are positive integral powers of fractional bases and of negative bases: 33. In operating with powers we are governed by the following Principles: (i.) All powers of positive bases are positive. 34. Since a product is zero if one or more of its factors is zero, it follows that any positive integral power of zero is zero; that is, 0m = 0. 35. In order to indicate clearly and exactly what number is to be considered as the base, it is often necessary to enclose the number within parentheses, as in the following illustrations: (i.) The base a negative number. (−3)2 = (-3)(−3) = +(3 × 3) = +9. Observe that a2 is not the same as (-a)2. a2 is read "negative a square;" (a) is read "the square of negative a." We have while ̄a2 = ̄(a xa) = ̄a2, ( ̄a)2 = ( ̄a)( ̄a) = +a2. Whenever the symbol before the number or base is regarded as one of operation, as for example, -33, we may write (ii.) The base not a single number, but either a product or a quotient. |