The Commutative Law for Division. 7. In a chain of divisions, or of multiplications and divisions, the successive operations are to be performed, as has been stated, in order from left to right. E.g., -14 ÷ +2 x7 = −7 x −7 = +49. +8 ÷÷ ̄4 ÷ +2 = −2 ÷ +2 = −1. But, if the operations in the above examples be performed in a different order, the symbol of operation, x or, being carried with its proper constituent, we have -14 x 7÷+2+98÷+2+49, as above. = = +8 ÷ +2 ÷ ̄4 = +4 ÷ ̄4 = −1, as above. This example illustrates the Commutative Law: (i.) To multiply any number by a second number and then to divide the product by a third number, gives the same result as first to divide the given number by the third number and then to multiply the resulting quotient by the second number; and vice versa; or, stated symbolically, Nxb÷c=N÷c × b, or xb÷c = ÷ cxb. (ii.) If a given number be divided successively by two numbers, the result is the same whichever of the two divisions is first performed; or, stated symbolically, N÷b÷c = N÷c÷b, or ÷ b÷ c = ÷ c÷b. N be replaced by N÷c xc, = N, by (2), Art. 6, we have = = N÷c xbx cc, since × cx b = × b × c, = N÷cx b, since x cc = x +1, by (2), Art. 6. In like manner, it can be shown that the Commutative Law holds for any number of successive multiplications and divisions. The Associative Law for Division. 8. By Art. 4, −32 × +4 ÷−2 =128÷2 =+64, -32 x (+4÷-2)=-32 x-2=+64. while, Likewise, while, And, while, +32 +-4x+2=-8x+2=-16, +32 ÷(−4 ÷+2)=+32 ÷ 2 =−16. -32-4+2=+8÷-2=-4, -32÷(-4x-2)=-32 ÷ +8 =−4. These examples illustrate the Associative Law: (i.) A chain of multiplications and divisions may be inclosed within parentheses preceded by the symbol of multiplication, if the symbols of operation, × and ÷, preceding the numbers inclosed within the parentheses be left unchanged; or, stated symbolically, Nxa+b=Nx (a+b). (ii.) A chain of multiplications and divisions may be inclosed within parentheses preceded by the symbol of division, if the symbols of operation, x and, preceding the numbers inclosed within the parentheses be reversed from x to and from ÷ to x; or, stated symbolically, N ÷ a ÷ b = N ÷ (a × b), and N÷a × b The proof is as follows: = N÷ (a ÷ b). N be replaced by N ÷ (a × b) × (a × b), = N, by (2), Art. 6, we have N÷a÷b N ÷ (a × b) × (a × b) ÷ a ÷ b = = N ÷ (a × b) × b × a ÷ a ÷ b, since xa x b = × b × a, = N ÷ (a × b) × b÷b, since x a ÷ a = x +1, = N÷ (a × b), since x b÷b = x +1. In like manner the other principles are proved, and all can be extended to include any number of successive multiplications and divisions. 9. An even number is one whose absolute value is exactly divisible by 2; as 4, 6, etc. Since, by the Commutative Law, 2n÷22÷÷2 x n = 1 x n = n, = 2n is always an even number when n is an integer. EXERCISES X. Find, in the most convenient way, the values of: Find, in the most convenient way, the value of a ÷ b÷c × d, 7. When a = +170, b = −3, c = +17, d = −6. 8. When a = :-125, b = −7, c = +25, d=-14. Find the values of the following expressions, first removing the parentheses: 9. +25 × (+12 ÷−4). 10. 20÷(5÷+2). 11. +100-÷ (+25 ×−2). 12. -600(-200÷-25 ×÷3÷-4). 13. +300÷(-150÷+6x+8÷-4). § 5. ONE SET OF SIGNS FOR QUALITY AND OPERATION. 1. In conformity with the usage of most text-books of Algebra we shall in subsequent work use the one set of signs, + and -, to denote both quality and operation. For the sake of brevity the sign is usually omitted when it denotes quality; the sign is never omitted. Thus, instead of +2, we shall write +2, or 2; instead of -2, we shall write — 2. 2. We have used the double set of signs hitherto in order to emphasize the difference between quality and operation. It should be kept clearly in mind that the same distinction still exists. We now have N++2 = N + (+2) = N+ 2, omitting the sign of quality, +; N +2 = N + ( − 2), wherein + denotes operation, and — denotes quality. N-+2= N(+2) = N-2, omitting the sign of quality, +; N--2-N-(-2), wherein the first sign, -, denotes operation, the second sign, -, denotes quality. 3. In the chain of operations (+2) + (− 5) − ( + 2) − (− 11) the signs within the parentheses denote quality, those without denote operation. That expression reduces to (+ 2) − (+ 5) − (+ 2) + (+ 11), or 25-2+11, dropping the sign of quality, +. In the latter expression all the signs denote operation, and the numbers are all positive. 4. The following examples illustrate the double use of the signs and . Ex. 1. +4 ++3 = + 4 + ( + 3) = 4 + 3 = 7. Ex. 2. 5+2 = − 5 + (+ 2) = −5+2 = −3. · Find the values of the expressions in Exx. 1-8, first changing them into equivalent expressions in which there is only the one set of signs, + and -: Find the values of the expressions in Exx. 9-14, first changing them into equivalent expressions in which there is only one set of signs, -, and then removing the parentheses: and Find the values of the expressions in Exx. 15-22, first changing them into equivalent expressions in which there is only one set of signs, + and -: 15. +3 x +4. 19. +18+2. 16. 18 x -4. 17. 21. -9 × +11. 18. -4 x -7. 22. -96 −6. 23. When a = 3, b =- 5, c=- 8, d9, e = 7? 28. When a = — =-5, b -3, c=4, d5, e-7? 29. When a = 12, b2, c =~ 9, d 13, e = 28? Find the results of the following indicated operations: § 6. POSITIVE INTEGRAL POWERS. 1. A continued product of equal factors is called a Power of that factor. Thus, 2 x 2 is called the second power of 2, or 2 raised to the second power; aaa is called the third power of a, or a raised to the third power. ... In general, aaa to n factors is called the nth power of a, or a raised to the nth power. The second power of a is often called the square of a, or a squared; and the third power of a the cube of a, or a cubed. 2. The notation for powers is abbreviated as follows: a is written instead of aa; a instead of aaa; a" instead of aaa ... to n factors. 3. The Base of a power is the number which is repeated as a factor. E.g., a is the base of a2, a3, ..., a". |