# Lectures on topics in algebraic number theory by Ghorpade S.R.

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Example text

77 16 CHAPTER R The definition of division parallels the one for subtraction. Using this definition and the rules for multiplying, we can see how to handle signs when dividing. Review of Basic Algebra EXAMPLES Divide. 31. 10 = -5, because -5 # 1-22 = 10 -2 32. -32 = -8, because -8 # 142 = -32 4 33. -25 = 5, because 5 # 1-52 = -25 -5 34. 40 = -10 -4 35. -10 , 5 = -2 36. 25 -40 4 37. 3 -3 3 The rules for division and multiplication are the same. Divide. 39. To multiply or divide two real numbers: 1.

21; 21 10. 440; 440 11. 5 + y ; ba; mn + 8, or nm + 8, or 8 + nm 12. 5 Equivalent Algebraic Expressions 41 The expressions 81x + y2 and 8x + 8y in Example 10 are equivalent. This fact is the result of a law called the distributive law of multiplication over addition. THE DISTRIBUTIVE LAW OF MULTIPLICATION OVER ADDITION For any numbers a, b, and c, 13. Evaluate 101x + y2 and 10x + 10y when x = 7 and y = 11. 14. Evaluate 91a + b2, 1a + b29, and 9a + 9b when a = 5 and b = -2. a1b + c2 = ab + ac, or 1b + c2a = ba + ca.

Add the absolute values: ƒ -13 ƒ + ƒ -8 ƒ = 13 + 8 = 21. Make the answer negative : -21. One negative, one positive. 5. 4. 4. Add. 5. -7 + 1-112 6. 72 7. - 23 6 + a- b 5 5 8. - 2 3 + a- b 10 5 One negative, one positive. Find the absolute values: ƒ -48 ƒ = 48; ƒ 31 ƒ = 31. Subtract the smaller absolute value from the larger: 48 - 31 = 17. The negative number, -48, has the larger absolute value, so the answer is negative, -17. 10. 4 + 0 One positive, one negative. The numbers have the same absolute value.