By Sheldon Axler

Collage Algebra and Trigonometry will entice those that are looking to supply very important issues extra in-depth, higher-level insurance. this article deals streamlined strategy followed with obtainable definitions throughout all chapters to permit for an easy-to-understand learn. collage Algebra comprises prose that's unique, exact, and simple to learn, with elementary definitions of even the themes which are generally such a lot tough for readers.

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Understanding the logical pattern of thinking that goes into this proof can be a valuable asset in dealing with complex issues. No rational number has a square equal to 2. Proof: Suppose there exist integers m and n such that m n 2 = 2. By canceling any common factors, we can choose m and n to have no m factors in common. In other words, n is reduced to lowest terms. The equation above is equivalent to the equation m2 = 2n2 . This implies that m2 is even; hence m is even. Thus m = 2k for some integer k.

2 Algebra of the Real Numbers 13 Multiplicative Inverses and the Algebra of Fractions The multiplicative inverse of a real number b = 0 is the number b· 1 b such that 1 = 1. b The multiplicative inverse of b is sometimes called the reciprocal of b. The connection between division and multiplicative inverses is captured by the identity a 1 =a· . b b In fact, the equation above can be taken as the definition of division. You need to be comfortable using various identities that involve multiplicative inverses and division.

1 5 3 7 30 chapter 1 The Real Numbers The next example goes in the other direction, starting with a set and then writing it as a union of intervals. example 3 Write the set of nonzero real numbers as the union of two intervals. solution The set of nonzero real numbers is the union of the set of negative numbers and the set of positive numbers. In other words, the set of nonzero real numbers equals (−∞, 0) ∪ (0, ∞). Absolute Value The absolute valueof a number is its distance from 0; here we are thinking of numbers as points on the real line.