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m25-hw3

# m25-hw3 - i = √-1 Complex numbers are added according to...

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Math 25: Advanced Calculus Fall 2010 Homework # 3 Due Date: Friday, October 15 Reading: Finish reading sections 1.1-1.3. Read sections 1.4-1.6. Complete the following problems from the textbook: 1.3.7, 1.4.1, 1.4.3, 1.6.1, 1.6.2, 1.6.10, 1.6.17 In addition, please complete the following problems: 3.1: Use the field axioms in § 1.3 and the ordered field axioms in § 1.4 to prove the following statements for an ordered field F . As before, you must justify each of your steps. (I). Show that x 2 0 for any x F . (II). Show that 1 > 0. (II). Show that if x F and x > 1, then 0 < x - 1 < 1. 3.2: The complex numbers are the field C = { a + bi : a, b R } , where
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Unformatted text preview: i = √-1. Complex numbers are added according to the rule ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d ) i, and complex numbers are multiplied according to the rule ( a + bi ) · ( c + di ) = ( a-d ) + ( b + c ) i. (Notice that the multiplication is just obtained by FOILing.) Deﬁne an order on the complex numbers by declaring that a + bi ≤ c + di, if a ≤ c AND b ≤ d . Does this turn C into an ordered ﬁeld? Why or why not? Be sure to justify your answers! (You only need to check axioms O1-O4. Don’t worry about showing that C is a ﬁeld.) 1...
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