Studying Rudin's Principles of Mathematical Analysis Through ...

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$Math Department Presentation$

8 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS 1.4.3 Theorem: Uniqueness of y n = x and n√ x Prove that for every real x > 0 and every integer n > 0 there is one and only one positive real y such that y n = x. This number y is written n√ x or x 1/n . Corollary: (ab) 1/n = a 1/n b 1/n Prove that if a and b are positive real numbers and n is a positive integer, then (ab) 1/n = a 1/n b 1/n . 1.4.4 Decimals Develop a system to write decimals in terms of integers by defining x = sup E. 1.5 The Extended Real Number System 1.5.1 Definition: Extended Real Number System What is the extended real number system? Does the extended real number system form a field? Why or why not? 1.6 The Complex Field 1.6.1 Definition: Complex Numbers Use an ordered pair of real numbers to define a complex number and addition and multiplication of complex numbers. 1.6.2 Theorem: Complex Numbers Form a Field Use your definitions above to prove that addition and multiplication turn the set of all complex numbers into a field, with appropriate description of 0 and 1 in terms of complex numbers. 1.6.3 Theorem Prove that for any real numbers a and b, we have (a, 0) + (b, 0) = (a + b, 0), and (a, 0)(b, 0) = (ab, 0). 1.6.4 Definition: i Define i. 1.6.5 Theorem: i 2 = −1 Use the definition of i to show i 2 = −1.
1.6. THE COMPLEX FIELD 9 1.6.6 Theorem: (a, b) = a + bi Prove that if a, b ∈ R, (a, b) = a + bi. 1.6.7 Definition: z, Re[z], Im[z] What are Conjugate Real part, Imaginary part of z? 1.6.8 Theorem If z and w are complex, prove that (a) z + w = z + w (b) zw = z · w (c) z + z = 2Re(z), z − z = 2iIm(z) (d) zz ≥ 0 and real. 1.6.9 Definition What is absolute value of a complex number z? 1.6.10 Theorem Let z and w be complex numbers. Prove that (a) |z| > 0 unless z = 0, |0| = 0 (b) |z| = |z| (c) |zw| = |z||w| (d) |Rez| ≤ |z| (e) |z + w| ≤ |z| + |w| 1.6.11 Notation What is the summation notation? 1.6.12 Theorem: Schwartz Inequality What is the Schwartz inequality? State and prove it for complex numbers.
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Studying Rudin's Principles of Mathematical Analysis Through ...

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