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

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.