Real Analysis

Math 310  MWF 10:30-11:25am
Fall 1996  127 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1:30-3:30pm Wed & Thu 1:30-2:30pm

Class #2

SUMMARY Natural, Rational and Algebraic Numbers READING §1, §2 (pp 1-9) \underline{Warm-Up} What are the Peano Postulates (a.k.a. Peano Axioms)? Write down as many of them as you can remember. (How many of them are there?)

\underline{ℤ, ℚ and ℝ}

One of the main goals of this course is to understand the construction of the real numbers. But before we get there we will be dealing with other types of numbers first. Give three examples of each type of number below:

• Natural

• Integer

• Rational

• Algebraic

• Real What’s an algebraic number?

\underline{Formal Definition:} A number is called an algebraic number if it satisfies a polynomial equation

a_{n}x^{n} + a_{n−1}x^{n−1} + ⋯ + a_{0} = 0

where the coefficients a_{0}, a_{1}, a_{2}, …, a_{n} are integers, a_{n} ≠ 0 and n ≥ 1. What polynomial does the number b = \sqrt{4 − ^{3} \sqrt{3}} satisfy? Is b an algebraic number?

Suppose we want to check if a number is rational or not. What can we do?

\underline{Rational Zeros Theorem:} Suppose that a_{0}, a_{1}, …, a_{n} are integers and that r is a rational number satisfying the polynomial equation

a_{n}x^{n} + a_{n−1}x^{n−1} + ⋯ + a_{0} = 0

where a_{0} ≠ 0, a_{n} ≠ 0 and n ≥ 1. Write r = p/q where p, q are integers having no common factors and q ≠ 0. Then q divides a_{n} and p divides a_{0}.

What does this mean?

\underline{Example}

1. Using RZT, show that \sqrt{2} cannot represent a rational number.

2. Using RZT, show that ^{3} \sqrt{6} cannot represent a rational number.

\underline{At Home} DO: §2, page 9: 2.3, 2.6 READ: §3, pages 10-14