Mathematics 242 – Principles of Analysis Discussion 1: Algebraic and Order Properties of Rational and Real Numbers February 2, 2011

Background

The purpose of today’s discussion is to start to think about what properties number systems should have in order for us to be able to solve equations, analyze functions, and ultimately “do calculus.” You have used the set of rational numbers or fractions:

Q = {m/n : m, n integers, n ≠ 0}

since grade school.

Discussion Questions

A) Linear Equations. Consider a simple linear equation ax + b = c with a, b, c ∈ Q or R more generally. We can solve for x by elementary algebra:

(1)  ax + b = c implies x = \frac{c}{a} + \frac{(−b)}{a}

Now of course the a, b, c here are covered by the algebraic Axioms 1 - 5 on page 12 of the text. (Note: an “axiom” is a property that is assumed to be true, without proof, as part of the description or definition of a mathematical object.)

1) Which of the algebraic Axioms for real numbers are used in solving the equation in (1) and writing the solution in this form?

2) Show carefully that if a = m/n, b = p/q, c = r/s, are rational numbers with m, n, p, q, r, s all integers with n, p, s ≠ 0, then the solution in (1) can also be written as a quotient of integers with nonzero denominator. Give a reason why every step in your algebraic manipulations is valid, using known properties of fractions. (See Theorem 1.3.1 on page 13.)

B) The absolute value of a real number x is defined by

|x| = \{ \begin{matrix}x & if x ≥ 0 \\ −x & if x < 0. \end{matrix}

If x, y are reals, then |x − y| can also be interpreted as the distance from x to y along the number line.

1) Find all solutions of the equation |x − 2| + |x + 6| = 12.

2) What is the smallest c for which |x − 2| + |x + 6| = c has a solution? Prove your assertion about this c.

3) Let max(a, b) be the larger of the two real numbers a, b, and similarly min(a, b) be the smaller. (If a = b, then that single value is both the max and the min.) Show that

max(a, b) = \frac{a + b}{2} + \frac{|a − b|}{2} and min(a, b) = \frac{a + b}{2} − \frac{|a − b|}{2}.

C) In this and the next question, we will “move up” to quadratic equations.

1) Some quadratic equations with rational number coefficients can also be solved in the rational numbers. For instance, x^{2} − 5x + 6 = 0 has solutions x = 2 and x = 3. Show how we know this, using the algebraic Axioms. (Hint: If a, b are real numbers with a · b = 0 and a ≠ 0, then why must b = 0?)

2) Now consider the quadratic equation x^{2} − 2 = 0, with solutions x = ±\sqrt{2}. What is special about the decimal expansion of a rational number. But can you tell anything by looking at a finite part of the decimal expansion \sqrt{2} = 1.4142136 ⋯? (Note: a finite part is all a human being could ever see!)

3) Here’s a different way to try to answer our question. Suppose \sqrt{2} is in Q, and write \sqrt{2} = m/n > 0, with positive integers m, n. We can assume that the fraction m/n is in lowest terms (no common factors between m and n). Then

(m/n)^{2} = 2, or m^{2} = 2n^{2}.

Can you find integers m, n with no common factors satisfying this equation? Why or why not? (You may use without proof the fact that if m is an integer and m^{2} is even, then m must also be even.)

D) Here’s another way to get approximations to \sqrt{2}. Note that \sqrt{2} solves the equation x = 2/x.

1) Explain why for all positive x_{0}, \sqrt{2} lies between x_{0} and 2/x_{0} – either x_{0} ≤ \sqrt{2} ≤ 2/x_{0} or 2/x_{0} ≤ \sqrt{2} ≤ x_{0}. (Sketching the graphs y = x and y = 2/x on the same set of axes may be useful! Both of the possibilities above occur for some x_{0}. Say when each is true.)

2) Thinking of x_{0} as a first approximation, show that the average

x_{1} = (1/2)(x_{0} + 2/x_{0})

also lies between x_{0} and 2/x_{0}.

We can repeat this as many times as we like to make

(2)  x_{n+1} = (1/2)(x_{n} + 2/x_{n})

for n = 0, 1, 2, 3, . . ..

3) Using (2), construct the x_{1}, …, x_{5} starting with x_{0} = 1, then restart from x_{0} = 3 and find x_{1}, …, x_{5}. What seems to be happening in either case? (We will return to this example in a couple of weeks!)

4) Which of the Axioms for real numbers and other results from section 1.3 in the text have you used in this question?

Writeups: Group writeups due in class Monday, February 7.