- Let S = . Find supS. Now let S = [2,5). Find supS. Is supS always a number in S? Give conditions under which supS ∈ S and then give conditions under which inf S ∈ S.
- Show that if S≠∅ and is bounded above (below) then supS (inf S) is unique. That is, there is only one least upper bound and only one greatest lower bound. If S = ∅ can you conclude that 7 is an upper bound? Can you conclude 7 is a lower bound? What about 13.5? What about any other number?
- Let S be a set which is bounded above and let −S denote the set . How are infand suprelated? Hint: Draw some pictures on a number line. What about supand inf S where S is a set which is bounded below?
- Which of the field axioms is being abused in the following argument that 0 = 2? Let
x = y = 1. Then
and so

Now divide both sides by x − y to obtain

- Give conditions under which equality holds in the triangle inequality.
- Prove by induction that n < 2
^{n}for all natural numbers, n ≥ 1. - Prove by the binomial theorem that the number of subsets of a given finite set
containing n elements is 2
^{n}. - Let n be a natural number and let k
_{1}+ k_{2}+k_{r}= n where k_{i}is a non negative integer. The symboldenotes the number of ways of selecting r subsets of

which contain k_{1},k_{2}k_{r}elements in them. Find a formula for this number. - Prove by induction that ∑
_{k=1}^{n}k^{3}=n^{4}+n^{3}+n^{2}. - Prove by induction that whenever n ≥ 2,∑
_{k=1}^{n}>. - Prove by induction that 1 + ∑
_{i=1}^{n}i=! . - Is it ever the case that
^{n}= a^{n}+ b^{n}for a and b positive real numbers? - Is it ever the case that = a + b for a and b positive real numbers?
- Is it ever the case that =+for x and y positive real numbers?
- Derive a formula for the multinomial expansion,
^{n}which is analogous to the binomial expansion. - Suppose a > 0 and that x is a real number which satisfies the quadratic
equation,
Find a formula for x in terms of a and b and square roots of expressions involving these numbers. Hint: First divide by a to get

Then add and subtract the quantity b

^{2}∕4a^{2}. Verify thatNow solve the result for x. The process by which this was accomplished in adding in the term b

^{2}∕4a^{2}is referred to as completing the square. You should obtain the quadratic formula^{2},The expression b

^{2}− 4ac is called the discriminant. When it is positive there are two different real roots. When it is zero, there is exactly one real root and when it equals a negative number there are no real roots. - Suppose f= 3 x
^{2}+ 7x − 17. Find the value of x at which fis smallest by completing the square. Also determine fand sketch the graph of f. Hint: - Suppose f= −5x
^{2}+ 8x− 7. Find f. In particular, find the largest value of fand the value of x at which it occurs. Can you conjecture and prove a result about y = ax^{2}+ bx + c in terms of the sign of a based on these last two problems? - Show that if it is assumed ℝ is complete, then the Archimedean property can be
proved. Hint: Suppose completeness and let a > 0. If there exists x ∈ ℝ such that
na ≤ x for all n ∈ ℕ, then x∕a is an upper bound for ℕ. Let l be the least
upper bound and argue there exists n ∈ ℕ ∩. Now what about n + 1?
- For those who know about the trigonometric functions from calculus or
trigonometry and the complex numbers, De Moivre’s theorem says
for n a positive integer. Prove this formula by induction. Does this formula continue to hold for all integers n, even negative integers? Explain. Hint: i is the complex number satisfying i

^{2}= −1. I am assuming the reader has seen complex numbers. You do arithmetic with them just like you do for algebraic expressions and whenever you see i^{2}, you put −1. I also assume the reader knows the standard formulas for trig. functions like the sine and cosine of the sum of two variables. This is discussed later in the book. This problem is really about math induction. - Using De Moivre’s theorem from Problem 20, derive a formula for sinand one for cos. Hint: Use Problem 18 on Page 69 and if you like, you might use Pascal’s triangle to construct the binomial coefficients.
- De Moivre’s theorem of Problem 20 is really a grand thing. I plan to use it now for
rational exponents, not just integers.
Therefore, squaring both sides it follows 1 = −1 as in the previous problem. What does this tell you about De Moivre’s theorem? Is there a profound difference between raising numbers to integer powers and raising numbers to non integer powers?

- Review Problem 20 at this point. Now here is another question: If n is an
integer, is it always true that
^{n}= cos− isin? Explain. - Suppose you have any polynomial in cosθ and sinθ. By this I mean an expression of
the form ∑
_{α=0}^{m}∑_{β=0}^{n}a_{αβ}cos^{α}θ sin^{β}θ where a_{αβ}∈ ℝ. Can this always be written in the form ∑_{γ=−(n+m) }^{m+n}b_{γ}cosγθ + ∑_{τ=−(n+m ) }^{n+m}c_{τ}sinτθ? Explain. - Let z = 5 + i9. Find z
^{−1}. - Let z = 2 + i7 and let w = 3 − i8. Find zw,z + w,z
^{2}, and w∕z. - Give the complete solution to x
^{4}+ 16 = 0. - Graph the complex cube roots of 8 in the complex plane. Do the same for the four fourth roots of 16. ▸
- If z is a complex number, show there exists ω a complex number with = 1 and ωz =.
- If z and w are two complex numbers and the polar form of z involves the angle θ
while the polar form of w involves the angle ϕ, show that in the polar form for zw
the angle involved is θ + ϕ. Also, show that in the polar form of a complex number
z, r = .
- Factor x
^{3}+ 8 as a product of linear factors. - Write x
^{3}+ 27 in the formwhere x^{2}+ ax + b cannot be factored any more using only real numbers. - Completely factor x
^{4}+ 16 as a product of linear factors. - Factor x
^{4}+ 16 as the product of two quadratic polynomials each of which cannot be factored further without using complex numbers. - If z,w are complex numbers prove zw = zw and then show by induction that
∏
_{j=1}^{n}z_{j}= ∏_{j=1}^{n}z_{j}. Also verify that ∑_{k=1}^{m}z_{k}= ∑_{k=1}^{m}z_{k}. In words this says the conjugate of a product equals the product of the conjugates and the conjugate of a sum equals the sum of the conjugates. - Suppose p= a
_{n}x^{n}+ a_{n−1}x^{n−1}++ a_{1}x + a_{0}where all the a_{k}are real numbers. Suppose also that p= 0 for some z ∈ ℂ. Show it follows that p= 0 also. - Show that 1 + i,2 + i are the only two zeros to
so the zeros do not necessarily come in conjugate pairs if the coefficients are not real.

- I claim that 1 = −1. Here is why.
This is clearly a remarkable result but is there something wrong with it? If so, what is wrong?

- De Moivre’s theorem is really a grand thing. I plan to use it now for rational
exponents, not just integers.
Therefore, squaring both sides it follows 1 = −1 as in the previous problem. What does this tell you about De Moivre’s theorem? Is there a profound difference between raising numbers to integer powers and raising numbers to non integer powers?

- Suppose p= a
_{n}x^{n}+ a_{n−1}x^{n−1}++ a_{1}x + a_{0}is a polynomial and it has n zeros,listed according to multiplicity. (z is a root of multiplicity m if the polynomial f

=^{m}divides pbutfdoes not.) Show that - Give the solutions to the following quadratic equations having real coefficients.
- x
^{2}− 2x + 2 = 0 - 3x
^{2}+ x + 3 = 0 - x
^{2}− 6x + 13 = 0 - x
^{2}+ 4x + 9 = 0 - 4x
^{2}+ 4x + 5 = 0

- x
- Give the solutions to the following quadratic equations having complex coefficients.
Note how the solutions do not come in conjugate pairs as they do when the
equation has real coefficients.
- x
^{2}+ 2x + 1 + i = 0 - 4x
^{2}+ 4ix − 5 = 0 - 4x
^{2}+x + 1 + 2i = 0 - x
^{2}− 4ix − 5 = 0 - 3x
^{2}+x + 3i = 0

- x
- Prove the fundamental theorem of algebra for quadratic polynomials having
coefficients in ℂ. That is, show that an equation of the form ax
^{2}+ bx + c = 0 where a,b,c are complex numbers, a≠0 has a complex solution. Hint: Consider the fact, noted earlier that the expressions given from the quadratic formula do in fact serve as solutions.

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