- Prove by induction that ∑
- Prove by induction that whenever n ≥ 2,∑
- Prove by induction that 1 + ∑
- The binomial theorem states
n = ∑
Prove the binomial theorem by induction. Next show that
- Let z = 5 + i9. Find z−1.
- Let z = 2 + i7 and let w = 3 − i8. Find zw,z + w,z2, and w∕z.
- Give the complete solution to x4 + 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
- De Moivre’s theorem says
n = rn for
n a positive integer.
Does this formula continue to hold for all integers n, even negative integers? Explain.
- You already know formulas for cos and sin
and these were used to prove De Moivre’s
theorem. Now using De Moivre’s theorem, derive a formula for sin
and one for cos
- 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 x3 + 8 as a product of linear factors.
- Write x3 + 27 in the form
x2 + ax + b cannot be factored any more using
only real numbers.
- Completely factor x4 + 16 as a product of linear factors.
- Factor x4 + 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=1nzj = ∏
Also verify that ∑
k=1mzk = ∑
k=1mzk. 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
- Suppose p =
anxn + an−1xn−1 + +
a1x + a0 where all the ak 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
- De Moivre’s theorem is really a grand thing. I plan to use it now for rational exponents, not just
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 10 at this point. Now here is another question: If n is an integer, is it always true
n = cos
− isin? Explain.
- Suppose you have any polynomial in cosθ and sinθ. By this I mean an expression of the form
β=0naαβ cosαθ sinβθ where aαβ ∈ ℂ. Can this always be written in the form
m+nbγ cosγθ + ∑
n+mcτ sinτθ? Explain.
- Show that ℂ cannot be considered an ordered field. Hint: Consider i2 = −1.
- Suppose p =
anxn + an−1xn−1 + +
a1x + a0 is a polynomial and it has n zeros,
listed according to multiplicity. (z is a root of multiplicity m if the polynomial f =
divides p but
f does not.) Show that
- Give the solutions to the following quadratic equations having real coefficients.
- x2 − 2x + 2 = 0
- 3x2 + x + 3 = 0
- x2 − 6x + 13 = 0
- x2 + 4x + 9 = 0
- 4x2 + 4x + 5 = 0
- 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
- x2 + 2x + 1 + i = 0
- 4x2 + 4ix − 5 = 0
- 4x2 +
x + 1 + 2i = 0
- x2 − 4ix − 5 = 0
- 3x2 +
x + 3i = 0
- Prove the fundamental theorem of algebra for quadratic polynomials having coefficients in ℂ. That is,
show that an equation of the form ax2 + 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.
- Prove the Euclidean algorithm: If m,n are positive integers, then there exist integers q,r ≥ 0 such
that r < m and
Hint: You might try considering
and picking the smallest integer in S or something like this. It was done in the chapter, but go
through it yourself.
- Recall that two polynomials are equal means that the coefficients of corresponding powers of λ are
equal. Thus a polynomial equals 0 if and only if all coefficients equal 0. In calculus we usually think
of a polynomial as 0 if it sends every value of x to 0. Suppose you have the following
where it is understood to be a polynomial in ℤ2. Thus it is not the zero polynomial. Show, however,
that this equals zero for all x ∈ ℤ2 so we would be tempted to say it is zero if we use the conventions
- Prove Wilson’s theorem. This theorem states that if p is a prime, then ! + 1 is
p. Wilson’s theorem was first proved by Lagrange in the 1770’s. Hint: Check
directly for p = 2,3. Show that p − 1 = −1 and that if a ∈
Thus a residue class a and its multiplicative inverse for a ∈ occur in pairs.
Show that this implies that the residue class of
! must be
−1. From this, draw the
- Show that in the arithmetic of ℤp,
p, a well known formula among
∈ ℤp for p a prime, and suppose
≠1,0. Fermat’s little theorem says that
p−1 = 1.
In other words
p−1 − 1 is divisible by p. Prove this. Hint: Show that there must exist
r ≥ 1,r ≤ p− 1 such that
r = 1. To do so, consider 1,
. Then these all have values in
, and so there must be a repeat in
, say p − 1 ≥ l > k and
k. Then tell why
l−k −1 = 0. Let r be the first positive integer such that
r = 1. Let
. Show that every residue class in G has its multiplicative inverse in G. In
r−k = 1. Also verify that the entries in G must be distinct. Now consider the sets
bG ≡ where
b ∈. Show that two of these sets are either
the same or disjoint and that they all consist of
r elements. Explain why it follows that p− 1 = lr for
some positive integer l equal to the number of these distinct sets. Then explain why
lr = 1.
- Let p and
q be polynomials. Then by the division algorithm, there exist polynomials
r equal to 0 or having degree smaller than
p such that
If k is the greatest common divisor of
, explain why k must divide
k is also the greatest common divisor of
r. Now repeat the
process for the polynomials
r. This time, the remainder term will have degree
r. Keep doing this and eventually the remainder must be 0. Describe
an algorithm based on this which will determine the greatest common divisor of two
- Consider ℤm where m is not a prime. Show that although this will not be a field, it is a commutative
ring with unity.
- This and the next few problems are to illustrate the utility of the limsup. A sequence of numbers
ℂ is called a Cauchy sequence if for every ε > 0 there exists m such that if k,l ≥ m,
< ε. The complex numbers are said to be complete because any Cauchy
sequence converges. This is one form of the completeness axiom. Using this axiom, show
k=0∞rk ≡ limn→∞∑
k=0nrk = whenever
r ∈ ℂ and
< 1. Hint: You
need to do a computation with the sum and show that the partial sums form a Cauchy
- Show that if ∑
j=1∞ converges, meaning that lim
j=1n exists, then
converges, meaning limn→∞∑
j=1ncj exists, this for cj ∈ ℂ. Recall from calculus, this says that
absolute convergence implies convergence.
- Show that if ∑
j=1∞cj converges, meaning limn→∞∑
j=1ncj exists, then it must be the case that
limn→∞cn = 0.
- Show that if limsupk→∞
1∕k < 1, then ∑
k=1∞ converges, while if limsup
1∕n > 1,
then the series diverges spectacularly because limn→∞ fails to equal 0 and in fact has a
subsequence which converges to
∞. Also show that if limsupn→∞
1∕n = 1, the test fails because
there are examples where the series can converge and examples where the series diverges. This is an
improved version of the root test from calculus. It is improved because limsup always exists.
Hint: For the last part, consider ∑
n. Review calculus to see why the first diverges and
the second converges.
- Consider a power series ∑
n=0∞anxn. Derive a condition for the radius of convergence using
1∕n. Recall that the radius of convergence R is such that if
< R, then the series
converges and if
> R, the series diverges and if =
R is it not known whether the series
converges. In this problem, assume only that x ∈ ℂ.
- Show that if an is a sequence of real numbers, then liminf n→∞ =