Question

Let w be a non-real complex number. Show that every complex
number z can be written in the form

? = ? + ?? (?, ? ∈ ?)

Furthermore, prove that a and b are uniquely determined by w and
z.

Answer #1

Complex Variables:
(a) Describe all complex numbers 'z' such that e^z = 1.
(b) Let 'w' be a complex number. Let 'a' be a complex number
such that e^a = w. Describe all complex numbers 'z' such that e^z =
w.

1. Show that |z − w| ≤ |z − t| + |t − w| for all z, w, t ∈
C.
2.Does every complex number have a multiplicative inverse?
Explain
3.Give a geometric interpretation of the expression |z − w|, z,
w ∈ C.
4.Give a lower bound for |z + w|. Show your result.
5.Explain how to compute the inverse of a nonzero complex number
z geometrically.
6.Explain how to compute the conjugate of a complex number z
geometrically....

Let Z be the integers.
(a) Let C1 = {(a, a) | a ∈ Z}. Prove that
C1 is a subgroup of Z × Z.
(b) Let n ≥ 2 be an integer, and let Cn = {(a, b) | a
≡ b( mod n)}. Prove that Cn is a subgroup of Z × Z.
(c) Prove that every proper subgroup of Z × Z that contains
C1 has the form Cn for some positive integer
n.

Use strong induction to prove that every natural number n ≥ 2
can be written as n = 2x + 3y, where x and y are integers greater
than or equal to 0. Show the induction step and hypothesis along
with any cases

Definition:In the complex numbers, let J denote the set, {x+y√3i
:x and y are in Z}. J is an integral domain containing Z. If a is
in J, then N(a) is a non-negative member of Z. If a
and b are in J and a|b in J, then N(a)|N(b) in Z. The units of J
are 1, -1
Question:If a and b are in J and ab = 2, then prove one of a and
b is a unit. Thus,...

Determine the real and imaginary parts of the complex number
z=5.2∠−62∘z=5.2∠-62∘.

. In this question, i ? C is the imaginary unit, that is, the
complex number satisfying i^2 = ?1. (a) Verify that 2 ? 3i is a
root of the polynomial f(z) = z^4 ? 7z^3 + 27z^2 ? 47z + 26 Find
all the other roots of this polynomial. (b) State Euler’s formula
for e^i? where ? is a real number. (c) Use Euler’s formula to prove
the identity cos(2?) = cos^2 ? ? sin^2 ? (d) Find...

Prove that every prime greater than 3 can be written in the form
6n+ 1 or 6n+ 5 for some positive integer n.

Prove that every prime greater than 3 can be written in the form
6n + 1 or 6n + 5 for some positive integer n.

The definition of a rational number is a number that can be
written with the form a/b with the fraction a/b being in lowest
form. Prove that √27 is an irrational number using a proof by
contradiction. You MUST use the approach described in class (and on
the supplemental material on cuLearn) and your solution MUST
include a lemma demonstrating that if ? 2 is divisible by 3 then ?
is divisible by 3. Hint: reduce √27 to the product...

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