For the complex number given as: z = a + bi / c+di where i =...
For the complex number given as:
z = a + bi / c+di
where i = √−1 is the imaginary unit. The parameters are defined as a = √2, b = 0, c = 0.5 and d = −0.5.
(a) Find the real and the imaginary parts of z, and then draw
the Argand dia-
gram. (Hint: Use the conjugate of the denominator.) 2.5
(b) Based on the Argand diagram, find the distance r of the
complex number
z from the origin and the argument θ. Now starting from the polar
form
z = reiθ calculate the power z4. 10
(c) Rewrite the complex number in trigonometric form (Hint:
Using the cosθ
and sin θ). Based on the latter form, recalculate independently
from question
(b) the power z4. State the formula used in this case.
Homework Answers
The polar form of a complex number z = a+bi is z = r(cosθ+isinθ) , where r = |z| = sqrt(a^2+b^2) , a = rcosθ and b = r...
The polar form of a complex number z = a+bi is z = r(cosθ+isinθ) , where r = |z| = sqrt(a^2+b^2) , a = rcosθ and b = rsinθ and θ = tan^−1(b/a) for a > 0 and θ = tan^−1(b/a ) + π or θ = tan^−1(b/a) +180° for a < 0. What is the value for θ = tan^−1(b/a) for a = 0? Example: Express z = 0 + i in polar from with the principal argument. The...
A complex number is a number of the form a + bi, where a and b...
A complex number is a number of the form a + bi, where a and b are real numbers √ and i is −1. The numbers a and b are known as the real and the imaginary parts, respectively, of the complex number. The operations addition, subtraction, multiplication, and division for complex num- bers are defined as follows: (a+bi)+(c+di) = (a+c)+(b+d)i (a+bi)−(c+di) = (a−c)+(b−d)i (a + bi) ∗ (c + di) = (ac − bd) + (bc + ad)i (a...
JAVA PROGRAMMING A complex number is a number in the form a + bi, where a...
JAVA PROGRAMMING
A complex number is a number in the form a + bi, where a and b are real numbers and i is V-1. The numbers a and b are known as the real part and imaginary part of the complex number, respectively. You can perform addition, subtraction, multiplication, and division for complex numbers using the following formulas: a + bi + c + di = (a + c) + (b + di a + bi - (c +...
Al. Practice with complex numbers: Every complex number z can be written in the form z...
Al. Practice with complex numbers: Every complex number z can be written in the form z r + iy where r and y are real; we call r the real part of z, written Re z, and likewise y is the imaginary part of z, y - Im z We further define the compler conjugate of z aszT-iy a) Prove the following relations that hold for any complex numbers z, 21 and 22: 2i Re (2122)(Re z) (Re z2) -...
A complex number is a number in the form a + bi, where a and b...
A complex number is a number in the form a + bi, where a and b
are real numbers and i is sqrt( -1). The numbers a and b are known
as the real part and imaginary part of the complex number,
respectively.
You can perform addition, subtraction, multiplication, and division
for complex numbers using the following formulas:
a + bi + c + di = (a + c) + (b + d)i
a + bi - (c + di)...
c++ 2) Complex Class A complex number is of the form a+ bi where a and...
c++
2) Complex Class A complex number is of the form a+ bi where a and b are real numbers and i 21. For example, 2.4+ 5.2i and 5.73 - 6.9i are complex numbers. Here, a is called the real part of the complex number and bi the imaginary part. In this part you will create a class named Complex to represent complex numbers. (Some languages, including C++, have a complex number library; in this problem, however, you write the...
4. (15 marks) Consider the following equation: where i denotes the complex number satisfying i2--1 (a)...
4. (15 marks) Consider the following equation: where i denotes the complex number satisfying i2--1 (a) Rewrite the number -i in the exponential form and transform equation (5) into (b) Solve (6) to get the five solutions wo, ..., wa and draw them on the Argand diagramme (c) Show that wo··· , ua are the eigenvalues of the following real-valued matrix 0 0 0 0 0 cos(2m/5) A-10 -sin2(2π/5) 0 0 cos(2π/5) 0 0 0 2cos(4π/5) 2 Hint: compute the...
JAVA PROGRAMMING
A complex number is a number in the form a + bi, where a and b are real numbers and i is V-1. The numbers a and b are known as the real part and imaginary part of the complex number, respectively. You can perform addition, subtraction, multiplication, and division for complex numbers using the following formulas: a + bi + c + di = (a + c) + (b + di a + bi - (c +...
Al. Practice with complex numbers: Every complex number z can be written in the form z r + iy where r and y are real; we call r the real part of z, written Re z, and likewise y is the imaginary part of z, y - Im z We further define the compler conjugate of z aszT-iy a) Prove the following relations that hold for any complex numbers z, 21 and 22: 2i Re (2122)(Re z) (Re z2) -...
A complex number is a number in the form a + bi, where a and b
are real numbers and i is sqrt( -1). The numbers a and b are known
as the real part and imaginary part of the complex number,
respectively.
You can perform addition, subtraction, multiplication, and division
for complex numbers using the following formulas:
a + bi + c + di = (a + c) + (b + d)i
a + bi - (c + di)...
c++
2) Complex Class A complex number is of the form a+ bi where a and b are real numbers and i 21. For example, 2.4+ 5.2i and 5.73 - 6.9i are complex numbers. Here, a is called the real part of the complex number and bi the imaginary part. In this part you will create a class named Complex to represent complex numbers. (Some languages, including C++, have a complex number library; in this problem, however, you write the...
4. (15 marks) Consider the following equation: where i denotes the complex number satisfying i2--1 (a) Rewrite the number -i in the exponential form and transform equation (5) into (b) Solve (6) to get the five solutions wo, ..., wa and draw them on the Argand diagramme (c) Show that wo··· , ua are the eigenvalues of the following real-valued matrix 0 0 0 0 0 cos(2m/5) A-10 -sin2(2π/5) 0 0 cos(2π/5) 0 0 0 2cos(4π/5) 2 Hint: compute the...
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