Complex Numbers and Functions

Question 1

The conjugate, z*, of the complex number z = a + jb is..

It’s the mirror image of the complex number across the horizontal axis

Answer 1

z* = a - j*b

Question 2

z =z₁*z₂ = (a₁+jb₁)(a₂+jb₂)

Answer 2

z = (a1*a2 - b1*b2) + j(a1b2+b1a2)

Question 3

z = z1/z2 =

Answer 3

[(a1*a2+b1*b2)/(a2² + b2²)] + j[(a2*b1-a1*b2)/(a2² + b2²)]

Question 4

Euler’s Idenity

Answer 4

e^jΘ = cos(Θ) + jsin(Θ)

Question 5

Let Θ be any real number. Then cos(Θ) & sin(Θ) equal

Answer 5

cos(Θ) = .5(e^jΘ+e^-jΘ)

sin(Θ) = (1/2j)(e^jΘ-e^-jΘ)

Question 6

z = |z|e^jΘ

z* = ?

Answer 6

z* = |z|e^-jΘ

Question 7

z = z₁*z₂

(exponential form)

Answer 7

z = |z₁|*|z₂|e^{j(Θ1 +Θ2)}

Question 8

z = z1/z2

Exponential Form

Answer 8

z = (|z₁|/|z₂|)e^j(Θ1-Θ2)

Question 9

To take the inverse of a complex number expressed in exponential form…

Answer 9

take the inverse of the magnitude and negate the phase

Question 10

What is the formula to find roots of complex numbers?

Answer 10

Question 11

j, j², j³, j⁴

Answer 11

Question 12

Prove that cos(-x) = cos(x)

&

sin(-x) = -sin(x)

Answer 12

Question 13

Convert this into cartesian coordinates

Answer 13