Complex Numbers with In-Depth Explanation

Solve $(6 + 5 i) \div (7 + 3 i) =$ ?

Kason Chang 2022-04-30

Simplify $\frac{2 + 2 i}{4 + 7 i}$ .

Rowan Huynh 2022-04-30

Find $(6 + 4 i) \times (6 + 4 i) =$ ?

hadaasyj 2022-04-24

Perform the indicated operation with complex numbers. (Simplify your answe completely.)
$(4 - 8 i) (9 + 7 i)$

Ellie Castro 2022-04-23

Show that if $z_{1}$ and are complex numbers then
$\overset{―}{\frac{z_{1}}{z_{2}}} = \frac{\overset{―}{z_{1}}}{\overset{―}{z_{2}}}$

Perla Galloway 2022-04-22

If z is complex variable, how to prove $lim_{z \to 0} \frac{z^{4}}{{| z |}^{2}} = 0$
I have simplified to $lim_{z \to 0} \frac{z^{4}}{z} \overset{―}{z} = lim_{z \to 0} \frac{z^{3}}{\overset{―}{z}}$
because ${| z |}^{2} = z \overset{―}{z}$
But how to progress?

Saul Cochran 2022-04-15

For what values of m and n does $\int_{C} z^{m} z^{- n} d z = 0$ and for what values does $\int_{C} z^{m} z^{- n} d z = 2 i π$ ?
I am stuck on this problem, any hint?

Leonard Montes 2022-04-15

Find roots of a complex variable such that $1 + ω^{m} + ω^{2 m} + \dots + ω^{(n - 1) m} = 0$
I need some help understanding the intuition behind the following question:
Consider the root of $z^{n} = 1$ given by $ω = \cos \frac{2 π}{n} + i \sin \frac{2 π}{n}$ . For which integers m is $1 + ω^{m} + ω^{2 m} + \dots + ω^{(n - 1) m} = 0$ ?

Jefferson Pacheco 2022-04-12

Find the following products of complex numbers
$(- 1 - 2 i) (2 + i)$

Rosa Townsend 2022-04-12

Find the distance between the complex numbers in the complex plane.
$- 8 + i$
$- 3 + 8 i$

George Michael 2022-04-09

I was trying to prove that

where w is a complex number with absolute value smaller than 1 and $r \in R$ , r<1. I saw some solution online, which says we substitute $w = m e^{i x}$ so that the LHS becomes ${(m - r)}^{2}$ . How is that true?
We have
$(w - r) (\overset{―}{w} - r) = (m e^{i x} - r) (m e^{- i x} - r) = e^{i x} e^{- i x} (m - r e^{- i x}) (m - r e^{i x}) = (m - r e^{- i x}) (m - r e^{i x})$

Jasiah Vincent 2022-04-09

Use complex variables to find the sum of the Fourier Series:
$\sin (θ) + \frac{\sin (2 θ)}{2^{2}} + \frac{\sin (3 θ)}{2^{3}} + \dots$
where $θ$ is a real variable.

Quinn Dean 2022-04-08

Find the product of the complex numbers $z_{1} = 1 + i$ and $z_{2} = 2 + 2 i$ . Leave answers in polar form.

Blaine Jimenez 2022-04-08

Using complex variables to evaluate real trig integral
$\int_{0}^{2 π} \frac{\cos (2 θ)}{1 - 2 a \cos (θ) + a^{2}} d θ, - 1 < a < 1$
I did the substitution $z = e^{i θ}$ and then got $\cos (2 θ) = \frac{1}{2} (z^{2} + \frac{1}{z^{2}})$ ,so
$\int_{0}^{2 π} \frac{\cos (2 θ)}{1 - 2 a \cos (θ) + a^{2}} d θ = \frac{1}{i} \int_{| z | = 1} \frac{\frac{1}{2} (z^{2} + \frac{1}{z^{2}})}{1 - a z - \frac{a}{z} + a^{2}} d z .$
Did I make a mistake here?

Max Williams 2022-04-07

Find the sum and product of the complex numbers $1 - 4 i$ and $- 1 + 6 i$

Karla Thompson 2022-04-07

Find the product of the complex numbers
$z_{1} = 4 ({\cos 50}^{\circ} + i {\sin 50}^{\circ})$
and $z_{2} = 7 ({\cos 100}^{\circ} + i {\sin 100}^{\circ})$
Leave the answer in polar form.

Izabelle Walsh 2022-04-07

I want to find this limit for complex variable z
$lim_{z \to - 1} (z + 1) \sin (\frac{1}{z + 1})$
In the real case I know sin(z) is bounded by -1,1, and the limit is 0. But in the complex case $\sin (z)$ is not bounded. How can I find the limit or prove that it doesn't exist?

Alan Zavala 2022-04-07

I need to find all complex number z such that $z^{2} + (- 7 - 7 i) z + 49 i = 0$ while using the quadratic formula. I’ve got $z = \frac{(7 + 7 i) \pm \sqrt{- 147 i}}{2}$ and then I don’t know what to do next. Can anyone help?

Algebra is never easy even for those that claim to know it well. Regardless if you are solving quadratic equations with complex numbers or require someone to help you eliminate possible mistakes in your assignment, our list of equations will help you to find the most common answers.

The list of questions that you can see below serves as the basic examples of similar tasks. Solving complex tasks with the help of various examples also starts with probability, which is why remember to check your original problem twice to ensure that nothing is missed!

Algebra II