Practice Problems and Examples for Complex Numbers

For the following exercises, solve the equations over the complex numbers.
$x (x - 4) = 20$

Joanna Benson 2022-01-18

Find the product of the complex numbers. Leave the answer in polar form.
$z_{1} = 6 ({\cos 40}^{\circ} + i {\sin 40}^{\circ})$
$z_{2} = 5 ({\cos 20}^{\circ} + i {\sin 20}^{\circ})$

percibaa8 2022-01-18

Find the quotient $\frac{z_{1}}{z_{2}}$ of the complex numbers
$z_{1} = 10 ({\cos 10}^{\circ} + i {\sin 10}^{\circ})$
and $z_{2} = 5 ({\cos 5}^{\circ} + i {\sin 5}^{\circ})$
Leave answers in polar form.

Karen Simpson 2022-01-18

Find the product of the complex numbers.Leave answers in polar form.
$z_{1} = 4 ({\cos 15}^{\circ} + i {\sin 15}^{\circ})$
$z_{2} = 7 ({\cos 25}^{\circ} + i {\sin 25}^{\circ})$

Sandra Allison 2022-01-18

Graph the complex numbers in the complex plane.
a) $- 1 + 3 i$
b) $- 4 - 5 i$
c) 3
d) 4i

2022-01-17

Problem finding zeros of complex polynomial
$z^{2} + (\sqrt{3} + i) | z | {\overset{―}{z}}^{2} = 0$

2022-01-17

Determine the set of values of
$\exp (\frac{1}{z})$ for $0 < | z | < r$

2022-01-17

Why is $\sqrt{- x^{2}} = | x | i$
So, $i = \sqrt{- 1}$

2022-01-17

Is there a problem in defining a complex number by
$z = x + i y ?$

2022-01-17

What are the rules for complex-component vectors and why?

2022-01-17

How to solve that quadratic complex equation?
$\frac{z^{2} - 1}{z^{2} + z + 1}$
So, there cant

2022-01-17

Find $| w |$ without finding w?
w is a complex number that satisfy:
$5 w^{3} - 3 i {| w |}^{2} - 2 i = 0$
Can we find $| w |$ without finding w or easy root?

2022-01-17

How can you find the complex roots of i?
A variation of the Root of Unity problem.
I want to find all possible answers to this:
$z^{n} = i$
Where
$i^{2} = - 1$

2022-01-17

Solve $z^{2} - 6 z + 25$ into complex conjugate

2022-01-17

Argument of the summ of two complex numbers
Let r, s be positive real numbers and $θ, ϕ$ real numbers with $| θ - ϕ | < π$ . Then an argument of $r e^{i θ} + s e^{i ϕ}$ lies between $θ$ and $ϕ$
Can someone give a short, clean proof of the statement above that doesn't rely on geometric intuition?

2022-01-17

Complex-number inequality
$| z_{1} z_{2} \dots z_{m} - 1 | \leq e^{z_{1} - 1 | + \dots + | z_{m} - 1 ∣} - 1$
Can anybody tell me how to prove the following inequality?

2022-01-17

Prove that
$\prod_{k = 1}^{n - 1} \sin \frac{k π}{n} = \frac{n}{2^{n - 1}}$
Using $n^{t h}$ root of unity
${(e^{\frac{2 k i π}{n}})}^{n} = 1$

Regardless if you’re dealing with complex numbers problems or need to find the answers to the complex numbers equation, do not let it bring you down as we shall provide you with complex numbers examples that will help you find the answers. Our complex numbers problems will offer you various solutions that can be matched to your instructions and the problems that you have. It is recommended to start with the complex numbers answers by going straight to your instructions to understand how exactly complex numbers differ. Have a look at our complex numbers questions and come up with solutions.

Precalculus

For the following exercises, solve the equations over the complex numbers.
$x (x - 4) = 20$

Find the product of the complex numbers. Leave the answer in polar form.
$z_{1} = 6 ({\cos 40}^{\circ} + i {\sin 40}^{\circ})$
$z_{2} = 5 ({\cos 20}^{\circ} + i {\sin 20}^{\circ})$

Find the quotient $\frac{z_{1}}{z_{2}}$ of the complex numbers
$z_{1} = 10 ({\cos 10}^{\circ} + i {\sin 10}^{\circ})$
and $z_{2} = 5 ({\cos 5}^{\circ} + i {\sin 5}^{\circ})$
Leave answers in polar form.

Find the product of the complex numbers.Leave answers in polar form.
$z_{1} = 4 ({\cos 15}^{\circ} + i {\sin 15}^{\circ})$
$z_{2} = 7 ({\cos 25}^{\circ} + i {\sin 25}^{\circ})$

Graph the complex numbers in the complex plane.
a) $- 1 + 3 i$
b) $- 4 - 5 i$
c) 3
d) 4i

Problem finding zeros of complex polynomial
$z^{2} + (\sqrt{3} + i) | z | {\overset{―}{z}}^{2} = 0$

Whats

Determine the set of values of
$\exp (\frac{1}{z})$ for $0 < | z | < r$

Why is $\sqrt{- x^{2}} = | x | i$
So, $i = \sqrt{- 1}$

Is there a problem in defining a complex number by
$z = x + i y ?$

What are the rules for complex-component vectors and why?

How to solve that quadratic complex equation?
$\frac{z^{2} - 1}{z^{2} + z + 1}$
So, there cant

Find $| w |$ without finding w?
w is a complex number that satisfy:
$5 w^{3} - 3 i {| w |}^{2} - 2 i = 0$
Can we find $| w |$ without finding w or easy root?

How to prove Eulers

Adding powers of i
Ive

How can you find the complex roots of i?
A variation of the Root of Unity problem.
I want to find all possible answers to this:
$z^{n} = i$
Where
$i^{2} = - 1$

Solve $z^{2} - 6 z + 25$ into complex conjugate

Complex-number inequality
$| z_{1} z_{2} \dots z_{m} - 1 | \leq e^{z_{1} - 1 | + \dots + | z_{m} - 1 ∣} - 1$
Can anybody tell me how to prove the following inequality?

Prove that
$\prod_{k = 1}^{n - 1} \sin \frac{k π}{n} = \frac{n}{2^{n - 1}}$
Using $n^{t h}$ root of unity
${(e^{\frac{2 k i π}{n}})}^{n} = 1$

Understanding Complex Numbers - Examples and Equations