Complex Numbers with In-Depth Explanation

How to prove that $\cos^{2} (z) + \sin^{2} (z) = 1$ , where z is a complex variable (if it is true)?

Blaze Shepherd 2022-04-06

For every real number $a > 0$ . Find all complex number z, satisfying the equation $z | z | + a z + i = 0$

Gretchen Barker 2022-04-06

Find the quotient $\frac{z_{1}}{z_{2}}$ of the complex numbers $z_{1} = 20 ({\cos 75}^{\circ} + i {\sin 75}^{\circ})$
$z_{2} = 4 ({\cos 25}^{\circ} + i {\sin 25}^{\circ})$
Leave answers in polar form.

Willie Kelley 2022-04-06

Find the product $z_{1} z_{2}$ and the quotient $\frac{z_{1}}{z_{2}}$ . Express your answers in polar form.
$z_{1} = \frac{7}{8} (\cos (35^{\circ}) + i \sin (35^{\circ})), z_{2} = \frac{1}{8} (\cos (135^{\circ}) + i \sin (135^{\circ}))$
$z_{1} z_{2} = ?$
$\frac{z_{1}}{z_{2}} = ?$

Kali Thomas 2022-04-06

Find the product of the complex numbers.Leave answers in polar form.
$z_{1} = 4 (\cos \frac{3 π}{7} + i \sin \frac{3 π}{7})$
$z_{2} = 10 (\cos \frac{4 π}{7} + i \sin \frac{4 π}{7})$

Rubi Riggs 2022-04-05

Find the radius of convergence of the power series:
$\sum_{n \geq 0} \frac{1}{{(n + 3)}^{2}} z^{n}$
Describes the convergence domain $(Ω)$ and study the convergence point on the border $(Ω)$ .

measgachyx5q9 2022-04-04

The distance from the origin to the point that corresponds to a complex number is known as the modulus of that complex number. There is an interesting relationship involving the moduli of two complex numbers and the modulus of their product. Spread the work around everyone try a few examples and try to figure out what this relationship is.

Makayla Stevens 2022-04-01

Subtract the following two complex numbers: $(- 5 + 3 i) - (4 + 7 i)$ . What do you notice?

Oxinailelpels3t14 2022-03-30

I need helping with simplifying expression:
$\frac{{(1 + i)}^{33}}{{(1 - i)}^{33}} + {(1 - i)}^{10} + (2 + 3 i) (2 - 3 i) - i^{- 7}$
What I got is:
$\frac{{(1 + i)}^{33}}{{(1 - i)}^{33}} + {(1 - i)}^{10} + (2 + 3 i) (2 - 3 i) - i^{- 7} = \frac{{(1 + i)}^{33}}{{(1 - i)}^{33}} + {(1 - i)}^{10} + 13 - i^{- 7}$
My problems begin with
$\frac{{(1 + i)}^{33}}{{(1 - i)}^{33}}$
is it possible to simplify it without de Moivre or Euler's formulas?

Paula Good 2022-03-27

Consider a function $f : C \mapsto R$ . As far as I know, for this function $f (x, y) = u (x, y) + i v (x, y)$ to be analytic, it must satisfy the Cauchy-Riemann conditions, which state:
$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$
$\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$
Of course, v(x,y)=0. This implies that u(x,y)=C where C is constant, and thus the only real function of a complex variable is f(z)=C.
This cannot be right, surely?

Adolfo Hebert 2022-03-26

Convert the following Complex Numbers into Polar and Exponential Form
a) $\frac{1}{2} + \frac{\sqrt{3}}{2} i$
b) $\sqrt{3} - i$

markush35q 2022-03-25

Given $f (z) = \frac{1}{z^{2} (1 - z)}$ I am to find two Laurent series expansions. There are two singularities, z=0 and z=1. So for the first expansion, I used the region 0<|z|<1 and I got $\sum_{n = 0}^{\infty} z^{n} + \frac{1}{z} + \frac{1}{z^{2}}$ . The second expansion is for the region $1 < | z | < \infty$ . I don't know how to approach this, the explanation in my book is confusing. Any help?

cleffavw8 2022-03-23

Let $a_{0}, a_{1}, \dots, a_{n - 1}, a_{n}$ complex numbers with $a_{n} \neq 0$ . If
$f (z) = | a_{n} + \frac{a_{n - 1}}{z} + \dots + \frac{a_{0}}{z^{n}} |$
Exist $lim_{| z | \to \infty} f (z)$ ?

Aiden Delacruz 2022-03-17

a) What is the sum of: $(3 - 4 i)$ and $(- 21 - 6 i)$ ?
1) $- 24 - 10 i$
2) $18 - 2 i$
3) $- 18 - 10 i$
4) $24 + 2 i$
b) Simplify $\frac{2 - 3 i}{2 + 2 i}$
1) $- \frac{1}{24}; - (\frac{5}{4}) i$
2) $- \frac{1}{24}; + (\frac{5}{4}) i$
3) $\frac{1}{22}; - \frac{1}{22}; i$
4) $\frac{1}{22}; + \frac{1}{22}; i$

Max Grant 2022-03-16

A question about the number of solutions for the following equation:
$z^{2} + (1 - i) z - 3 i = 0$
So the solutions are:
$z_{1, 2} = \frac{- 1 + i \pm \sqrt{10 i}}{2}$
But $\sqrt{10 i}$ has two options with itself: $\sqrt{5} + \sqrt{5} i$ and $- \sqrt{5} - \sqrt{5} i$ , so the equations has 4 solutions?

Ben Shaffer 2022-03-16

Calculating probabilities for complex random variables
I am having some trouble understanding/formulating how one computes probabilites given a (somehow complex) continuous random variable. For example, if I define a random variable Z as: $Y = 10 (2 + μ + σ X)$ where X has standard normal distribution. How would I, for example, formulate/define P(Y>0) for some $μ$ and $σ$ ?

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

How to prove that $\cos^{2} (z) + \sin^{2} (z) = 1$ , where z is a complex variable (if it is true)?

For every real number $a > 0$ . Find all complex number z, satisfying the equation $z | z | + a z + i = 0$

Find the quotient $\frac{z_{1}}{z_{2}}$ of the complex numbers $z_{1} = 20 ({\cos 75}^{\circ} + i {\sin 75}^{\circ})$
$z_{2} = 4 ({\cos 25}^{\circ} + i {\sin 25}^{\circ})$
Leave answers in polar form.

Find the product $z_{1} z_{2}$ and the quotient $\frac{z_{1}}{z_{2}}$ . Express your answers in polar form.
$z_{1} = \frac{7}{8} (\cos (35^{\circ}) + i \sin (35^{\circ})), z_{2} = \frac{1}{8} (\cos (135^{\circ}) + i \sin (135^{\circ}))$
$z_{1} z_{2} = ?$
$\frac{z_{1}}{z_{2}} = ?$

Find the product of the complex numbers.Leave answers in polar form.
$z_{1} = 4 (\cos \frac{3 π}{7} + i \sin \frac{3 π}{7})$
$z_{2} = 10 (\cos \frac{4 π}{7} + i \sin \frac{4 π}{7})$

Find the radius of convergence of the power series:
$\sum_{n \geq 0} \frac{1}{{(n + 3)}^{2}} z^{n}$
Describes the convergence domain $(Ω)$ and study the convergence point on the border $(Ω)$ .

(6+4i)-(7+2i)=?

Solve. (5+5i)+(4+4i)=?

What is 6+7i plus 2+4i?

What is 2+5i plus 3+6i?

Subtract the following two complex numbers: $(- 5 + 3 i) - (4 + 7 i)$ . What do you notice?

Convert the following Complex Numbers into Polar and Exponential Form
a) $\frac{1}{2} + \frac{\sqrt{3}}{2} i$
b) $\sqrt{3} - i$

Let $a_{0}, a_{1}, \dots, a_{n - 1}, a_{n}$ complex numbers with $a_{n} \neq 0$ . If
$f (z) = | a_{n} + \frac{a_{n - 1}}{z} + \dots + \frac{a_{0}}{z^{n}} |$
Exist $lim_{| z | \to \infty} f (z)$ ?

Complex numbers Explained: In-Depth Examples, Equations, and Expert Insights