Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

19229 questions
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Roots of $(z^2-z)^4=81$ in polar form

Question : Find the roots of $(z^2-z)^4=81$ in polar form. What i've done : I'm thinking about using De Moivre's Theorem. Let $w=z^2-z \Rightarrow w^4=81$ $\begin{align} &\Leftrightarrow w^4=3^4\,\text{cis $2\pi$}\\ &\Leftrightarrow w=3\,\text{cis…
user516076
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Find the roots of $z^4-3z^2+1=0$ in polar form.

Question : Prove that the solutions of $z^4-3z^2+1=0$ are given by : $$z=2\cos{36^\circ},2\cos{72^\circ},2\cos{216^\circ},2\cos{252^\circ}$$ My work : First of all, i want ro find the roots with quadratic…
user516076
  • 2,200
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Module & argument of complex number exp(ix) +exp(2ix)

I tried unsuccessfully to solve the following complex expression and get the module and the argument. ${e}^{ix}+{e}^{2ix}$ I converted the whole expression to trigonometric function cos and sin but it got more complex than it looks at the…
SAM.Am
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Complex number sine-cosine simplification: I can't understand this step.

Good evening, I'm working on this step coming from a differential equation. I have: $A\cos(\frac{kL}{2})+Ai\sin(\frac{kL}{2})+B\cos(\frac{kL}{2})-Bi\sin(\frac{kL}{2})=0$ $(A+B)\cos(\frac{kL}{2})+(A-B)i\sin(\frac{kL}{2})=0$ This expression is set…
muserock92
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Find real numbers c and d from $\frac{1}{a+bi}$

Suppose $a$ and $b$ are real numbers, both not 0. Find real numbers $c$ and $d$ such that $$ \frac{1}{a+bi} = c + di$$ I am not really sure what the question is asking me to do. Am I supposed to represent $c$ and $di$ both in terms of $a$ and $b$…
Evan Kim
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Is this set region?

$$ |z^2 - 1| < 1 $$ Hint: use polar coordinates. the answer is not a region. I don't know how to start. Whenever I am trying to do, it failed. *. z is complex number.
jakeoung
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Complex numbers, polar and rectangular form

Express z=(6-2i)(1-3i) in polar form and calculate z^4. Express results in both polar and rectangular form. Workings: (6-2i)(1-3i) 6-20i-6 0-20i. z^4=(-20i)(-20i)(-20i)(-20i) z^4=0+160000i -> Rectangular Form. Tan…
Alan Marsh
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Can square roots be negative?

1955 AHMSE Problem 20 asks when $\sqrt{25 - t^2} + 5 =0.$ I know square root of real numbers cannot be negative. So t cannot be real. But I don't know whether imaginary numbers' square root can be negative or not. I think square roots can never be…
Ram Keswani
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Finding product of trigonometric values (cos/sin) using complex number/roots of unity

Using the ninth root of unity of 1, show that $cos\frac{π}{9}cos\frac{2π}{9}cos\frac{4π}{9}=\frac{1}{8}$. Here is my solution. Let $\omega=cos\theta+isin\theta$, As $\omega^9=1$, $$cos9\theta+isin9\theta=1$$ so $$9\theta=2nπ {(n\in…
Winson
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For real part of $z$ and $w >0$, prove $\left|\frac{z - w}{\bar z + w}\right|< 1$?

So I use the rule $$|z|^2 = z \times\bar z$$ and then in the end I got $$\frac{(|z|^2 + |w|^2 - z\times\bar w-w\times\bar z)}{(|z|^2 + |w|^2 + \bar z\times\bar w+w\times z)}$$ Any help to continue with that what I got? Thank you for all your…
Niki Sedlarevič
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Find all complex numbers z, for which is $(\frac{(z - i - 1)}{(iz + 1)})^2$ real number?

When I put in $z = x + iy,$ in the end I got: $${x^2 - 2x - y^2 + 2y + i (2 - 2y - 2x + 2xy) \over -x^2 + y^2 - 2y + 1 + i (2x - 2xy)}$$ Imaginary part is equal to $0$ and when I did that in the end I got: $x - xy + y = 1$ and $x (1-y) = 0.$ From…
Niki Sedlarevič
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Is it possible to express $(\sin(x))^x$ as a complex number, if it is, then how is it done?

I have looked at the graph and I know it is real for all $x$ that satisfy $2\pi \mathbb{Z}
Ekadh Singh - Reinstate Monica
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Prove $32 \sin^6 x = -15 \cos (2x) + 6 \cos (4x) - \cos (6x) + 10$?

I already tried to do it, but my answer is incorrect! This is how I do it. assuming $$2 \cos x = z + \frac1z\space\space\text{ and }\space\space 2j \sin x = z - \frac1z$$ $$(2j \sin x)^6 \cdot \frac12 = \left(z - \frac1z\right)^6 \cdot…
rosorosoo
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Multiplying polar complex numbers ( Check my work )

I am multiplying complex numbers, but not sure if I am doing it right. Are the answers below to the questions b) and d) correct? These answers came back as wrong. Can anyone see where i went wrong?? Thought they was correct.
Kyle Anderson
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Value of $|z_{1}|^2+|z_{2}|^2+|z_{3}|^2$ is

If $|z_{1}+z_{2}|=|z_{1}|-|z_{2}|=2$ and $|2z_{2}+2i(z_{3}-z_{2})|=|2iz_{3}+(1-2i)z_{2}|=10$ Where $z_{1}=3+4i.$ Then value of $|z_{1}|^2+|z_{2}|^2+|z_{3}|^2$ is Try: Let $z_{2}=a+ib$ and $z_{3}=c+id$. Then we have $(a+c)^2+(b+d)^2=4.$ and…
DXT
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