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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Inequality involving modulus of a complex number

Show that $$ \left\vert z^{3}+\frac{1}{z^{3}}\right\vert \leq1\Rightarrow\left\vert z+\frac{1}{z}\right\vert \leq1. $$ I have tried with the triangle inequality and the reverse triangle inequality, i.e. $$ |a+b|\le|a|+|b| \text{ and }…
Mihai
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Write $(1+i )$ in polar form, then use this to compute $(1+i)^{10}$

I understand how to write $(1+i)$ in polar form, but how do I use it to compute $(1+i)^{10}$? Thanks for the help!
Nick
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Can we write a complex number z like this Re(z)+Im(z)?

I mean without the i. Re(z) and Im(z) would be real numbers right?
GniruT
  • 977
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sketch set satisfying $|z-2|+|z+2|\le5$

In the complex $z$ plane, $z = x+iy$, sketch the set satisfying the inequality: $|z-2|+|z+2|\le5$ I know from experience that this is an ellipse, but if I just wanted to find the $x$ and $y$ intercepts, is there a relatively quick way to do this,…
JackReacher
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solve complex equation for $x$ and $y$

How to solve this? I have tried to put all $z$s on one side, but I don't have an idea to continue. $$z^3-i(z-2i)^3=0$$
zivce
  • 213
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Finding real numbers such that $(a-ib)^2 = 4i$ Prove that $(a^2 - b^2) = 0$

I sometimes find myself overcomplicating my life... overthinking simple concepts. Here I don't use what's given, i.e., $$(a − ib)^2 = 4i$$ So I might say let $a = 1$ and $b = 1$ then $a = b$ and $a^2 = b^2$ thus $a^2 - b^2 = 0$ Now that seems…
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(-8)^(4/3) is equals with 16 or (-16)*(-1)^(1/3)?

1. $(-8)^{4/3}=\bigl((-8){^4\bigr)^{1/3}}=4096^{1/3}=16$. 2. $$ \begin{align*} (-8)^{4/3} &= (-8)^{1+1/3} \\ &= -8\times(-8)^{1/3} \\ &= -8\times (-1)^{1/3}\times 8^{1/3} \\ &= -2\times 8\times…
Botond
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Find $x$ such that $\frac{(1+ 2i)^2}{x+i}$ is purely imaginary

If $x$ is a real number, $\displaystyle\frac{(1+ 2i)^2}{x+i}$ is purely imaginary, what is the value of $x$? So I expanded the numerator to $-3+4i$, which turns the imaginary number into $\displaystyle\frac{-3+4i}{x+i}$ from here though I'm not…
Lincoln
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Fourth complex roots of $i$

Solve $z^4 = i$. I cannot figure out why the angle of $i$ is $\frac{\pi}{2}$ and how to determine the values of $k$. If someone could show step-by-step that would be great! Thanks.
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Why is it valid to multiply both sides of an equation by its complex conjugate?

This is the closest explanation I can find, although I still don't fully understand. We know what’s happening: division is subtracting an angle and shrinking the magnitude. By multiplying top and bottom by the conjugate, we subtract by the…
halcyon
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Complex numbers - locus of a point

Question: If $z \neq 1$ and ${z^2} \over {z-1}$ is real, then find the locus of the point represented by the complex number $z$. I'm not sure how to approach this question. I attempted to substitute $z = x + iy$, however, that didn't solve the…
Gummy bears
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What is the meaning of Euler's identity?

I know that euler's identity state that $e^{ix} = \cos x + i\sin x$ But e is a real number. What does it even mean to raise a real number to an imaginary power. I mean multiplying it with itself underoot $-1$ times? What does that mean?
mehulmpt
  • 324
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Solve complex equation $z^4=a^{16}$

Let $a$ be some complex number. I have to solve equation $$z^4=a^{16}$$ One is tempted to "simplify" it to $z=a^4$, so it is the solutions. But somebody told me, there are more solutions than that. Is it true and why? It seems counterintuitive.
adm34
  • 45
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Finding complex number defined by 3 equations

Let $z$ be a complex number satisfying $$\DeclareMathOperator{\Re}{Re}\Re[z^4]=1/2$$ $$z\bar{z}+2|z|-3=0$$ $$\arg z \leq \frac{\pi}{4}.$$
Desperado
  • 361
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4 answers

Question about Euler's formula

I have a question about Euler's formula $$e^{ix} = \cos(x)+i\sin(x)$$ I want to show $$\sin(ax)\sin(bx) = \frac{1}{2}(\cos((a-b)x)-\cos((a+b)x))$$ and $$ \cos(ax)\cos(bx) = \frac{1}{2}(\cos((a-b)x)+\cos((a+b)x))$$ I'm not really sure how to get…
Jackson Hart
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