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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Least possible polynomial deegree of complex roots

What is the least possible deegree of polynomial with real coefficients having roots $2\omega , 2+3\omega , 2+3\omega ^2 , 2-\omega -\omega ^2$ As there are four roots so the deegree should be four but the answer is given as five . how ?
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How do I find value of $i^{1/i}$ and $i^{\sqrt{i}}$?

How do I find value of $i^{1/i}$ and $i^{\sqrt{i}}$ ?
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Question about complex numbers (what's wrong with my reasoning)?

Can someone point out the flaw here? $$e^{-3\pi i/4} = e^{5\pi i/4}$$ So raising to $\frac{1}{2}$, we should get $$e^{-3\pi i/8} = e^{5\pi i/8}$$ but this is false.
Paul
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Finding when a complex expression is equal to $0$

How would I check when the expression $$\Sigma_{i=1}^{j-1} (1+\zeta_j^i)^n$$ is equal to $0$, where all $\zeta_j^i$ are the $j$th roots of unity not equal to $1$? Of course, then I would be looking for an expression of $n$ in terms of $j$ or vice…
qt.
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Upper bounds for moduli of contour integral.

Without evaluating the integral, show that $$\left | \int_{C}\frac{dz}{\left ( z^2-1 \right )} \right |\leq \frac{\pi}{3}$$ where $C$is the arc of a circle $\left | z \right |=2$, from $z=2$ to $z=2i$ that lies in the first quadrant.I know …
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Stewart's Theorem on Complex Numbers

Can you prove $$|z_1|^2\cdot [(1-r)\cdot|z_1 - z_2|] + |z_2|^2\cdot(r\cdot|z_1-z_2|)= |z_1-z_2|\cdot(|(1-r)z_1+rz_2|^2+r\cdot(1-r)\cdot|z_1-z_2|^2) $$ (such that $z_1,z_2 \in \mathbb{C}$ and $r\in\mathbb{R}$, $0 \leq r \leq 1$) without constructing…
u123435
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Complex numbers - find real and imaginary parts of $z=(1+i)^{100}$

Currently studying for my calculus exam when I stumbled upon this example: Find the real and imaginary parts of the following $$z=(1+i)^{100}$$ Following the answer to this problem it's first stated that $1+i$ can be written in polar form as…
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What should be the range of the principal value of a complex number?

My lecturer stated that the principal argument is $[0,2\pi)$. However my tutor (my tutor is different to my lecturer) states that it is $(-\pi, \pi]$. Wikipedia article on complex numbers mentions both of these. In my test I used $[0,2\pi)$…
user39193
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Which complex exponent can turn x into a -x?

Actually that is, I would like to use a complex exponent to turn a number x into an -x just by applying some complex exponent to x. I think I would have to use some ln, π and so one, but I am actually stack and cannot find the correct path to…
webdeb
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Expressing the value of $(2\sqrt 3 − 2 i)^4$ in Cartesian form

I got pretty far into this question, but the further I got the more convoluted my answer was becoming to the point that I figured I was doing something wrong. I converted the equation into polar form using de moivre, and ended up getting $z= 4^{1/4}…
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Find complex number(s) $z$ for which $|z|$ has maximum and minimum value if $|z-2+2i|=1$

Find complex number(s) $z$ for which $|z|$ has maximum and minimum value if $|z-2+2i|=1$ My try: I know that $|z-2+2i|=1$ is a circle centered at $(2,-2)$ and having unit radius. Also $|z|$ is the modulus of moving point on this circle and I have…
Rayees Ahmad
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The expression $\sqrt{13+3\sqrt{\frac{23}{3}}} +\sqrt{13-3\sqrt{\frac{23}{3}}} $ is which type of number?

(a) A natural number, (b) A rational number but not a natural number, (c) An irrational number not exceeding 6, (d) An irrational number exceeding 6. Please help with this, i can't manage to simplify it. Thanks in advance.
Idkwoman
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How many solutions does $z + i|z|= 0$ have?

I figured out that it can be transformed in $|z| = iz$ and using the trigonometric method I get: $|z| = |z|(\cos(x) + i\sin(x))(\cos(\pi/2) + i\sin (\pi/2))$ which becomes $|z| = |z|(\sin(x) - i\cos(x))$ I delete $|z|$ from both sides and get $1 =…
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What is $(1 − i)e^{{i\pi}/4}$ equal to?

I don't know where to start... It's a multiple-choice question: I can choose from $\sqrt{2}, 0, 2, 1$ Thank you!
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Find a complex number $w$ such that $w^2=-\sqrt{3} - i$

This is a problem in my undergrad foundations class. \begin{equation} w^2=-\sqrt{3} - i \\w=(-\sqrt{3}-i)^{\frac{1}{2}} \\w=\sqrt{2}\bigg(\cos\frac{5\pi}{12}+i\sin\frac{5\pi}{12}\bigg) \end{equation} So I get to here and the next step is…
Lanous
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