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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Conceptual question about the imaginary number $i$

One of the first things we see in our first complex analysis class is the standard way of introducing us to the imaginary unit $i$ which is to think of a solution to the equation $$x^2=-1$$ Obviously, since a real number has the same sign,…
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Proving that $O^{51} + \bar{O}^{51}$ is real

Let $O \in \mathbb{C}$. How can I prove that $O^{51}+\bar{O}^{51}$ is a real number, or in other words: $\Im(O^{51}+\bar{O}^{51}) = 0$?
Some1
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Complex number equations

I cannot solve two problems regarding complex equations. 1)Let $z^2+w^2=0$, prove that $$z^{4n+2}+w^{4n+2}=0, n \in \mathbb{N^{*}}$$ What I tried; $$z^2 \cdot z^{4n}+w^2 \cdot w^{4n}=0 \iff w^2(w^{4n}-z^{4n})=0$$ but it doesn't really prove…
UserX
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If $ \frac {z^2 + z+ 1} {z^2 -z +1}$ is purely real then $|z|=$?

If z is a complex number and $ \frac {z^2 + z+ 1} {z^2 -z +1}$ is purely real then find the value of $|z|$ . I tried to put $ \frac {z^2 + z+ 1} {z^2 -z +1} =k $ then solve for $z$ and tried to find |z|, but it gets messy and I am stuck. The…
raj
  • 669
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Complex equations with no complex solutions?

Are there complex equations that admit no complex solutions, but rather quaternions or hypercomplex solutions, for example, in complete analogy to, say, the equation $x \times x = -1$ when restricted to the real line? Edit: I am indeed using…
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De Moivre Theorem for Fractional Power: k & n explanation

I don't understand the explanation for k & n in my text book. What are they trying to say by "... for any n consecutive values of k"? In the 1st formula, it says $k=0,1,2,\ldots,n-1$ then next it says $k=n,n+1,\ldots,2n-1$, that will mean…
Jiew Meng
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Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$

Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$ I went wrong somewhere, this is what I have so far (this is in polar): $z=4\left(\cos\left(\frac{11\pi}{6}\right)+\sin\left(\frac{11\pi}{6}\right)\right)…
Joshhw
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Proving a Complex Number is Real

I have the following question. Let $z_1$ and $z_2$ be complex numbers. Assumptions: $|z_1|=|z_2|=1$ $ z_1z_2 \neq -1$ What I have to prove is that: $$\frac{z_1+z_2}{1+z_1z_2}$$ is real. My thoughts: First, I multiplied the numerator and…
Alan
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Set of Points in the Complex Plane

I'm having trouble describing the set: $\{z\in\mathbb{C}:|z-a|=r|z-b|\}$ where $r$ is a positive real number and $a,b$ are fixed complex numbers. I worked out the algebra and it seems to be a (real) equation in two variables each with maximum degree…
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Compute the square root of a complex number

This is a follow up to a previous question. I solved the equation $z^4 - 6z^2 + 25 = 0$ and I found four answer to be $z = \pm\sqrt{3 \pm 4i}$. However someone in the comment said that the answer is going to be $2+i$, $2-i$, $-2+i$, $-2-i$. I…
bman
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Finding the roots of the sum of complex numbers

Find the roots of $(z-1)^6 +(z+1)^6$. So far we've tried binomial expansion, but where to go now, as it is a non-calculator question?
Miss
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Find sum of roots of complex equation $z^6 = z^{18} = -1$

Find sum of roots of equation $z^6 = z^{18} = -1$ Roots of $z^6=-1$ Roots of $z^{18}=-1$ From the circle I can see, that: $\{z : z^6=-1\} \cap \{z : z^{18}=-1\} = \{z : z^6=-1\}$ So, I can notice, that sum of roots $z^6=-1$ is equal to…
stil
  • 383
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If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is

If $11z^{10}+10iz^9+10iz-11 = 0$. Then possible value of $\mid z \mid,$ is $\bf{My\; Try::}$ Given $11z^{10}+10iz^9+10iz-11 = 0\Rightarrow \displaystyle z^9 = \frac{11-10iz}{11z+10i}.$ Now Put $z = x+iy\;,$ we get $\displaystyle (x+iy)^9 =…
juantheron
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Complex logarithm $\log(zw)\neq\log(z)+\log(w)$

Can anyone help me out with explaining why $\log(zw)\neq\log(z)+\log(w)$?
Roos Jansen
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Help with hard complex numbers

We had the topic of complex numbers for my senior math team meet this week, and I wasn't able to solve two of the problems. 1.) $z=i^{\displaystyle \left(i^{\displaystyle \left(i^{(2)}\right)}\right)}$ and $a$ is the real part of $z$, find…