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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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The image of the curve with equation $z\bar z = 2\operatorname{Im} z $ under the map $w=1/z$

Let $z \in \mathbb{C}$ satisfy $z \overline{z} = 2\Im (z)$ Let $w=\frac{1}{z}$ Find the equation describing the curve $w$ forms on the complex plane and the $z$ that has the minimun distance from that curve. I found that $z$ forms a circle with…
UserX
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Given $w=e^{2i\pi/7}$, perform algebraic manipulations with complex numbers like $w+w^2+w^4$

It is not my own homework and I forgot how to solve this kind of things. Anyway, the following are the statement of the homework and my attempts: The homework: Let $w=e^{2i\pi/7}$, $u=w+w^2+w^4$, and $v=w^3+w^5+w^6$. 1) Calculate $u+v$ and then…
Chiba
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How do you divide complex numbers in polar form?

A question in my textbook asks: Find $\frac{z_1}{z_2}$ if $z_1=2\left(\cos\left(\frac{\pi}3\right)+i\sin\left(\frac{\pi}3\right)\right)$ and $z_2=\cos\left(\frac{\pi}6\right)-i\sin\left(\frac{\pi}6\right)$. I converted $z_2$ to…
user175978
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How find this $a$ such $(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$

Question: find all the complex $a$,such for every complex $z_{1},z_{2}(|z_{1}|,|z_{2}|<1,z_{1}\neq z_{2})$,such $$(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$$ My idea: let $$a=x+yi$$ then $$(z_{1}+z_{2}+2a)(z_{1}-z_{2})\neq…
math110
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Proof of equality of complex numbers needed?

Is $a+bi=c+di\iff a=c, b=d$, where $a,b\in\mathbb{R}$ something that requires proof? My instinct is telling me that proof is required to demonstrate that playing around with real numbers (using field operations) cannot make them 'escape' into the…
Trogdor
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Solving complex equation for z?

How do you solve equations involving $z = a + bi$ and imaginary units? The one I am looking at right now: $$\frac{z-2}{z+1} = 3i$$ If you could help me with this one, I think I can do the rest by myself.
Oscar
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Show that if $z,w\in\mathbf{C}$, $|z|<1$ and $|w|<1$, then $\left|\frac{z-w}{1-\overline{z}w}\right|$<1?

Every way I try to approach this turn it into proving the inequality $|z-w|<|1-\overline{z}w|$. Not sure at all how to approach it at the moment.
user163916
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Solving a complex equation z^6 =- i

I am doing some simple complex equations. z^6 = -i. Allright so i cant get angle from tan of angle = y / x because x = 0. So i can get it either from x = z * cos of angle or y = z * sin of angle. First gives me 90 degress, second one -90. Second one…
goblinsly
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Finding conjugate of a complex number

I am stuck with a really silly question : What is the conjugate of $a\bar c-\bar ac$ ? I calculated it as $\bar ac-a\bar c$ but according to my lecture notes, its conjugate is $a\bar c-\bar ac$ itself, i don't understand how's this happening ? Can…
johny
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What does the Fraktur-R or Re stand for in math?

I have an assignment where we are suposed to find the angle between two complex vectors and we have been given the formula to try to work out the problem and were told by the prof. that he got it from wikipedia. I went to the page to try to figure…
Morg Man
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A tricky complex numbers if and only if proof

For complex numbers $z$ and $w$ prove that $$|z|^2w -|w|^2z = z-w\quad \iff\quad z=w\quad\text{or}\quad z\bar{w}=1.$$ How would you go about solving this problem?
user34304
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Geometric interpretations of an equality.

I need to prove that for complex numbers $w_1, w_2$ and $w_3$ if: $$\frac{w_2-w_1}{w_3-w_1}=\frac{w_1-w_3}{w_2-w_3}$$ then: $$|w_2-w_1|=|w_3-w_1|=|w_2-w_3|$$ by geometric interpretation of the given equality. Thanks.
user135582
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Principal angle and Euler form of cube root of unity.

The principal angles of the complex arguments are in the range $-\pi \le x\le \pi $. If so why does the comlex number $\omega =\frac{\left(-1-\sqrt{3}i\right)}{2}$ is represented by $e^{\frac{4\pi i}{3}}$ = $\cos\left(\frac{4\pi…
Hijaz Aslam
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Failure of De Moivre's Theorem

I know that De Moivre's Theorem does not necessarily work for non-integer powers. The classic counter-example is by considering $\left (\cos \theta + i \sin \theta \right )^n=\cos n\theta + i \sin n \theta$ when $n=\frac{1}{2}$ and setting…
Trogdor
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square of complex numbers

I have this equation from here: but it is not equal to: $$(a + bi)^2 = a^2 + 2abi + (bi)^2.$$ could someone explain me what is the difference between this two calcultion?
Kaja
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