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.

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Complex numbers question - sum of three complex numbers

Seems an easy one but i can't figure it out: $z_1+z_2+z_3=0$ $|z_1|=|z_2|=|z_3|=1$ Need to prove the following: $z_1^2+z_2^2+z_3^2=0$ Thanks!
user3343396
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A set in a disc

Let $z_1,z_2$ be any two given complex numbers and $r$ be a nonnegative real number. I am trying to find a smallest closed disc containing the set $\left\{z\,\big|\,\left|\frac{z-z_1}{z-z_2}\right|\leq r\right\}.$ May I seek your help in proceeding…
user159888
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Can we show a sum of symmetrical cosine values is zero by using roots of unity?

Can we show that $$\cos\frac{\pi}{7}+\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{5\pi}{7}+\cos\frac{6\pi}{7}=0$$ by considering the seventh roots of unity? If so how could we do it? Also I have observed…
UnsinkableSam
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Find the complex numbers $z$ such that $w=\frac{2z-1}{2+iz}$ is real

Find the complex numbers $z$ such that $w=\dfrac{2z-1}{2+iz}$ is real. I have been trying to separate the imaginary part from the real one, so I can cancel the imaginary one. Trouble is that, by manipulating $w$, it just seems to get much worse.…
Sebas Martinez Santos
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minimum value of $\bigg||z_{1}|-|z_{2}|\bigg|$

If $z_{1}\;,z_{2}$ are two complex number $(|z_{1}|\neq |z_{2}|)$ satisfying $\bigg||z_{1}|-4\bigg|+\bigg||z_{2}|-4\bigg|=|z_{1}|+|z_{2}|$ $=\bigg||z_{1}|-3\bigg|+\bigg||z_{2}|-3\bigg|.$Then minimum of $\bigg||z_{1}|-|z_{2}|\bigg|$ Try: Let…
DXT
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Meaning of the sign of a complex number

What is the meaning and the practical uses of the sign (signum function) of a complex number $z$ defined as $\frac{z}{|z|}$? Does it also extend to quaternions?
plasmacel
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How to solve for $z$: $|z|=z+\bar{z}$

I'm not sure how to figure this out (how to solve for $z$): $$|z|=z+\bar{z}$$ What I did was, Let $$z=a+bi $$ $$\sqrt{a^2+b^2}=(a+bi)+(a-bi)$$ $$\sqrt{a^2+b^2}=2a$$ $$a^2+b^2=4a^2$$ I'm not sure what to do from here to solve for $z$...
user565804
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If the complex number Z satisfies $ |Z^2 - 9| + |Z^2| = 41 $ then the true statements among the following are?

If the complex number Z satisfies $ |Z^2 - 9| + |Z^2| = 41 $ then the true statements among the following are ? $A)$ $|Z+3| + |Z-3| = 10$ $B)$ $|Z+3| + |Z-3| = 8$ $C)$ Maximum value of $|Z|$ is $5$ $D)$ Maximum value of $|Z|$ is 6 (More than one…
SmarthBansal
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How to prove that at least two of three complex numbers have the same absolute value?

Let $a, b,c$ complex numbers such that $a + b + c =0$ and $a^n + b^n + c^n =0$ for some integer $ n \geq 2$. Then two of them have the same absolute values. I try a recursive method but doesn't work. Can you give me an idea? Thank you!
Almath
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Proving that a complex number is nonreal.

Let $m$ be a nonzero complex number such that and $z=-1+im$ and $w=-1-im$. Prove that the number $$\frac{m-w}{z-w}$$ is nonreal. I've tried all sorts of approaches to this question but there seems to be something I'm missing. Any help would be…
fuzzypixelz
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If $u = 2 + 3i$ and $v = -5 + 4i$, then $u + v =$?

I have an exercise, it says: If $u = 2 + 3i$ and $v = -5 + 4i$, then $u + v =$ ? My answer: $$u + v = (2 - 5) + (3 + 4)i = -3 + 7i$$ but the correct answer is $7 - 3i$. What is wrong with my solution?
Jav. dev
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Splitting the square root of complex function into real and imaginary parts

I have these functions below: $$\sqrt{(x+iy)^2-a^2}$$ $$\frac{b(x+iy)}{\sqrt{(x+iy)^2-a^2}}$$ How do I split these to get the real and imaginary parts of these functions? If anyone could help me out, that really would be helpful!!!! It will help me…
MousumiG
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Complex numbers: With conjugate

I've just started calculating complex numbers (last time I calculated with complex numbers was in high school) and I've already got stuck at this exercise: $$3z-i\bar z = 7-5i$$ where $\bar z$ is the conjugate of z. What I've tested so far is to…
Rob
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Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root

Let $P$ be a degree $3$ polynomial with complex coefficients such that the constant term is $2010$. Then $P$ has a root $a$ with $|a| > 10$. how can I show that the above statement is true/false? Can anyone help?
rahu
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Complex number $i^{{i^{i^{.^{.^.}}}}}$

If $A+iB=i^{{i^{i^{.^{.^.}}}}}$ Principal values only being considered, Prove that (a)tan $ \frac {\pi}{2} $A= $\frac{B}{A}$ (b) $A^2 + B^2 = e^{-\pi B}$ I tried the concept A+iB= $y=i^y$ $i= e^{ \frac{i\pi}{2}}$ $\ln(A+iB)=i…
Samar Imam Zaidi
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