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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An interesting inequality involving complex numbers

Let $a,b,c$ be any three complex numbers lying on or inside the unit circle $|z|= 1.$ Let $|a-b|=r, |b-c|=s, |c-a|=t, $ with $r\leq t,$ then the inequality $$r^2s^2+s^2\leq 2r^2+2t^2$$ seems to be true. Whether the circumradius of the triangle …
user159888
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Math question please ? Complex numbers?

I have to solve this equation $5z^2+6z+2=0$ where $z$ is a complex number.. I tried writing $z=\alpha+\beta i$ but still nothing..I tried finding the roots but the discriminant is negative $= 36-4\cdot 5 \cdot 2 =-4$ what do I do?
einsteinwannabe
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Evaluate $\sum_{p=1}^{32}(3p+2)\left[\sum_{q=1}^{10}\left(\sin\frac{2q\pi}{11}-i\cos\frac{2q\pi}{11}\right)\right]^p$

Evaluate$$\sum_{p=1}^{32}(3p+2)\left[\sum_{q=1}^{10}\left(\sin\frac{2q\pi}{11}-i\cos\frac{2q\pi}{11}\right)\right]^p$$ I wanted to convert this problem in the form $e^{i\theta}$ but not able to proceed
Samar Imam Zaidi
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Is the square root of negative 1 equal to i or is it equal to plus or minus i?

I didn't see a duplicate.. The motivation is you tube. Tanton lectures of which one is titled " The Complex number i is NOT the square root of negative one". Does anyone have a clue why this may be true. I did not follow his one line explanation…
Sedumjoy
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How to solve $ z^6 + z^3 + 1 = 0 $?

Can someone explain to me how to solve the equation $ z^6 + z^3 + 1 = 0 $ ? I started working it out and obtained the following: $ z^6 + z^3 = -1$ $ z^2 + z = -1 $ $ (x+iy)^2 + (x+iy) = -1 $ $ x^2 + 2xiy -y^2 = -1 $ And now I'm stuck. What would be…
Math420
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Square root of complex numbers

I’m studying complex analysis. I was reading a section about roots of complex numbers, and I found that $\sqrt{i}$ has two values. ($i$ : imaginary unit) However, for a non-zero real number, $\sqrt{a}$ is always one value. Moreover, $\sqrt{a}$ is a…
ylh0501
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writing Cosines using De Moivre's formula

Given the question: Use De Moivre’s formula to find a formula for $\cos(3x)$ and $\cos(4x)$ in terms of $\cos(x)$ and $\sin(x)$. Then use the identity $\cos^2(x) + \sin^2(x) = 1$ to express these formulas only in terms of $\cos(x)$. I started out…
mathminutemaid
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Polar form of Complex numbers

I'm doing some work with complex numbers and I've come across this exercise in the "Polar form" section. $$(1/2+i(\sqrt{3}/2))^{100}$$ Of course this exercise is manageable with the help of Pascal's triangle and numerous hours of calculating, but I…
Rob
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$|z+w|=|z|+|w|$ iff $z=cw$

I'm trying to prove that if $z,w$ are non non-zero complex numbers then $|z+w|=|z|+|w|$ iff there exists a positive real $c$ st $z=cw$. Now I've proved the leftward implication but I'm having difficulties with the rightward one, so I'd appreciate…
lorenzo
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which of the following statement is true..?

If the cross ratio $Z_1, Z_2, Z_3$ and $Z_4$ is real, then which of the following statement is true? 1)$Z_1, Z_2$ and $Z_3$ are collinear 2)$Z_1, Z_2$ and $Z_3$ are concyclic 3)$Z_1, Z_2$ and $Z_3$ are collinear when atleast one $Z_1, Z_2$ or…
jasmine
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Finding the real and imaginary parts of $\frac{e^{i\theta}}{2-e^{3i\theta}}$

To find the real and imaginary parts of $$\frac{e^{i\theta}}{2-e^{3i\theta}}.$$ I multiplied the expression by the conjugate of the denominator to get $$\frac{2e^{i\theta}+e^{4i\theta}}{4-e^{-\theta^2}}.$$ This then simplified to give…
iza
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Why are the complex numbers unordered?

first post on here! I have been learning about complex numbers, and how they do not satisfy the trichotomy like real numbers do. For example, there is no way to say $i<3$, $i>3$, or $i=3$. But consider this: If $x
Sam
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Complex Number Inequality(Prove)

If $\ z_1$ and $\ z_2$ are two complex numbers such that | $\ z_1$ | < 1 < | $\ z_2$ | then prove that |$(\frac{1-\ z_1 \overline z_2}{\ z_1 -\ z_2})$ |<1 My initial approach $\ z_1=\ r_1 e^{i\alpha}$ where $\ r_1$ $\in$ (0,1) and $\ z_2=\ r_2…
Samar Imam Zaidi
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Calculating the magnitude of a complex exponential function

$$x(t)=5 e^{-i 20\pi t} +5$$ I need to find the magnitude of this function, $|x(t)|$. My idea was to convert from the first have from polar to rectangular form. $$x(t)=5[\cos(-20\pi t)+j \sin(-20\pi t)]+5$$ $$=5\cos(20\pi t)+5-j5 \sin(20 \pi t)$$ I…
cdignam
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The complex conjugate of a complex function.

Hi I'm trying to work out the complex conjugate of: $Ae^{(-a(mx+it))}$. Generally when I have tried to work out a complex conjugate of a complex number I just replace $i$ with $-i$. However I didn't know if it would work this time due to this being…
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