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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If $z_1z_2$ is real and non-zero, then $z_1=p\overline{z_2}$ for some real $p$.

Given that the product of two complex numbers $z_1$ and $z_2$ is real and different from $0$, show that there exists a real number $p$ such that $z_1 = p\overline{z_2}$. Please help!!
Sophie
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using phasors to handle complex numbers

I have been reading the post: Simple Complex Number Problem: $1 = -1$ and other similar posts based on the manipulation of complex numbers. If I want to simplify $\sqrt{(-1) \times (-1)}$, I would, using phasor algebra, do $$\sqrt{(-1) \times (-1)}…
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Can i be written as the multiplication of two complex numbers?

Possible Duplicate: what is the square root of i? I know that $i^2=-1$, and so $i$ necessarily equals $\sqrt{-1}$. But is it possible to write $i$ as the multiplication of two complex numbers, i.e. can we find a complex number $z$ so that…
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The Imaginary Unit $i$ as an Operator

Let us consider the real line, and a point $n$ on it which corresponds to the the real number $n$. Let us also imagine a line segment that represents this number $n$ by possessing the length equal to $n$ units to the right of $0$. If we multiply the…
Samama Fahim
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If $(\cos \alpha + i \sin \alpha )^n = 1$ then $(\cos \alpha - i \sin \alpha )^n = 1$

Prove that if $(\cos \alpha + i \sin \alpha )^n = 1$ then $(\cos \alpha - i \sin \alpha )^n = 1$. What should I use? De Moivre's formula? Exponential form? I tried, but It doesn't work.
Thomas
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What steps are taken to make this complex expression equal this?

How would you show that $$\sum_{n=1}^{\infty}p^n\cos(nx)=\frac{1}{2}\left(\frac{1-p^2}{1-2p\cos(x)+p^2}-1\right)$$ when $p$ is positive, real, and $p<1$?
Amory
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Find the circle equation passing through i, 1+i, 2-i in the form of |z-p| = r|z-q|

I know that the circle equation becomes $|w - c/1-r^2| = |c| \frac r{1-r^2}$. I started by assigning $i$, $1+i$, $2-i$ to the variables $z$, $p$, and $q$ respectively. I am not sure I am doing this right. Can anyone help with the steps on how to…
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complex conjugate; prove that $|z+w| \leq |z|+|w|$

Prove that $|z+w| \leq |z|+|w|$. Okay so I labeled $z$ as $z=a+bi$ and $w$ as $w=c+di$. Substituting and solving for everything I got $\sqrt{(a+c)^2 + (b+d)^2} \leq \sqrt{a^2 +b^2} + \sqrt{c^2 +d^2}$. I know if I plug in numbers that this statement…
Ruth Gutierrez
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Is this exercise in complex correctly done?

Find all $z \in \mathbb C$ such: $$|z|=1, Re(z^4)=-Im(z^4) $$ So what I thougt was: First, let $z$ be $$z = |z|e^{ix+2k\pi}, |z|=1$$ then, $$z^4 = e^{i4x+8k\pi}$$ given $$Re(z^4)=-Im(z^4)$$ this only happens if $4x+8k\pi = \pi/4$ or $4x+8k\pi =…
FranckN
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Find product of roots of complex equation $z^{12}=-i$

Find product of roots of complex equation $z^{12}=-i$ Let's apply modulus on both sides: $|z^{12}| = |-i|$ It turns out, that $|z|^{12}=1 \implies |z|=1 = r$ Now, use the polar form: $z=re^{i\phi}$ The argument is: $$\phi =…
stil
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A simple question related to Complex Numbers?

Ok so this was the equation given in my text book $$\implies\sqrt{-a}\sqrt{-a} $$$$= (-1)a $$$$= -a $$ so my question is why can't i solve it this way $$\implies\sqrt{-a}\sqrt{-a}$$$$=\sqrt{(-a)(-a)}$$$$=\sqrt{a^2}$$$$=a$$ so what is wrong with my…
Deiknymi
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Graphing $\sqrt { -x}$

how does my calculator graph ($\sqrt { -x}$. Since I can't graph a complex number, how does my calculator graph the $\sqrt { -x}$ ?
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Dilations - Complex Numbers

Not really sure where to start...hints are appreciated, thanks. In this problem, we will show that the composition of two dilations is, in general, another dilation. (a) Let $ z_0$ be an arbitrary complex number. We perform a dilation on $z_0,$…
Freedom
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Value of the complex expression

How can I calculate the exact value of something like that: $|e^{\sqrt{i}}|$
jaranna
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Determine all complex number for which: $ \arg(Z^6) = \arg(-Z^2),\ \mathrm{Re}(Z^3) = 2 $

While preparing for the next semester, I stumbled upon this complex number problem which kind of confuses me. I know it has something to do with this - but I simply can't think of any proper way to solve it. Here it is: Determine all complex number…
Jinx
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