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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Find the modulus of $|z-5|/|1-3z|$ when z is given

If $z = 3-2i$ then find $$\frac { \left| z-5 \right| }{ \left| 1-3z \right| } $$ I've substituted z by $|z|^2/z$ conjugate but still cant figure out what to do, Thanks in advance
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Square root of $i$

What's my error ? $i^1$ means rotation of $90°$ in anticlockwise manner from positive real axis. So $i^{1/2}$ means rotation of $45°$. So square root of $i$ must have both part positive (real and imaginary). But answers in all books contain negative…
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Logarithmic function in complex number

Show that: $$\cos[i\log(2+\sqrt3)]=2$$ I attempted by taking$(2+\sqrt3)$ into trigonometrical form but i am stuck Please help me out.
Rahul
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What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane?

What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane? Is it $$\frac{-1}{4i}[z_1(z_2^* - z_3^*)-z_1^*(z_2-z_3)+{z_2(z_3^*)-z_3(z_2^*)}]$$ where $w^*$ denotes the complex conjugate?
sai saandeep
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How to simplify $\sqrt{-8}$

How would I go about simplifying square root of $-8$? I know I can rewrite that as $\sqrt{(-1)(8)}$, and then I would get $i\sqrt{8}$, but how do I simplify that $8$ further? Thanks for your help.
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Roots of unity whose sum and product are known

Is cube root of unity is a complex number I know the sum is 0 and product is -1 but I am somewhat confused please give me some idea. Thanks in advance
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Least value of complex expression

If $z_{1},z_{2},z_{3},z_{4}$ are $4$ points on a circle $|z| = 1$ such that $z_{1}+z_{2}+z_{3}+z_{4}=0\;,$ Then least value of expression $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{4}|^2+|z_{4}-z_{1}|^2$ is $\bf{My\; Try::}$ Here…
juantheron
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A fallacy in the imaginary numbers.

$$\sqrt{-5}*\sqrt{-3}=\sqrt{-1*5}*\sqrt{-1*3}$$ $$\sqrt{-1*-1}*\sqrt{5*3}=\sqrt{5*3}$$ $$=\sqrt{15}$$ But we all know that this below is right, $$\sqrt{5}i*\sqrt{3}i=-\sqrt{15}$$ So, please explain the formal result. And any confusion that have been…
mobifz96
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complex number quotient and proof

I am working on a complex number problem set and I have had success so far except for the following, which makes me think that I am missing something rather simple in the initial steps that would clear everything up for me. I have made attempts in…
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Circumference in a complex plane

In this question I learnt that the circumference of a unit circle in a complex plane is just the circumference of any normal circle $2\pi r$. Now I would like to take into account the imaginary character of intervals on the imaginery axis. For this…
Moonraker
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Prove (∀z∈ℂ\{1,-1} : |z|=1)(∃x∈ℝ) where z=(x+i)/(x-i)

I am having trouble proving the next problem: Prove that (∀$z$∈ℂ \ {1,-1} : |$z$|=1)(∃x∈ℝ) where $z=\frac{x+i}{x-i}$ What have I done: I observed complex number $z$ as dots on a circle with radius of 1, because |$z$|=1=$r$ (but $z$ cannot be 1 or…
Asleen
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How to go from $\frac{1}{1+2j}$ to $\frac{1}{5} - \frac{2}{5}j$, where $j^2=-1$?

I am reading a book (DSP First), or mainly skipping through the pages trying to solve various exercises. At some point I came across this How exactly did we go from the second to the last step?
dearn44
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How can this Complex problem?

The problem is $e^z=i+\sqrt{3}$. I put $z= x+yi$ ,then $i+\sqrt{3}$ is $2(\cos\left(\frac \pi 6\right)+\sin\left(\frac \pi 6\right)\cdot i)$. So $e^x$ is $2$ and $y$ is $\frac \pi 6+2k\pi$. ($k$ is integer) Is this right answer?
Parkji
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Number of solutions of a complex equation

Question: Find the number of solutions of the equation: $$z^3 + \frac{3(\bar z)^2}{|z|} = 0$$ I substituted $z = re^{i\theta}$ to convert the equation into: $$r^3e^{i3\theta} + 3re^{i2\theta}=0$$ This can be rearranged to be written as:…
Gummy bears
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What exactly does a quaternion represent?

I am wondering a few things about quaternions. Firstly, besides the $ijk=-1$, etc definition, what exactly is a quaternion? What is their geometric representation and how does this differ from the representation of complex numbers on the complex…
thecat
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