# Tagged Questions

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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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 ...
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### Argument of complex numbers

If $z=re^{i\theta}$ and $w=\rho e^{i \phi}$ are two complex numbers, then $arg(zw)=arg (z)+arg (w)$ But if $z=-1$ and $w=-1$, we get $0= 2\pi$ which is not correct. So why it gives us this ...
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### Why does the Mandelbrot shape show up in other fractals?

In the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fractal ...
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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.
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### A geometric approach to this problem?

Question: A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z$, where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane ...
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### What is the geometric interpretation of $|z-1|^2+|z+1|^2=4$ for all $z$ such that $|z|=1$?

Show that $|z-1|^2+|z+1|^2=4$ for all z such that $|z|=1$. [Note that $|z|$ refers to the magnitude of z where $z=a+bi$]. I was able to 'prove' the question; however, I cannot think of a geometric ...
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### What is a geometric interpretation of multiplication/division in the complex plane? [duplicate]

How can one visualize the multiplication/division of a complex number, z, by a real number, an imaginary number, or another complex number?
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### What is the kernel of $\phi$?

Let $\phi: \mathbb{C}^* \to \mathbb{R}^*$ with $z \mapsto |z|$ be a homomorphism. What is the kernel of this homomorphism? We know the identity in $\mathbb{R}^*$ is $1$. So we need to find the ...
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### Show the limit exists.

For $|z|\neq 1$,show that the following limit exists: $$f(z)=\lim_{n\to\infty}\frac{(z^n -1)}{(z^n+1)}$$ Is it possible to define f(z) when $|z|\neq 1$ in such a way as to make $f$ continuous? ...
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### Simple Proof of the Euler Identity $\exp{i\theta}=\cos{\theta}+i\sin{\theta}$

my question is too simple. We know all that if we define the exponential function on $\mathbb{C}$ then we define the real part and imaginary part of $\exp{it}$ as $\cos{t}$ and $\sin{t}$. So if we ...
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### Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
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### Are complex numbers complete in every way?

I was told many times a story. Indeed a fascinating one to me as a student learning mathematics. First there were natural numbers. People started adding things and finding solutions to finding the ...
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### What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of $-1$. When I ...
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