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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Why do $e^i$ and $i^e$ both have absolute value 1?

Why do $e^i$ and $i^e$ both have absolute value 1? I don't know how to view $i^e$ as a complex number. What is its real part and imaginary part?
christine
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Find all complex numbers $z$ such that $|z| = 1$ and $\Im((z+1)^{2020}) = 0$

I've stumbled upon an interesting problem. The task is to find all complex numbers $z$ such that $$|z| = 1$$ and $$\Im((z+1)^{2020}) = 0$$ So far I found, that it's possible to follow these steps: $$u = z+1$$ $$\Im(u^{2020}) = 0$$ $$\sin(2020x) =…
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Argand Digram when arguments are equal

I am helping a student to draw the following equation on complex plane: $$\arg\left[z-\left(3+i\right)\right]=\arg\left[z-\left(1+3\,i\right)\right]$$ Can anyone explain why the dashed line to be neglected.
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How to find the complex number based on its argument?

Suppose I've the following: $$\arg(2+i5-z) = 0.9$$ How can I find $z$? Wolfram says it's $-1.96$, but I didn't understand how to get this value. EDIT: I wrote originally that the answer was 2, but it's in fact 0.90.
FY Gamer
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Sketching a set of $Re(z^2(1+i))$ on the Complex plane.

So I am trying to sketch a set of complex numbers that is characterized by inequality: $ Re((1+i)z^2) < 0 $. I know that if I take $ z = a + bi $ , then: $$Re(z^2) \implies (a + bi)^2 = a^2 + 2abi - b^2 \implies Re(z^2) = a^2 -…
theman
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Solution of |x| + 3x = 2 + 6i. Where is an error in my thinking?

So I solved the equation: $|x| + 3x = 2 + 6i$ but I don't know where my error is. And I know there is an error because Wolfram Alpha shows that the only solution is $ x = 2i $ . In my calculations I have 2 solutions. $$|x| + 3x = 2 +…
cocacola
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Complex number inequality plot - modulus of a fraction

I have a question. Consider the following: $$|\frac{z-2}{z+1-i}|\ge1$$ Here's what I did: $$\frac{|z-2|}{|z+1-i|}\ge1$$ $$ \frac{\sqrt{(x-2)^2+y^2}}{\sqrt{(x+1)^2+(y-1)^2}} \ge1$$ If we square both sides we get $$ \frac{(x-2)^2+y^2}{(x+1)^2+(y-1)^2}…
john doe
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How do I prove |z-i|=2 with $z - i = 2\cos\theta - 2i\sin\theta $?

I have the following question. It's basically my first day doing complex numbers, so I am absolutely lost here. I have read that the modulus-arg form is $$ z = r(\cos\theta + i \sin\theta)$$ Now, in this case, I tried expanding the equation given…
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Can the argument of a complex number be negative? If yes, what is meant by a negative argument if observed graphically?

Please find the modulus and argument of the complex number given by $$z = \frac{(1 + cosθ + isin)^5}{cos3θ + sin3θ}$$ My solution :- $$z = (2cos^2\frac{θ}{2} + isinθ)^5(cos3θ - isin3θ)$$ $$z = ((2cos\frac{θ}{2})(cos\frac{θ}{2} +…
Aman Jain
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How to find $\operatorname{Re}\left(1+e^{i\triangle\varphi}+e^{i2\triangle\varphi}+...+e^{i\triangle\left(N-1\right)\varphi}\right) $

So, actually this is a calculation I'm struggling to make in physics exercise. I have to find $\operatorname{Re}\left(1+e^{i\triangle\varphi}+e^{i2\triangle\varphi}+...+e^{i\triangle\left(N-1\right)\varphi}\right) $ and it's supposed to be something…
FreeZe
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Minimum value of magnitude of complex expression

If $|2z_1 + \bar z_2| = 2\sqrt2$ and $|1 + 2z_1z_2 | = 3 $ then minimum value of $(|z_1|^2 + 4|z_2|^2)$, is:- Now $|2z_1 + \bar z_2|^2 = 8$ $(2 z_1 + \bar z_2)(2\bar z_1 + z_2) = 8$ $4z_1\bar z_1 + 2 z_1z_2 + 2\bar z_1\bar z_2 + z_2\bar z_2 =…
Mathking
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Calculations with complex numbers

So I encountered this one question today in my math book, and I don't know how to get the right answer even though it seems really easy, I just wanna know how to do it so i can get some sleep. This is the question: When the complex number…
mikejacob
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What is the value of real number n to the power of i?

More specifically, what is $e^i$ (exact value)? Can I use regular exponent rules for imaginary numbers? Do regular exponent rules apply to both (like a complex number)?
John Liu
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Solve complex equation with $\overline z$

I need help solving this task, if anyone had a similar problem, it would help me. The task is: Solve the equation in a set of complex numbers. $z^3=\overline z$ I tried this : $z^3=\overline z\\\frac{z^3}{\overline…
LogicNotFound
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Solving $z^{n}Re(z)=z^{-n}Im(z)$

So, I came across this equation and started playing around with it to see if I could solve it exactly. The first thing I noticed is that if I assume $z$ lies in the real or the imaginary axis the only possible solution is $0$. But this cannot be a…