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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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Prove the following identities ..complex numbers

I found this question on my Algebra book but i couldn't answer it Can you please explain step by step Prove the following identities ..explain its geometric meaning $|1+z_1\bar z_2|^2 + |z_1-z_2|^2 = (1+|z_1|^2)(1+|z_2|^2)$
Ahmed emad
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Center of mass of random points in a general disc

I want to prove that for a fixed positive integer $l,$ the center of mass of the $lth$ powers of $m$ randomly and uniformly chosen complex numbers from the disc $|z-c| \leq R,R>0 \quad \text{and} \quad c \in \mathbf{C} $ approaches $c,$ the…
AgnostMystic
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Dot product of two complex vectors where one of them has constant norm

I have two complex vectors, $\mathbf{a}$ and $ \mathbf{b}$ in $\mathbb{C}^n$. The vector $\mathbf{b}$ is a function from $\mathbb{C}^m$ to $\mathbb{C}^n$ with constant norm, that is there is a $c\in\mathbb{R}$ such that…
zdm
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If $|1+z|<\frac12$, then $|1+z^2|>1$

What is the best way to treat proofs like Let $z$ be a complex number. If $|1+z|<\frac12$, then $|1+z^2|>1.$ I have tried first by taking $z=x+iy$ and substitute in the given modulus inequalities and then by working with $z\bar{z}$ and taking…
Dimitris
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The four forth roots of $7 + 24i$ in Cartesian Form

I came across this problem today and need some direction for part (b). a) Show $\tan^{-1}\frac{24}{7} = 4\tan^{-1}\frac{1}{3}$. b) Hence find the four fourth roots of $7 + 24i$ in Cartesian form. Part a) is fine, since I used the fact that $Arg(z) =…
Stephan
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FInd the set of all complex numbers that satisfy argument inequality

The inequality given is as stated bellow: $$\frac{\pi}{2}< \arg\left(\frac{z-i}{z+i}\right) \le \frac{3\pi}{4}$$ There are not any other constraints except $z\ne-i$. I tried letting $z=x+yi$, which leads…
kesetovic
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Show $\cos3\theta = 4\cos\theta(\cos\theta -\cos\frac{\pi}{6})(\cos\theta -\cos\frac{5\pi}{6})$

I need some direction in part (c) of this problem. a) Find the 6 roots of $z^6 + 1 = 0$ (I have solved this using de Moivre's theorem) b) Hence show that $z^6 + 1 = (z^2 + 1)(z^2 - \sqrt{3}z + 1)(z^2 + \sqrt{3}z + 1)$ (no problems here) c) …
Stephan
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Show that $\bigg( \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} +...+ \frac{1}{n^2} \bigg) \frac{(2n+1)^2}{2n(2n-1)} > \frac{\pi^2}{6}$

I came across the last part of this question and need some direction to find the solution. The original problem was to use de Moivre's theorem to show: $$ \sin(2n+1)\theta = \binom{2n+1}{1}\cos^{2n}\theta \sin\theta -…
Stephan
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How To Compute Args and Mods of Complex Numbers That Satisfy Inequalities

A complex number $z$ satisfies the inequality $$|z + 2 - (2\sqrt{3})i|\le 2$$ Find the least possible value of $|z|$ and the greatest possible value of $argz$ the answers given in the text book is $2$ and $\frac{5\pi}{6}$ respectively. Even when…
user866190
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Describing what $A \subseteq \mathbb{C}$ looks like

Given $A = \{z \in \mathbb{C}: \Im(e^{z}) > 0\}$, describe what the set looks like. Note: $\Im(z)$ is the imaginary part of the complex number $z$. So if $z = x+iy$, then according to what we were given, $\Im{(e^z)} = \Im{(e^{x+iy})} = e^y >0$.…
Oran
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Which field should I read to understand extending the natural numbers to integers, up to complex numbers?

I'm a computer science graduate with interest in mathematics, and I'm trying now to read some textbooks as self learning. I want to read about the extension of natural numbers to integers, fractions, real numbers and up to the complex numbers. I…
Loai Ghoraba
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If $\frac{3+2i\cos\theta}{1-3i\cos\theta}$ is purely imaginary, find $\sin^2(3\theta)+\cos^2{\theta}$

Let $$z=\frac{3+2i\cos\theta}{1-3i\cos\theta},$$ such that $z$ is purely imaginary and $\theta \in\left(0, \frac{\pi}2\right) $, then find $$\sin^2(3\theta)+\cos^2{\theta}.$$ Let $z=ni$, where $n$ is any real…
math and physics forever
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If $z$ is a non-real complex number , find the minimum value of $\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5}$

If $z$ is a non-real complex number , find the minimum value of $$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5}$$ where $\operatorname{Im}(z)$ is the imaginary part of a given complex number $z$. My attempt: Let $z =x+iy$ so we…
math and physics forever
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If a complex number, $z$, satisfies the equation $z+ \sqrt2 |z+1| +i = 0, \text{ find } |z|$

If $z+ \sqrt2 |z+1| +i = 0, \text{ find } |z|$ My attempt: As the RHS $= 0$, the sum of the real parts and imaginary parts are both $0$. As the amplitude of a complex number is always real, $z + i = 0 \iff z=i$. Moreover, $|i+1|=0$. Thus,…
math and physics forever
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Complex Logarithm Difference

We know that $\log z = \log\vert z\vert + i\arg(z)$. $\log z$ and $\log\vert z\vert$ are only different by a bounded function. (More specifically by $i\pi$ since $\vert i\arg(z)\vert\le\pi.)$ Now however I'm interested in the case $$\log(1+c\vert…
bochner.martinelli
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