📘 complex numbers

1. **Problem Statement:** Find the values of $n$ such that $\left(1 + \sqrt{3}i\right)^n$ is a real number. 2. **Formula and Important Rules:**

1. **Problem Statement:** Find the modulus and argument of the complex number

1. **Problem Statement:** Calculate the following complex expressions step-by-step:

1. **Problem:** Calculate the following complex expressions: i. $(1 - 2i)^4$

1. مسئله: یافتن ریشه‌های معادله $z^6 = -2 + 6 \sqrt{3} i$ در دستگاه اعداد مختلط است. 2. ابتدا عدد مختلط سمت راست را به فرم مثلثاتی تبدیل می‌کنیم. برای این کار، قدر مطلق و آرگومان آ

1. **State the problem:** Given $x = \cos \theta - i \sin \theta$, find the value of $x - \frac{1}{x}$. 2. **Recall the expression for $x$:** We have $x = \cos \theta - i \sin \the

1. The problem states: "The nth roots of unity form a regular polygon on joining on an Argand diagram." We need to determine if this statement is true or false. 2. Recall that the

1. The problem is to find the value of $z_1 + z_2 - z_3$ given $z_1 = 2 + j3$. 2. However, the values of $z_2$ and $z_3$ are not provided in the problem statement.

1. Problem 18. State the problem: In the Argand diagram the point P represents the complex number $z$ and we are given $|z-1-i|=\sqrt{2}$. 2. Interpretation and locus: The equation

1. Problem: Simplify the following complex quotients and express each result in the form $a+ib$. 2. (a) Compute $\frac{20}{3+i}$.

Problem statement: Work through the listed complex-number exercises: simplify complex fractions (a)–(j), simplify a general fraction in 5, solve quadratics in 6, find complex squar

1. Problem: Given complex numbers $z=3+2i$ and $w=1-4i$, find $z+w$, $z-w$, and $zw$. Step 1: Calculate $z+w$ by adding real and imaginary parts:

1. نبدأ بحل المعادلة المعطاة: $c + di = \frac{5 - i}{1 + i}$. 2. نضرب البسط والمقام في المرافق للعدد المركب في المقام لتبسيط الكسر:

1. The problem asks to show that a complex number $z$ belonging to the set $EC 1 2 - 2 1 5$ is an imaginary number. 2. First, let's clarify what it means for a number to be imagina

1. **State the problem:** Calculate $(-\sqrt{3} + i)^7$. 2. **Convert to polar form:** Let $z = -\sqrt{3} + i$. We find the modulus $r$ and argument $\theta$.

1. The problem is to evaluate the expression $100 \times i$ where $i$ is the imaginary unit defined by $i^2 = -1$. 2. Multiplying a real number by $i$ results in a purely imaginary

1. **பிரச்சினையை விளக்குதல்:** P1, P2 என்ற புள்ளிகள் z1, z2 என்ற சிக்கலெண்களால் ஆகணவரிப்பத்தில் குறிக்கப்படுகின்றன.

1. Énonçons le problème : On a trois nombres complexes $a$, $b$, et $c$ tels que $|a|=|b|=|c|$ et $a+b+c=0$. 2. Puisque $a+b+c=0$, cela signifie que la somme des trois vecteurs com

1. **Énoncé du problème** : Soient $a$, $b$, et $c$ trois nombres complexes tels que $|a| = |b| = |c| = 1$ et $a + b + c = 0$. Il faut démontrer que $a^3 = b^3 = c^3$. 2. **Analyse

1. **Énoncé du problème :** On considère les points A(1), B(3), M(z) et M'(z') dans le plan complexe avec l'application $$z' = \frac{2z - 4}{z - 1}$$ où $$z \neq 1$$.

1. **Énoncé du problème :** Écrire sous forme trigonométrique les complexes suivants et calculer l'expression $$\left(\frac{1 + \sqrt{2} + i}{1 + \sqrt{2} - i}\right)^{20}$$.