📘 complex numbers

1. **Problem:** Find the modulus $|z|$ of the complex number $z = \frac{2 - i}{2 + i}$. 2. **Formula:** The modulus of a complex number $z = \frac{a}{b}$ is $|z| = \frac{|a|}{|b|}$

1. **State the problem:** Simplify the expression $ (5 - 3i) - (-2 + 5i) $ where $ i = \sqrt{-1} $. 2. **Recall the rule:** Subtracting a complex number means subtracting both its

1. **State the problem:** Simplify the complex expression $$\left(\frac{1-\sqrt{3}i}{1+\sqrt{3}i}\right)^{12}$$. 2. **Recall the formula and rules:** To simplify powers of complex

1. **Problem statement:** Given a complex number $z = x + yi$ satisfying $$\left|\frac{z - (4 - 6i)}{z - (5 - 5i)}\right| = 6,$$

1. The problem is to evaluate the expression $\sin 30^\circ + i \cos 30^\circ$ where $i$ is the imaginary unit. 2. Recall the values of sine and cosine for $30^\circ$:

1. **Problem statement:** Express $ (1+i)^8 $ in the form $ a+bi $. 2. **Formula and rules:** Use the polar form of complex numbers and De Moivre's theorem:

1. **Problem statement:** Given complex numbers $z_1, z_2, z_3$ with magnitudes $|z_1|=1$, $|z_2|=2$, $|z_3|=3$ and sum $z_1 + z_2 + z_3 = 3 + \sqrt{5}i$, find the value of $\opera

1. مسئله: باید مقدار $$\cosh(2 + \frac{\pi i}{4})$$ را به صورت $$a + ib$$ که $$a$$ و $$b$$ اعداد حقیقی هستند، بیان کنیم. 2. فرمول مورد استفاده: تابع هذلولی کسینوس برای عدد مختلط $$

1. The problem involves the complex number in polar form: $z = r(\cos\theta + i\sin\theta)$. 2. This is the polar representation of a complex number, where $r$ is the magnitude (di

1. **Problem Statement:** Use De Moivre's Theorem to find $(\cos \theta + i \sin \theta)^n$ for given $\theta$ and integer $n$. 2. **Formula:** De Moivre's Theorem states that for

1. **Problem Statement:** Given a complex number $z_0$ on the positive real axis between 0 and 1, and points $A$, $B$, $C$, and $D$ representing complex numbers in the plane, deter

1. **State the problem:** We need to find the magnitude $|z|$ of the complex number $$z = \frac{(1 + i)^5}{(\sqrt{3} - i)^2}.$$\n\n2. **Recall the formula for magnitude:** For any

1. **Problem:** Given complex numbers $z_1 = 1 + i$, $z_2 = 3 - 2i$, and $z_3 = -2 + 3i$, calculate the following: 2. **Formula and rules:**

1. **Problem:** Given complex numbers $z_1 = 1 + i$, $z_2 = 3 - 2i$, and $z_3 = -2 + 3i$, calculate $z_1 \times z_2 + 2z_3^*$. 2. **Formula and rules:**

1. مسئله: اعداد مختلط داده شده را به شکل دکارتی بنویسید. 2. فرمول‌ها و قوانین مهم:

1. **Problem 1:** Compute $$\frac{z_1^3 + z_2^3 + z_3^3}{z_1 z_2 z_3}$$ where $$z_1 = 1 + i\sqrt{3}, z_2 = \sqrt{3} - i, z_3 = -1 + i$$. 2. **Step 1:** Calculate each cube.

1. The question "why it -j?" likely refers to the use of the imaginary unit in electrical engineering or complex numbers. 2. In mathematics and engineering, the imaginary unit is d

1. **State the problem:** We are given a complex number $z$ that satisfies the equation $$z + i|z| = 12 + 9i.$$ We need to find the imaginary part of $z$. 2. **Express $z$ in terms

1. **Problem:** Write $\frac{3 + 2i}{4 - 3i}$ in the form $a + bi$. 2. **Formula and rules:** To write a complex fraction in the form $a + bi$, multiply numerator and denominator b

1. Problem: Pretvoriti kompleksni broj $z = -\cos\left(\frac{7\pi}{10}\right) - i\sin\left(\frac{7\pi}{10}\right)$ u trigonometrijski zapis. 2. Formula: Kompleksni broj u trigonome

1. **Problem:** Write each expression in the standard form $a + bi$. **a)** $(-3 - 2i)^2 + \frac{1 + i}{8 - 5i}$