📘 complex numbers

1. **Problem Statement:** Understand the polar form of a complex number and how to multiply and divide complex numbers using this form. 2. **Polar Form of a Complex Number:** A com

1. **Problem Statement:** Given two complex numbers in polar form, $z_1 = r_1(\cos \theta + i \sin \theta)$ and $z_2 = r_2(\cos \varphi + i \sin \varphi)$, we want to understand ho

1. Problem: Divide complex numbers and find real and imaginary parts. 2. Formula: To divide complex numbers $\frac{a+bi}{c+di}$, multiply numerator and denominator by the conjugate

1. The problem is to understand the effect of placing a conjugate bar only on the numerator of a complex fraction. 2. Given the original expression:

1. **Problem Statement:** Show the following properties of complex conjugates for two complex numbers $z_1$ and $z_2$:

1. **State the problem:** Calculate the following complex number expressions given: $z_1 = -2 + 4i$, $z_2 = 3 + 6i$, $z_3 = -7 + 4i$, $z_4 = 3 + 2i$, $z_5 = -1 - 2i$, $z_6 = -4 - 2

1. **State the problem:** We want to simplify the expression $$\sqrt{2} \left(\cos \frac{5\pi}{24} + i \sin \frac{5\pi}{24}\right)^6$$.

1. The problem is to find the value of the expression $11i$, where $i$ is the imaginary unit defined by $i^2 = -1$. 2. Since $i$ is the imaginary unit, multiplying it by a real num

1. **Énoncé du problème :** Calculer $Z^2$ pour $Z = \frac{1 + i\sqrt{3}}{1 - i}$, déterminer le module et un argument de $Z^2$, puis en déduire ceux de $Z$.

1. **Énoncé du problème :** Calculer $Z^2$ pour $Z = \frac{1 + i\sqrt{3}}{1 - i}$, déterminer le module et un argument de $Z^2$, puis en déduire ceux de $Z$. Ensuite, déduire la va

1. The problem is to evaluate and understand the function $f(1) = \left(\frac{12}{17} + \frac{13}{17}i\right)^1$. 2. The formula used here is the power of a complex number. Since t

1. The problem is to evaluate the function $f(1) = \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\right)^1$ and understand its value. 2. The formula used here is the power of a com

1. **Énoncé du problème :** On donne le nombre complexe $Z = (1 + i)(\sqrt{3} + i)$.

1. **State the problem:** Express $$\frac{2(2-3i)^i}{1+5i}$$ in polar form and then evaluate $$\left(\frac{2(2-3i)}{1+5i}\right)^8$$. 2. **Convert complex numbers to polar form:**

1. **State the problem:** We are given two complex numbers $z = -17 - 6i$ and $w = 3 + 1i$. We need to find the value of $u$ that satisfies the equation:

1. **Problem statement:** We are given two complex numbers:

1. The problem is to understand the meaning or context of "ii" in mathematics or related fields. 2. In mathematics, "ii" often refers to the imaginary unit $i$ raised to the power

1. **State the problem:** Find the value of $ (1 - i)^4 $ using De Moivre's theorem. 2. **Recall De Moivre's theorem:** For a complex number in polar form $ r(\cos \theta + i \sin

1. **State the problem:** Convert the complex number $\frac{\sqrt{6}}{2}(i-1)$ to polar form. 2. **Recall the polar form:** A complex number $z = x + yi$ can be written in polar fo

1. **State the problem:** Find the roots of the complex equation $$z^3 + 27 \operatorname{cis}\frac{3\pi}{4} = 0$$ where $$\operatorname{cis}\theta = \cos\theta + i\sin\theta$$. 2.

1. **Stating the problem:** We are given two complex numbers $z_1 = 3 - i$ and $z_2 = -3 + i$. We need to find their modulus and principal argument, and express them in polar form.