📘 complex numbers

1. Problem (c): Compute $ (2 + 2i)^4$. Note that $2+2i=2(1+i)$.

1. **Add the Complex Numbers:** (a) Given $z_1 = 1 + 2i$ and $z_2 = 2 - 3i$, their sum is:

1. Problem 1: Perform operations on complex numbers.\n(i) Add (3,2) + (9,3):\n\nAdd the real parts: $3 + 9 = 12$.\nAdd the imaginary parts: $2 + 3 = 5$.\nAnswer: $12 + 5i$.\n\n(ii)

1. **Problem (iii): Simplify** $\left( \frac{1}{1 + i} \right)^2$. 2. Start by simplifying $\frac{1}{1 + i}$. Multiply numerator and denominator by the conjugate of denominator, $1

1. সমস্যাটি হল: -17 + 6i\sqrt{2} এর নিয়মিত রূপ (মডিউলাস এবং আর্গুমেন্ট) নির্ণয় করা। 2. একটি যেকোনো জটিল সংখ্যার নিয়মিত রূপ হল $$r(\cos\theta + i\sin\theta)$$ যেখানে $r$ হচ্ছে মড

1. Problem: Simplify each complex fraction to its simplest form. 2. Simplify (3 - 6i)/3i:

1. We start with the given equation: $$\frac{(4 - i)(4 + i)}{2 - i} = x + yi$$

1. The input "ii" looks like the imaginary unit "i" raised to the power of 2, that is $i^2$. 2. Recall that $i$ is defined as the imaginary unit with the property $i^2 = -1$.

1. We start with the given values: $x=1$ and $y=1$. 2. Calculate $z = \sqrt{x^2 + y^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$.

1. **State the problem:** Express the complex number $$z=\frac{(1 - i\sqrt{3})^4}{(1 + i)^3}$$ in polar form. 2. **Find polar form of numerator:**

1. **State the problem:** We are given a complex number $z = x + iy$ that satisfies the equation $$\frac{z}{z+2} = 2 - i.$$ We want to find the values of $x$ and $y$. 2. **Write th

1. **Stating the problem:** Given the complex number $z=x-iy$, we want to find $x$ such that $$|z+3|-|z|=0.$$ 2. **Rewrite the problem:** This means $$|z+3| = |z|.$$

1. **Stating the problem:** We want to find the form of a complex number $z$ if $z$ is equal to the multiplicative inverse of its conjugate $\overline{z}$. That is, $$z = \frac{1}{

1. The problem asks us to find the complex number obtained by rotating the complex number $2+i$ by 90° anticlockwise. 2. A rotation of 90° anticlockwise in the complex plane corres

1. **State the problem:** Find all complex solutions to the equation $$(z+1)^3 = (\overline{z} - 2 + \sqrt{3}i)^3.$$\n\n2. **Rewrite the equation:** Since both sides are cubes and

1. **State the problem:** Express the complex number $1 + i$ in polar form. 2. **Recall the polar form:** A complex number $z = x + yi$ can be represented in polar form as $$z = r(

1. Stating the problem: Simplify the complex numbers $Z_1 = \frac{-1}{i}$ and $Z_2 = \frac{1}{2} + i\frac{\sqrt{3}}{2} \div -\sqrt{3} - i$.\n\n2. Simplify $Z_1$: Recall that $\frac

1. **State the problem:** We want to express the roots of $\left(4-3i\right)^{-\frac{2}{3}}$ in polar form. 2. **Convert the base to polar form:** The complex number $4-3i$ has mod

1. **Énoncé du problème** : Nous devons déterminer l'ensemble des points $M$ d'affixe $z$ tels que :