📘 complex numbers

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 $$\left(\frac{1 + \sqrt{2} + i}{1 + \sqrt{2} - i}\right)^{20}$$.

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:**