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Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

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Let $z=x+iy$, where $x,y\in \mathbb{Z}$. Find the area of the octagon whose vertices are roots of $(z\bar z)|z^2-\bar z^2|=1200$

$$|z^2||z-\bar z||z+\bar z|=1200$$ $$(x^2+y^2)(|2x|)(|2y|)=1200$$ $$(x^2+y^2)(|xy|)=300$$ I don’t think there is any realistic way to obtain $x$ and $y$ other than hit and trial. I am asking this question to know if there is one. Thanks! If not…
Aditya
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finding the principal arguement

I have $(1+i)^{13}$ and I need to find the principal argument. I did this: $(1+i)^{13}$ = $(2)^{13/2}(cos(\frac{13pi}{4}) + isin(\frac{13pi}{4}))$ using De Moivre's Theorem, but I dont know where to go from here to find the principal arguement
user556713
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Show that $\sqrt[n]{z}=\{|\sqrt[n]{z}|\left(\cos\left(\frac{\theta}{n}\right)+i\sin\left(\frac{\theta}{n}\right)\right)\zeta^{k}|k= 0,1,\ldots,n−1\}.$

Show that $$\sqrt[n]{z}=\{ |\sqrt[n]{z}| \cdot\left(\cos\left(\frac{\theta}{n}\right) +i \sin\left( \frac{\theta}{n} \right) \right) \zeta^{k} \mid k= 0,1,\ldots,n−1\}.\tag{1}$$ My attempt: Let $z=r(\cos{ \theta }+i\sin\theta )$ and let $w = \rho…
Chairman Meow
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Solve $z^{10}=i$ for $i$ with restrictions

Solve $z^{10}=i$ whose argument is strictly between 120° and 180°. I have no idea how to do these ones we were only taught how to evaluate.
Sarah Roberson
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complex numbers help

Given the following complex numbers: $$ z=1+i\sqrt{3} \qquad w = 0.707 - 0.707i $$ find the cartesian forms of the following expressions: $$ z^2 \bar{w}\qquad\text{and}\qquad \frac{z^3}{w^9} $$ The first one i found the answer to be 1.414 -…
hoyes127
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Proving basic complext numbers properties.

I was given a task to prove the following properties of $\mathbb {C}$ identities $x + 0 = x $ and $ x1 = x$ for all $x\in\mathbb {C} $ additive inverse $\forall a \in \mathbb {C}$ there exists a unique $b \in \mathbb {C}$ such that $a + b =…
Rustem Sadykov
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Complex exponentiation and its meaning

We've come across exponential functions like 2^x ,3^x etc. Let me take one of them, say y=2^x here; if we consider y to be the length of the tree after X years, it has a meaningful explanation, that is, say X=2, then we can say that after 2 years…
user794763
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Finding the difference of square root of conjugate complex number

Find the imaginary part of $\left( {{{\left( {3 + 2\sqrt { - 54} } \right)}^{\frac{1}{2}}} - {{\left( {3 - 2\sqrt { - 54} } \right)}^{\frac{1}{2}}}} \right)$ (1) $-\sqrt 6$ (2) $-2\sqrt 6$ (3) $\sqrt 6$ (4) $6$ My Approach is as follow and none of…
Samar Imam Zaidi
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Finding $a$ such that the complex solutions of $z^4 - 6z^3 + 11az^2 - 3(2a^2 + 3a - 3) z + 1 = 0$ form a parallelogram in the complex plane

Find all values of the real number $a$ so that the four complex roots of $$z^4 - 6z^3 + 11az^2 - 3(2a^2 + 3a - 3) z + 1 = 0$$ form the vertices of a parallelogram in the complex plane. I set $z = x + yi$ due to the given knowledge that $z$ is…
questionasker
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Find the argument of $z = {\left( {2 + i} \right)^{3i}}$

$z = {\left( {2 + i} \right)^{3i}}$ My approach is as follow $z = {\left( {2 + i} \right)^{3i}} = {\left( {{{\left( {2 + i} \right)}^3}} \right)^i} = {\left( {8 + {i^3} + 12i - 6} \right)^i} = {\left( {2 + 11i} \right)^i}$ $\ln z = i\ln \left( {2 +…
Samar Imam Zaidi
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Maximizing the value of $|z|$ where $az^2 + bz + c = 0$

Let $a,$ $b,$ $c,$ $z$ be complex numbers such that $|a| = |b| = |c| > 0$ and $$az^2 + bz + c = 0.$$Find the largest possible value of $|z|.$ I immediately set up equations which told me that $$a_1^2 + a_2^2 = b_1^2 + b_2^2 = c_1^2 + c_2^2$$ and…
questionasker
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How to write in polar form

To write in polar form you use this formula $$z=a+bi=r \left(\cos \theta+i\sin\theta \right)$$ I want the polarform for this rectangular function$$4\sqrt2(-1+i)$$ See this for more information Complex number from a region
user1838781
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Solving complex expression

The value of $(-8\sqrt{-1})^{\left(\frac 16\right)}+(8\sqrt{-1})^{\left(\frac 16\right)}$ is a pure real number. Can you find the real number? My try: $$(-8\sqrt{-1})^{\left(\frac 16\right)}+(8\sqrt{-1})^{\left(\frac 16\right)}\\ =i^{\frac…
Ankita Pal
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$a,b,c,d$ such that $a^n=b^n=c^n=d^n=1$ and $a+b+c+d=1$

Natural number $n$ is said to be Beautiful if there exists four complex number $a,b,c,d$ such that $a^n=b^n=c^n=d^n=1$ and $a+b+c+d=1$ . Show that there exists beautiful number and explain whether 28 is beautiful or not I'm more concerned with how I…
lio
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If $z \in \mathbb C$ such that $|z|+|z-2019|=2019$ then $z \in \mathbb R$

Let $z$ be a complex number such that $|z|+|z-2019|=2019$. Note that $$|z+(2019-z)|=2019=|z|+|z-2019|=|z|+|2019-z|$$ This equality occurs when $0,z,2019-z$ are collinear. But, how to show that z is a real number from that? Note. By using the…
Agung Izzul Haq
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