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.

19229 questions
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Can someone explain how the square root of two negative numbers multiplied together doesn't equal a positive/negative number.

On a test I got a question asking to simplify $\sqrt{-25}\times \sqrt{-3}$ I answered $5\sqrt 3$ but apparently it's only $-5\sqrt 3$ because of some rule of imaginary numbers, could someone please explain to me why $\sqrt{-25}\times \sqrt{-3} \ne…
Dalton
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Problem solving complex equation with high power

I have to solve :$$(i+z)^{16}-(7+7i)(i+z)^8+25i=0$$ I consider this equation like a quadratic equation and find its delta and solutions. But i can't solve two equations :$$(i+z)^8=4+3i$$ $$(i+z)^8=3+4i$$ I really need some helps to solve completely…
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Help Understanding Complex Roots

I was reading a graphical explanation of complex roots, and between Figures 7 and 8 I became confused. The roots appear in the imaginary plane, but I don't understand why the original function must be inverted before the graphical representation…
User3910
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How to express $\ln\left(\frac{1}{1+j}\right)$ in rectangular form?

Trying to understand how to express this complex number $\ln\left(\frac{1}{1+j}\right)$ into rectangular form. I thought to use $x = r\cos(\theta)$ and $ y = r\sin(\theta)$, but I am pretty sure that is wrong.
Geno C
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Given a complex number, convert it to the form $a + bi$

So I'm trying to convert a complex number to the form of $a + bi$. The complex number in question is $$\bbox[5px,border:2px solid #C0A000]{\large 2e^{\frac{-3\pi}{4i}+\ln(3)}}$$ I'm not quite sure how to tackle this to be honest, I would appreciate…
JakeDrone
  • 563
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Finding the smallest $n$ that satisfies $\left(\frac{1-i}{\sqrt{2}}\right)^n=1$

Maybe I'm just stupid, but right now I'm in a dilemma so please help. I had the question $$\left(\frac{1-i}{\sqrt{2}}\right)^n=1$$ where I need to find the smallest number for $n$ ($n> 0$) that would satisfy the problem. My first step was to…
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Complex numbers order property

As a complex number set/field isn't an ordered set/field. Now $1 \in \mathbb{C} $ & $2 \in \mathbb{C} $ . How is $2>1$ ?
user78743
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The maximum value of $\left| {\operatorname{Arg}\left( {\frac{1}{{1 - z}}} \right)} \right|$ for $|z|=1$,$z\ne1$

The maximum value of $\left| {\operatorname{Arg}\left( {\frac{1}{{1 - z}}} \right)} \right|$ for $|z|=1$,$z\ne1$=_____ My approach is as follow Already this question is solved Maximum value of argument but I would like to slve by my approach which…
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What does it mean to evaluate $i^{786}$?

In my book , To evaluate $i^{786}$. They did. $i^{4*196 + 2}$ Why do we have to do it this way? I have to write in the exam in a subjective format. So want to know is this the only correct way to write it. Since if the exponent is sometimes bigger,…
user864449
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How do I determine all complex solutions of $|z|z=-i(\bar{z})$, for all $z\in \mathbb{C}$

Let $z\in \mathbb{C}$. How do I determine all complex solutions of $|z|z=-i(\bar{z})$? My approach: We can see that $$|x+yi|(x+yi)=y-ix$$ Then I came up with the real part $$-y^2+x^2+x^4-y^4$$ and the imaginary part $$2ixy+x^2(2iyx)+y^2(2xiy)$$ I…
Jon
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Solve geometrically: Arg(z(z+1)) = 0 mod(pi)

Solve geometrically the following equation : $arg(z) +arg(z+1)=0 \pmod{\pi}$ My solution: Let: $z=x+iy$ Then: $\arg(z)+\arg(z+1)=0 \pmod{\pi} \Longleftrightarrow z(z+1) \in \mathbb{R}$ Then: $\operatorname{Im}((x+iy) (x+1+iy))=0$ After…
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Understanding Needham's derivation of equivalence between symbolic and geometric rules for complex multiplication

Tristan Needham's Visual Complex Analysis contains a proof that the symbolic (i.e. $(a+ib)(c+id) = (ac-bd) + i(bc+ad)$) and geometric (i.e. "multiply the lengths and add the angles") rules for complex multiplication are equivalent (pages 8-10). The…
IssaRice
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Complex number question?

Are there only finitely many complex numbers $z$ such that $z^{634}=1$? I think there are only finitely many; is that not so?
Logan
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$z^3 = i * \frac{|z^5|}{z* \bar z}$ Did I do it right?

I just want to check whether I understand the basic algebra of complex numbers. I have to find solution to: $z^3 = i * \frac{|z^5|}{z* \bar z}$. So I transform that expression into: $z^3 = i * \frac{|z^5|}{|z^2|} \iff z^3 = i * |z^3|$. Then I take…
theboyboy
  • 397
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qq, can I solve equations with complex numbers like that?

So I have 2 examples: $\frac{27|z|}{z}=\frac{1}{3}\bar z *z^3 \implies 27|z|=\frac{1}{3}\bar z *z^4 \implies 27|z|=\frac{1}{3} |z|^2*z^3 \implies \frac{27}{|z|}=\frac{1}{3} z^3 \implies \frac{27}{ \sqrt{a^2 + b^2}}=\frac{1}{3} (a+bi)^3 $ On the left…
theman
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