Mathematics Stack Exchange
Mathematics Stack Exchange

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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What is the solution(s) to $1^i$?

$1^x$ is always $1$ with real numbers, but everything gets more complicated with complex numbers. Using Eulers formula, you know that $$e^{ix}=\cos(x)+i\sin(x)$$ If you make x=2π into this you'd get $$e^{i2π}=1$$ and raising each side to the power…
Banbadle
  • 265
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Why is $\sqrt{xy}=\sqrt{x}\sqrt{y}$ also true when $x=-1$, making it $i$?

A : If: $$\sqrt{xy}=\sqrt{x}\sqrt{y}$$ only when $x,y>0$, B : Then why can I do this: $$\sqrt{-4}=\sqrt{4\times-1}=\sqrt{4}\sqrt{-1}=2i$$ which violates A since $y<0$ C : But why can I not do…
GambitSquared
  • 3,662
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The imaginary unit, $i,$ and an alternate representation.

Recently, I began working with both complex, and imaginary numbers, and I looked at the complex number $i^{n}.$ If $n = 0, i^{n} = 1,$ $n = 1, i^{n} = i = \sqrt{-1},$ $n = 2, i^{n} = i^{2} = i \cdot i = -1,$ $n = 3, i^{n} = i^{3} = i^{2} \cdot i =…
Taylor
  • 1,128
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Find all complex numbers so that $Im(\frac{z+2}{2-i})=1$ and $Re(z^2+1)=1$ and for $z$ which is in the first quadrant find $\sqrt{z}$

Find all complex numbers so that $Im(\frac{z+2}{2-i})=1$ and $Re(z^2+1)=1$ and for $z$ which is in the first quadrant find $\sqrt{z}$. If $z=x+iy$ then $$\frac{z+2}{2-i}=\frac{x+2+iy}{2-i}\times…
user300045
  • 3,449
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Why is $\sqrt{-i} \neq i\sqrt{i}$?

I wanted to figure out the square root of $-i$. Since $\sqrt{-x} = i\sqrt{x}$, $\sqrt{-i}$ should equal $i\sqrt {i}$, however, WolframAlpha said it was false. However, if I do say that $\sqrt{-i} = i\sqrt{i}$, I can replace $\sqrt{i}$ with…
ZTqvhI5vpo
  • 955
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Simplify $\sqrt{-3}$

I was reading about this known fallacy $$ -1 = i^2 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1 $$ and according to Wikipedia "The fallacy is that the rule $\sqrt{xy} = \sqrt{x}\sqrt{y} $ is generally valid only if both…
Alex T
  • 445
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Express $w=f(z)=\frac{1}{(1-z)^2}$ in the form $w=u(x,y)+iv(x,y)$

I start by writing $f(z)$ as $$\frac{1}{(1-(x+iy))^2}$$ and then I expand the bottom to get $$\frac{1}{(1-2x+x^2-y^2) + i(2y-2xy)}$$ The answer says $$w=\frac{(1+x^2-2x-y^2)-(i(2xy-2y)}{(1+x^2-2x-y^2)^2+(2xy-2y)^2}$$ How do I get to this stage?
Al jabra
  • 2,331
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Does $\lambda_1^n+ \lambda_2^n+ \dots +\lambda_k^n =0 $ for all $n$ imply that $\lambda_1= \lambda_2= \dots= \lambda_k = 0 $?

Suppose $\lambda_1, \lambda_2, \dots, \lambda_k $ are complex numbers that $\forall n \in \mathbb{N}$ satisfy $$\lambda_1^n+ \lambda_2^n+ \dots +\lambda_k^n =0.$$ Can we deduce that $\lambda_1= \lambda_2= \dots= \lambda_k = 0 $?
user120269
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Modulus of a complex number

I am now encountering a problem regarding on complex analysis Lets say we have $w=u+iv$ What would it be for $$|w|^{2}$$ I check a lot of videos and lecture notes, and realize the answer is $$u^{2}+v^{2}$$ Can someone explain to me why is it like…
samheihey
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Express in x + iy form.

Could somebody please verify or help me along with my answers? Thank you. My updated work(maybe someone can help me a little more now):
Jeff
  • 224
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4 answers

Minimum value of $|z+1|+|z-1|+|z-i|$

How to find the minimum value of $|z+1|+|z-1|+|z-i|$. I have tried geometrically etc but failed.
UNM
  • 1,177
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Complex number calculation

I'm supposed to show $ z^{10} $, when z = $ \frac{1+ \sqrt{3i} }{1- \sqrt{3i} } $ I can work it out to $ \frac{(1+\sqrt{3}\sqrt{i})^{10}}{(1-\sqrt{3}\sqrt{i})^{10}} $ However this is inconclusive because I need to show $ z^{10} $ in the form x+yi,…
user140878
  • 415
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Find the possible values of |A + B + C |

$ |A |= |B | = |C | = 1 $ ,where A B and C are complex nos $$ \frac{A^2}{BC}+ \ \frac{B^2}{ \ {CA}} \ +\ \frac{C^2}{ \ {AB}} + 1 = 0$$ Find the possible values of $ |A + B + C |$ Tried substituting cos(theta) + i sin(theta)
maths lover
  • 3,344
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How to transform the complex number $\frac{(1+i)^{29}}{1-i}$ to the form $a + bi$?

My problem is I have to transform $\displaystyle \frac{(1+i)^{29}}{1-i}$ on its binomial representation $(a + bi)$. I was thinking about transforming that into it polar representation and then reverse the process to get the binomial one, but it gets…
Juan Santiago Acevedo Correa
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Can a set containing $0$ be purely imaginary?

A purely imaginary number is one which contains no non-zero real component. If I had a sequence of numbers, say $\{0+20i, 0-i, 0+0i\}$, could I call this purely imaginary? My issue here is that because $0+0i$ belongs to multiple sets, not just…
chris
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