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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Is -i negative?

It's pronounced "negative i," yet I'm told complex numbers aren't negative or positive. Is the - just arbitrary and unreflective of -i's sign?
Alec
  • 354
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Maximum and minimum of $|z_1-z_2|$ whereas $|z_2|=2,z_1=3+4i$

What are the greatest and least values of $|z_1-z_2|$ whereas $|z_2|=2,z_1=3+4i$ & $z_1,z_2$ are complex numbers? $|z_2|=2$ $z_1=3+4i$ $|z_1-z_2|=?$ Let $z_2=a+bi$ After doing some manipulation I got $|z_1-z_2|=\sqrt{29-(6a+8b)}$ Now I have to…
Invnto
  • 169
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Magnitude of Complex Numbers

Let $\alpha \neq 1$ be a complex number such that the distance from $\alpha^2$ to 1 is twice the distance from $\alpha$ to 1, while the distance from $\alpha^4$ to 1 is four times the distance from $\alpha$ to 1. Enter all possible values of…
sumi
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What is the easiest way to get: $2+ \sqrt{-121} = (2+ \sqrt{-1})^3$

I was reading the book Seventeen equations have changed the world. At some point, while the book was talking about complex numbers, I see this equation: $2+ \sqrt{-121} = (2+ \sqrt{-1})^3$ Even if it's easy to proof the truth of this equivalence (it…
mattia.b89
  • 141
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Complex number set has least upper bound property?

Exercise 1.9 Suppose $z = a + bi, w = c + di$. Define $z
주혜민
  • 33
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Derivative of Complex Exponential

I would like help with this problem. Prove this is true by using complex analysis. This problem appeared in my workbook and there appears to be no solution in the back. $$\frac{d}{dx}e^{tx} = te^{tx}.$$ Edit: I think I am supposed to use chain…
Quaxton Hale
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Complex numbers with conjugate multiplication - field or ...?

I could have sworn that when we learned about complex numbers in signals and systems that they form a field in (at least) two ways, depending on multiplication, which is most intuitively described in polar coordinates: Normal multiplication adds the…
alancalvitti
  • 3,400
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For $z\in\mathbb{C}, |z|=1$, Prove that $\Big(\frac{1+ia}{1-ia}\Big)^4=z$ has all roots real and distinct

Given $z\in\mathbb{C}$ with $|z|=1$, then prove that the equation $\Big(\dfrac{1+ia}{1-ia}\Big)^4=z$ has all roots real and distinct My Attempt $$ z=e^{i\theta}\implies…
Sooraj S
  • 7,573
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What am I doing wrong in calculating $i^2$?

$$i^2 = i \cdot i = (\sqrt{-1}) \cdot (\sqrt{-1}) = ({\sqrt{-1}})^2 = -1 \tag{1}$$ $$i^2 = i \cdot i = \underbrace{\sqrt{-1}\cdot\sqrt{-1} = \sqrt{(-1)\cdot(-1)}}_{?} = \sqrt{(-1)^2} = 1 \tag{2}$$ What are the conceptual or logical errors…
raf
  • 155
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Set of all numbers

Does the set of complex numbers contain every number that exist? Around 600 years ago people thought the set of real numbers contain any number that one can think of. Slowly, with the introduction of $\sqrt{-1}$ and complex numbers that perspective…
Rob
  • 622
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Square root of a complex number

I was trying to find the square root of a complex number $w=u+iv$, I assumed $z^2=(x+iy)^2=w$ Now solving this equation I got two values of $x$ and two values of $y$ in terms of $u$ and $v$. So, I get a possible of 4 solutions. But fundamental…
Mathematician
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Prove $|\alpha-i|=|\beta-i|$ if $\alpha$ and $\beta$ are the roots of $z+\dfrac{1}{z}=2e^{i\theta}$

If $\alpha$ and $\beta$ are the roots of $z+\dfrac{1}{z}=2(\cos\theta+i\sin\theta),$ $0<\theta<\pi$, then prove that $|\alpha-i|=|\beta-i|$ My…
Sooraj S
  • 7,573
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How to deduce the following complex number problem

I am stuck with the following problem that says: Using the result $$x^n-1=(x^2-1)\prod_{k=1}^{(n-2)/2}[x^2-2x\cos \frac{2k\pi}{n}+1],$$ if $n$ be an even positive integer, deduce that $$\sin \frac{\pi}{32}\sin \frac{2\pi}{32}\sin…
learner
  • 6,726
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How to extend this inequality?

It can be easily verified that for any real $\theta$ and complex number $a$, $$\Re\left(\frac{e^{i\theta}}{e^{i\theta}-a}\right)\leq\frac12,\quad\text{ for }|a|\geq1.$$ I was trying to verify this result for any complex number $a$ with $0\leq|a|<1$.…
user159888
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Solving the complex equation $|z|+z=6-2i$

I have to tackle the following question. My thoughts so far are below it. Given that $z$ is the complex number $x+iy$ and that $|z|+z=6-2i$, find the value of $x$ and the value of $y$. What I only know is that $|z|=r=\sqrt{x^2+y^2}$ So how should…
busyyyy
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