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
8
votes
2 answers

Conjugate of complex polynomial?

Say I have a complex polynomial: $$a_0+a_1x+\cdots+a_nx^n,$$ where $a_0,\ldots,a_n$ are complex numbers. What is the conjugate of this polynomial? How is it defined? For example, if we have an inner product on the vector space $V$ defined by…
Justin
  • 2,813
8
votes
2 answers

Super hard complex numbers problem: there do not exist $n>1$ complex numbers $z_1, z_2, \ldots, z_n$, no two equal, such that for all $1 \le k \le n$

Prove that there do not exist $n>1$ complex numbers $z_1, z_2, \ldots, z_n$, no two equal, such that for all $1 \le k \le n$ $$ \prod\limits_{i\neq k} (z_k-z_i)=\prod\limits_{i\neq k} (z_k+z_i)$$ At the first look it seems too easy to solve but…
Taha Akbari
  • 3,559
8
votes
7 answers

Find the 2016th power of a complex number

Calculate $\left( \frac{-1 + i\sqrt 3}{1 + i} \right)^{2016}$. Here is what I did so far: I'm trying to transform $z$ into its trigonometric form, so I can use De Moivre's formula for calculating $z^{2016}$. Let $z = \frac{-1 + i\sqrt 3}{1 + i}$.…
George R.
  • 2,833
8
votes
3 answers

How to prove $|z_1-z_2| \geq |z_1|-|z_2|$ in other way than this?

How to prove $|z_1-z_2| \geq |z_1|-|z_2|$ in other way than this? I mean I tried to find on the internet but could not find. I ask for more straighforward way than the proof that is presented for item 3.
user2723
8
votes
3 answers

If $f(z)=\cfrac{z+1}{z-1}$ , then find $f^{1991}(2+i)$

If $f(z)=\cfrac{z+1}{z-1}$ , then find $f^{1991}(2+i)$ Forgive me if the question is too short but really I don't know how to do this one. That's what I have done so far: $\left(f(2+i)\right)^{1991}=\left(\cfrac{3+i}{1+i}\right)^{1991}$ So now If…
Mr. Y
  • 2,637
8
votes
9 answers

How to solve $z^3 + \overline z = 0$

I need to solve this: $$z^3 + \overline z = 0$$ how should I manage the 0? I know that a complex number is in this form: z = a + ib so: $$z^3 = \rho^3\lbrace \cos(3\theta) + i \sin (3\theta)\rbrace$$ $$\overline z = \rho\lbrace \cos(-\theta) + i…
Christian Giupponi
  • 687
7
votes
0 answers

Minimising a sum of roots of unity

Let $n$ be an integer, $n\ge2$. Let $m$ be a positive integer, $m\le n$, having no common factor with $n$. How can we select $m$ distinct complex $n$th roots of unity in such a way as to minimise the modulus of their sum? The condition that $m$…
David
  • 82,662
7
votes
5 answers

Why do we use Complex numbers and not other systems?

I just started studying Complex analysis and have in fact just switched field to mathematics recently and so please forgive me if this is question seems trivial for a mathematics student to ask. Question: Why do we use Complex numbers instead of…
Jean Valjean
  • 1,657
  • 12
  • 25
7
votes
3 answers

Elementary complex numbers question

I've just started learning about the quaternions and it's raised some interesting questions for me about the complex numbers. The $+$ sign in the expression $a+bi$ is confusing me. Usually when we use the $+$ sign it represents an operation…
Lammey
  • 1,363
7
votes
3 answers

Solving equation $(z-i)^3=(4-i\sqrt{48})z^3$ in $\mathbb C$.

I need to solve the following equation in $\mathbb C$. $$(z-i)^3=(4-i\sqrt{48})z^3.$$ I tried with trigonometric form , but having $z-i$ on the LHS is confusing me, since I get $(\cos\varphi+i\sin\varphi-i)^3$. On the RHS, I got $8\cdot\text{cis}…
Lazar Ljubenović
  • 2,896
7
votes
6 answers

Prove if $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} $ is a real number

If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $ i found one link that had a similar problem. Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$
vordep
  • 73
7
votes
1 answer

Let $\alpha$ and $\beta$ be any two distinct complex numbers,then $|\alpha-\sqrt{\alpha^2-\beta^2}|+|\alpha+\sqrt{\alpha^2-\beta^2}|=$

Let $ \alpha $ and $\beta $ be any two distinct complex numbers,then $|\alpha-\sqrt{\alpha^2-\beta^2}|+|\alpha+\sqrt{\alpha^2-\beta^2}|=$ My Attempt Let $z_1=\alpha-\sqrt{\alpha^2-\beta^2}$, $z_2=\alpha+\sqrt{\alpha^2-\beta^2}$ $z_1$ and…
Monocerotis
  • 321
7
votes
2 answers

Finding magnitude of a complex number

$$z = \dfrac{2+2i}{4-2i}$$ $$|z| = ? $$ My attempt: $$\dfrac{(2+2i)(4+2i)}{(4-2i)(4+2i)} = \dfrac{4+12i}{20} = \dfrac{4}{20}+\dfrac{12}{20}i = \dfrac{1}{5} + \dfrac{3}{5}i$$ Now taking its magnitude and we have that $$|z| = \sqrt{\biggr (\dfrac 1…
Hamilton
  • 360
7
votes
6 answers

Adding Absolute value to a complex number: $ z+| z|=2+8i$

I would like to know my error in this problem. Find the complex number such that: $$ z+|z|=2+8i$$ So far, I have: $$ \begin{split} a+bi+\sqrt{a^2+b^2} &= 2 + 8i\\ a^2-b^2+a^2+b^2&=4-64\\ 2a^2 -b^2 + b^2&=-60\\ a^2&=-30 \end{split} $$ But I should…
BouncySlime555
  • 135
  • 1
  • 7
7
votes
1 answer

Value of complex number

Let $a, b, c$ be distinct complex numbers such that $\frac{a}{1-b}=\frac{b}{1-c}=\frac{c}{1-a}=k$. Find the value of $k$. My approach: $a+b+c=k(3-(a+b+c))$, after this step not able to proceed
Samar Imam Zaidi
  • 8,800
1 2 3
…
99 100