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
2
votes
3 answers

Does a purely imaginary number have a corresponding "angle" in polar coordinate system?

Let's say we have a pure imaginary number with no real part, $i$. I know that complex numbers in the form $a+bi$ can be converted into the polar coordinate system using the following relations: $\theta = \arctan{Im/Re} $ $r = \sqrt{a^2+b^2}…
Bob Shannon
  • 1,561
2
votes
1 answer

How to solve complex number

how to solve below complex number problem . The points $A,B,C$ represent the complex numbers $z_1,z_2,z_3$ respectively, and $G$ is the centroid of the triangle $ABC$ . If $4z_1+z_2+z_3=0$, show that the origin is the mid point of…
Suraj
  • 125
2
votes
1 answer

Remainders with complex numbers

Let $ f(x) \in C [x] .$ Suppose $ f(-1+i) = 2+5i $ and $ f(-2-i)=-3. $ Determine the remainder of f(x) divided by $(x+1-i)(x+2+i). $ How would i begin with this question, like how would i determine what f(x) is to begin with?
user136088
  • 251
2
votes
1 answer

What is the limit (when it exists) of a complex number raised to an integral power?

What can be said about $\lim_{n\to\infty} z^n$, when $z$ is complex? Can it be expressed in terms of $a,b$ where $z = a + bi$? Is the formula $z^n = r^n(cos(n\theta) + i\sin(n\theta))$ helpful here -- if so, I'm not seeing it yet.
bosmacs
  • 997
2
votes
3 answers

Sketching on the complex plane $|z + 1| - |z - 1| = 4$

Sketch the set of points that satisfy $|z + 1| - |z - 1| = 4$ on the complex plane. Wolfram alpha gives me an empty graph, I end up with the equation of an ellipse but with the condition $x>4$. The ellipse is the same as $|z + 1| + |z - 1| = 4$…
Superbus
  • 2,136
2
votes
1 answer

Raising a complex number to integer and complex powers

This came out on today's test: "Under what condition(s) is $a^b$ a real number, if $a$ is a complex number and $b$ is a positive integer?" And of course, the bonus extension: "In general, under what condition(s) is $a^b$ a real number, if both $a$…
Yiyuan Lee
  • 14,435
2
votes
3 answers

Why can we assume that $z \in \mathbb{R}$?

I am looking at the solution to the following question: Let $z, w \in \mathbb{C}, |z|, |w| < 1$. Show that: $$ \left| \frac{z - w}{1 - z\overline{w}} \right| < 1 $$ The solution starts by assuming w.l.o.g that $z \in \mathbb{R}$ (from there it's…
Hila
  • 1,919
2
votes
3 answers

Can we deduce that $|z|≥|a|-|b|$

Let us consider three complex numbers $z,a,b$ such that the equality $z=a-b$ holds true. Can we deduce that $$|z|≥|a|-|b|$$
DER
  • 3,011
2
votes
2 answers

Evaluate equation $z^{13}=\overline{z}$

I need to find all solutions of such complex number equation: $z^{13}=\overline{z}$ We assume that $r = |z|$ and we use Euler's formula $z=|z|e^{i\phi}=re^{i\phi}$. Then, multiplying both sides by $z$, we have $z^{14}=|z|^2$ $z^{14}-|z|^2 =…
stil
  • 383
2
votes
2 answers

Evaluate $a^i$ where $a$ is real:

I have a question involving the evaluation of $3^i$, but I am unsure how to do this. I know how to solve such questions involving $e^{i\theta}$, but how does this work with a different base? (I understand that the angle is 1 radian). I have…
Ruben
  • 135
2
votes
3 answers

When are complex conjugates of solutions not also solutions?

I've heard that for "normal" equations (e.g. $3x^2-2x=0$), if $(a+bi)$ is a solution then $(a-bi)$ will be a solution as well. This is because, if we define $i$ in terms of $i^2=-1$ then we might as well define $i^\prime=-i$. Since ${i^\prime}^2=-1$…
spraff
  • 1,587
2
votes
1 answer

Solving complex eqautions

I have an equation: $\frac{8\sqrt{3}}{z^4+8}=-i$ I have no idea where to begin solving this! My book doesn't seem to give any hints on equations like this. How do I solve it and what can I search for on information on this? Roots of complex…
Paze
  • 1,956
2
votes
3 answers

Complex Numbers and exponential form and roots

The roots of $z^7 = -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}$ are $\text{cis } \theta_1, \text{cis } \theta_2, \dots, \text{cis } \theta_7,$ where $ 0^\circ \le \theta_k < 360^\circ $for all $ 1 \le k \le 7$. Find $\theta_1 + \theta_2 + \dots +…
Math Dude
  • 159
2
votes
3 answers

$ \overline{x} = x^2$

What $z \in \mathbb{C}$ numbers solve $ \overline{z} = z^2$? It seems obvious that $\left|z\right|$ can only be $1$, otherwise $\left|z^2\right| \ne \left|z\right|$. Since $\varphi$ arguments add up upon square, I suppose solutions for $\varphi +…
kdani
  • 277
2
votes
2 answers

The pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfies $|w1| =1$, $|w2| = 1$ and $\Re\left(w_1\overline{w_2}\right) =0$

If $z_1=a + ib$ and $z_2 = c + id$ are complex numbers such that $\left|z_1\right| = \left|z_2\right| = 1$ and $\Re\left(z_1\overline{z_2})\right)=0$, then Prove that the pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfies…
maths lover
  • 3,344
1 2 3
…
99 100