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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.

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Explain complex numbers

My cousin asked me if I could provide him with a practical example with complex numbers. I found it hard to do, so does anyone have a easy practical example with the use of complex numbers? I tried to show him that complex numbers is needed to solve…
tandberg
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Evaluating $\frac{(1+i)^{n+2}}{(1-i)^n}$

Evaluate $\dfrac{(1+i)^{n+2}}{(1-i)^n}$ I think that the meaning is that it need to be simplified. Thanks
gbox
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Set in the Complex Plane

How can I describe the set: $$ \left\vert z - {\rm i}\,\right\vert = 3\left\vert z\right\vert $$ It does appear quite unfamiliar. Attempt: $$ \left\vert\frac{z-i}{z}\right\vert = 3 $$ so, $$ \left\vert 1 - {\rm i}\,\frac{1}{z}\right\vert = 3 $$ But…
MadHatter
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Show that general bilinear transformation transforms circles into circles (or lines)

The exercise I got a bit stuck with is at the end of chapter on complex numbers that does not deal with transformations in general. So the notion of general bilinear transformation is introduced for the purpose of this exercise on the spot. I am…
Storyteller011
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For a polynomial $p(z)$ with real coefficients if $z$ is a solution then so is $\bar{z}$

I can "see" it intuitively, though I do not know how correct this is: in a complex conjugate we change the sign of all imaginary parts, and since the effect of all imaginary parts cancels out on the whole, this change of sign would not matter. I…
kuch nahi
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Ordering on complex numbers

So I just learned that complex numbers cannot be ordered from Total ordering on complex numbers. Now, does this mean that 4 - 3i < 4 + 4i isn't true? I haven't really seen any concrete examples and I don't even get what the proof means, but I know…
Don Larynx
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Something about $i^i$

Possible Duplicate: What is the value of 1^i? Note that I am absolutely not a mathematician, so this may be silly, but I saw this on Wikipedia's page about $i$: One definition of $i^i$ is : $i^i = \left( e^{i (2k \pi + \pi/2)} \right)^i =…
eje211
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How to say sum of complex number is non zero

Let $\omega$ be primitive nineth root of unity. Let $a= \omega^4+\omega^3+\omega^2-\omega-2$. Is there a nice way to show the sum is non zero? Given some linear combination of powers of primitive $n$-th roots of unity with coefficient from real, Is…
Cloud JR K
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Finding center and radius of circumcircle

Find the center and radius of the circle which circumscribes the triangle with (complex) vertices $a_1,a_2,a_3$. Express the result in symmetric form. I'm not sure where to start in this question. The circumcenter is the intersection of the…
Paul S.
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$n$ complex numbers in 1-sphere given, show that there exists a $w$ for which the sum of all differences is at least $n$

I've found the following interesting exercise. Let $z_1,...,z_n \in B(0,1) = \{ z \in \mathbb{C} \mid | z | \leq 1 \}$ be $n$ complex numbers in the unit sphere. Show that there exists a $w \in B(0,1)$ for which is $$\sum_{i=1}^n | z_i - w | \geq…
Lukas
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Is this a valid proof of Pythagoras theorem using complex numbers?

So basically this is a very simple algebraic proof of Pythagoras theorem, but I never saw it anywhere so I'm wondering if this is valid (or already presupposes the pythagoras theorem). Inspired by $a^2-b^2=(a+b)(a-b)$ you can write the following: $$…
bananenheld
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Find minimum $|z|$ satisfying $|z + 1/z |= 2$.

When I tried this using normal complex inequalities like $|z_{1} - z_{2}| \ge ||z_{1}| - |z_{2}||$. $\sqrt 2 - 1$ came up but the real answer seems to be $(3 - 2\sqrt 2)^{1/2}$. Some online answers on other sites support my answer as well, but I am…
Maths
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Suppose $2+7i$ is a solution of $2z^2+Az+B=0$, where $A,B \in \mathbb{R}$ . Find $A$ and $B$

The question is as follows; Suppose $2+7i$ is a solution of $2z^2+Az+B=0$, where $A, B \in \mathbb{R}$ . Find $A$ and $B$. My understanding is that this equation holds: $$2(2+7i)^2 + A(2+7i) + B = 0$$ which will eventually lead to: $$-90 + 2A…
X-men
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Does $i$ have one value or two values?

The imaginary number $i$ is defined as the solution to the equation $x²=-1$. Since the solution to this equation could be either positive or negative, does $i$ have two possible values? The original question asked if $i$ was a variable, however I…
ToMath
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Proving $\sum_{k=1}^{4} b_k |z_k|^2=0$ if $z_k$ are con-cyclic s. t. $\sum_{k=1}^4 b_k =0=\sum_{k=1}^{4} b_k z_k$

If four complex numbers $z_k$ are con-cyclic in Argand plane such that $\sum_{k=1}^4 b_k =0=\sum_{k=1}^{4} b_k z_k$ and $b_k$ are real, earlier it has been proved that $$b_1b_2|z_1-z_2|^2=b_3 b_4|z_3-z_4|^2~~~~~(1)$$ after a good number of…
Z Ahmed
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