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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Complex number $|z-w|$

On an Argand diagram, sketch the locus representing complex numbers satisfying $|z + i| = 1$ and the locus representing complex numbers w satisfying $\arg(w − 2) = \dfrac{3\pi}{4}$. Find the least value of $|z − w|$ for points on these loci. I…
Arodi007
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Complex number modulus identity (on unit circle)

For any three complex numbers $z_1, z_2, z_3$ on the unit circle, $|z_1 + z_2 + z_3| = |z_1 z_2 + z_1 z_3 + z_2 z_3|$. I am able to prove this by putting each number in modulus-argument form and then expanding algebraically, but this is somewhat…
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Complex Cubic Equation z^3+3z+2i=0

How we can solve the equation $z^3+3z+2i=0$ ? And is there exist a general method to solve similar equation?
SKMohammadi
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Proof for $|1 - z| \geq 1 - |z|$ for $|z| < 1$, $z \in \mathbb{C}$

I can prove it "by picture" by drawing a picture of a circle of radius $|z|$ centered at $(0, 1)$. Then $1 - |z|$ is the length from the origin to the intersection of the circle with the x-axis (to the left). $|1 - z|$ is length from the origin to…
MT_
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Evaluate and find the principal value of $(-1+i)^ {2-i}$

Can anyone please help me evaluate and find the principal value of $(-1+i)^{2-i}$ I got up to $=e^{2-i}(ln(-1+i))$ $=e^{(2-i)(1/2 ln(2)+i(3pi/4))}$
ru77
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Find square roots of $8 - 15i$

Find the square roots of: $8-15i.$ Could I get some working out to solve it? Also what are different methods of doing it?
Alex
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How to solve the equation $\exp(iz)=-e$

How do I solve the equation $$\exp(iz)=-e$$ Can anyone please explain the procedure to solve this kind of question to me please? Much appreatiate
ru77
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Proof the bijectivity of the exponential function mod $2 \pi i$

I am trying to show that the map $\Psi\ \colon \left\{\begin{align}\mathbb{R}/\mathbb{Z} & \longrightarrow S^1 \\ x & \longmapsto e^{2\pi i x}\end{align}\right.$ is a bijection. I am stuck on proving that it is injective. I suppose that…
Trajan
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Complex number problem becomes huge when using formulas - is there any workaround?

Here is an equation I need to solve for $z$ where $z$ is a complex number.(I need to show which complex numbers are solution for this problem): $$\left|\frac{1+z}{1-i\bar z}\right| = 1$$ Here are formulas I am using: $|a+bi| = \sqrt{a^2+b^2}$; …
qiubit
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calculate the absolute value of a complex number

I have to calculate the absolute value of $\lvert{i(2+3i)(5-2i)\over(2-3i)^3}\rvert$ solely using properties of modulus, not actually calculating the answer. I know that I could take the absolute value of the numerator and denominator and then take…
cele
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Show that $ \left|\sum _{S (\theta ) } z _k\right|= \left|\sum _{S (\theta ) } e ^{-i \theta } z _k \right|$.

Let $z _k = |z _k |e ^{i \alpha _k} $ and let $S(\theta ) $ be the set of all $k $ for which $\cos(\alpha _k - \theta) >0 $, $1 \le k \le n $. Then $$ \left|\sum _{S (\theta ) } z _k\right|= \left|\sum _{S (\theta ) } e ^{-i \theta } z _k…
Alexander
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Complex numbers proof problem

If $|z|,|w| \leq 1$, show that $|z-w|^2 \leq (|z|-|w|)^2 + (\arg(z)-\arg(w))^2$, where $z,w$ are complex numbers. How can I solve such a problem?
user34304
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Is product of two square roots of two integers square root of their products?

Suppose $a,b\in \mathbb{Z}$. Is it true $\sqrt{a}\sqrt{b}=\sqrt{ab}$. If so, then $\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1$ But we know $\sqrt{-1}=i$ and so $i^2=-1.$ Finally we get $i^2=-1=1.$ Which is not true. What…
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Simple identity involving complex numbers

We have to prove the following identity: $$z_1 \bar{z_2} = \frac{1}{4}(|z_1 + z_2|^2 + i|z_1 + iz_2|^2 - |z_1 - z_2|^2 - i|z_1 - iz_2|^2)$$ It says to use the identity we just proved, which is $\Re(z_1 \bar{z_2}) = \frac{1}{4}(|z_1 + z_2|^2 - |z_1 -…
MT_
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Why is $\frac 25$ the real part of $\frac{1}{2+i}$?

According to Wolfram Alpha, Re(1/(2+i))=2/5. How did it calculate that?
Hal
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