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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Geometrical interpretation $|z-1| = Re(z+1)$

I'd like to ask you about the example below (I have to draw a geometrical interpretation in an argand diagram). $|z-1| = Re(z+1)$ I know that: Re(z+1) = $Re(x + yi + 1) = x + 1$ But what is the most efficient way to solve it? What I did: $|x + yi -…
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Solve a complex equation

Solve the following equation $$(4-3i)z^2-25z+31-17i= 0 $$ Dividing by 4-3i gives me $$z^2 \frac{-100z-75zi + 124 + 93i -68i -51i^2}{25}$$ which goes to $$z^2 -4z-3zi + 7+i$$ then i collect the terms so $$z - \left(\frac{(4-3i)}{2}\right)^2 = -7 -i…
addde
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Solving complex numbers equation $z^3 = \overline{z} $

We have the following equation: $$z^3 = \overline{z} $$ I set z to be $z = a + ib$ and since I know that $ \overline{z} = a - ib$. I was trying to solve it by opening the left side of the equation. $$ z^3 = (a+ib)^3 \Rightarrow $$ $$ [a^2+b^2+i(ab +…
D_R
  • 987
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Imaginary fraction square root?

I have a fraction - $-\frac{1}{3}$ Which could either mean the value of fraction is $\frac{-1}{3}$ or $\frac{1}{-3}$ Note the minus sign Now, what is the sqaure root of the fraction? I tried and I got this…
Arulx Z
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Confused with imaginary numbers

In 9th grade I had an argument with my teacher that ${i}^{3}=i$ where $i=\sqrt{-1}$ But my teacher insisted (as is the accepted case) that: ${i}^{3}=-i$ My…
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Rules of imaginary numbers

I know $i=\sqrt{ -1}$; what I don't get is the results you get from raising $i$ to an exponent: $i^1 = i$ makes sense since anything to the first is itself. $i^2 = -1$ also makes sense since squaring a square root is the same as removing the…
Isaiah
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Comparing real and complex numbers

If I'm correct, a complex number can be interpreted as a set in the following manner: $$ \forall x, y \in \mathbb{R}, x + yi = \{(x,\ y)\}.\ \mathbf{(1)} $$ My question is, is it technically correct to say: $$ \forall a \in \mathbb{R},\ ( b = 0…
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Complex number question on proving an inequality.

If $|z_1|=1,|z_2|=1$, how can one prove $|1+z_1|+|1+z_2|+|1+z_1z_2|\ge2$
RE60K
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primitive n-th roots of unity

Show that the primitive n-th roots of unity have the form $e^{2ki\pi/n}$ for $k,n$ coprime for $0\leq k\leq n$. Since all primitive n-th roots of unity are n-th roots of unity by definition they all have that form, the question is, how to show $k$…
Emir
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What is the square root of "$i$"?

Where $i$ is the square root of negative one. And is there a generalization of the $n$th root of $i$? Also how would this look graphically on the real number axis? Thanks
Amour
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Can anyone tell me how $\frac{\pi}{\sqrt 2} = \frac{\pi + i\pi}{2\sqrt i}$

I was working out a problem last night and got the result $\frac{\pi + i\pi}{2\sqrt i}$ However, WolframAlpha gave the result $\frac{\pi}{\sqrt 2}$ Upon closer inspection I found out that $\frac{\pi}{\sqrt 2} = \frac{\pi + i\pi}{2\sqrt i}$ But I…
dingari
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Is the Complex Conjugate the Only Way to Get a Real Number?

Is the complex conjugate of a number (or a real multiple of it) the only complex number which, when multiplied with the original number, gives a real number?
user82004
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Is there such thing as an imaginary (imaginary number)?

In other words... is there such a thing that is to imaginary numbers what imaginary numbers are to real numbers? And could this be expressed as a "complex" type number? If a complex number is in the form x + yi, I guess this would be in the form of…
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How to solve $z^4 +6z^2 +25 = 0$ into complex conjugate?

How to solve $z^4 +6z^2 +25 = 0$ into complex conjugate? I started with $$(z^2 + 3)^2 + 16 = 0$$ $$(z^2 + 3)^2 = - 16$$ $$z^2 + 3 = \pm 4i$$ Is this the way to start solving the equation or am I completely of?
David
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About Gaussian Integers .

If $a+bi$ is a Gaussian integer with $(a,b)=1$ call it 'viewable' ( as a line from $a+bi$ to $0$ can be drawn intersecting no other gaussian integers). Are there any gaussian composites that are viewable?