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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How to compute $\sqrt{i + 1}$

Possible Duplicate: How do I get the square root of a complex number? I'm currently playing with complex numbers and I realized that I don't understand how to compute $\sqrt{i + 1}$. My math program, sage, returns $$1.09868411346781 +…
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Proving the following number is real

Let $z_i$ be complex numbers such that $|z_i| = 1$ . Prove that : $$ z\, :=\, \frac{z_1+z_2+z_3 +z_1z_2+z_2z_3+z_1z_3}{1+z_1z_2z_3} \in \mathbb{R} $$ This problem was featured on my son's final exam today, I tried helping him with this, but I guess…
John
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Why are complex numbers considered to be numbers?

I've had Dave's Short Course on Complex Numbers on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers. I'm frequently asked that question. This is not a question about the existence or…
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How to get principal argument of complex number from complex plane?

I am just starting to learn calculus and the concepts of radians. Something that is confusing me is how my textbook is getting the principal argument ($\arg z$) from the complex plane. i.e. for the complex number $-2 + 2i$, how does it get…
tb747
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What's bad about calling $i$ "the square root of -1"?

I vaguely recall a teacher telling me that he dislikes introducing the imaginary unit $i$ as "the square root of $-1$", but I can't remember why. Is there a lack of rigour in the statement, or is it a misleading statement in any other way? I suppose…
Andrey
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Equation of ellipse, hyperbola, parabola in complex form

Write the equation of an ellipse, hyperbola, parabola in complex form. For an ellipse, there are two foci $a,b$, and the sum of the distances to both foci is constant. So $|z-a|+|z-b|=c$. For a hyperbola, there are two foci $a,b$, and the absolute…
Paul S.
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Finding the square root of a complex number - why two solutions instead of four?

I want to find the square roots of a complex number, $w = a+ib \in \mathbb{C}$, i.e. I'm looking for solutions, $z = x + iy$, for the equation $z^2 = w$. This question has been asked here a couple of times, but I still don't get why there are only…
Marc
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Is there a formula for $(1+i)^n+(1-i)^n$?

I'm wondering if there is a formula for the value of $(1+i)^n+(1-i)^n$? I calculated the first terms starting with $n=1$ to be, in order, $2$, $0$, $-4$, $-8$, $-8$, $0$, $16$, $\dots$ So it seems to be some sequence of positive and negative powers…
hagar
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Help me prove $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$

Please help me prove this Leibniz equation: $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$. Thanks!
user39471
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Are properties of the imaginary unit assumed or proved?

Reading about the cyclic values returned by integral powers of $i$: $$\begin{align*} i^0&=\hphantom{-}1\\ i^1&=\hphantom{-}i\\ i^2&=-1\\ i^3&=-i\\ & \,\,\,\vdots \end{align*}$$ and the pattern continues. Now, a simple question popped up in my…
codetalker
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Real Numbers Raised to Imaginary Powers?

What is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples and explanations would be appreciated.
Nathan
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What's the thing with $\sqrt{-1} = i$

What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is arbitrary. So saying $\sqrt{-1} = i$ is plainly…
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Difference between imaginary and complex numbers

Recently I was talking to my teacher about complex and imaginary numbers and he told me basically that $i$ is a complex number; its real part is just 0. However, this has made me wonder; if you can see $i$ as a complex number because you could argue…
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Complex numbers: Calculate the multiplicative inverse for $z=a+bi \neq 0$

Complex numbers: Calculate the multiplicative inverse for $z=a+bi \neq 0$ $a$ is the real part and $i$ is the imaginary one. I tried this but not sure: Inverse of $z$ is $z^{-1}=\frac{1}{z}$ so $$\frac{1}{z}=…
kathelk
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how prove $\varepsilon_i \in \left \{ -1,1 \right \}$ such $\left | \sum_{i=1}^{n} \varepsilon_i z_i \right | \leqslant 1.$

Let $n\geqslant 3$ be an odd number, and $z_1,z_2,...,z_n$ be complex numbers such that $|z_i|=1$ for all $i$. Prove that there exist $\varepsilon_i \in \left \{ -1,1 \right \}$ satisfying $$\left | \sum_{i=1}^{n} \varepsilon_i z_i \right |…
math110
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