Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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5answers
1k views

Alternative to imaginary numbers?

In this video, starting at 3:45 the professor says There are some superb papers written that discount the idea that we should ever use j (imaginary unit) on the grounds that it conceals some ...
8
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5answers
569 views

Is there any way to represent an imaginary number?

Is there any way to represent an imaginary number? Like the square root of -1? Is there any possible way to do this? Sorry if you think this is a dumb question. I am a 7th grade student in ...
8
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5answers
290 views

Does there exist a field $K$ such that $\mathbb R \subsetneq K \subsetneq \mathbb C$?

I'm thinking of unions of $\mathbb R$ with some subset of $\mathbb C$ but am not sure how to approach this without ending up with all of $\mathbb C$. Doe anyone have any suggestions?
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5answers
572 views

What's the importance of a formula for the real and imaginary parts of a complex number?

I've learned that $$\bbox[8px,border:1px solid black]{\operatorname{Re}(z)= \frac{z+\overline{z}}{2} \qquad \qquad \operatorname{Im}(z)=\frac{z-\overline{z}}{2i}} $$ And that in the number $z=a+bi$, ...
8
votes
4answers
272 views

When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
8
votes
3answers
234 views

Putting ${n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ in a closed form

As the title says, I'm trying to transform $\displaystyle{n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ into a closed form. My work: $\displaystyle\left(1 + ...
8
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2answers
786 views

Summation of complex numbers

This is a series problem where the terms are complex numbers. I am looking for a better approach to solving this problem. If $\displaystyle z = \frac{1+i}{\sqrt2}$, Evaluate $1 + z + z^2 + ... + ...
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2answers
320 views

How to show that $\overline{zw}=\overline{z}\,\overline{w}$?

I thought about first multiplying two complex which aren't in the conjugate form: $$zw=a c+i a d+i b c-b d$$ Then multiply two complex conjugates: $$\overline{z}\,\overline{w}=a c\color{red}{-}i a ...
8
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2answers
401 views

Are there any elegant methods to classify of the Gaussian primes?

Out of curiosity, are there any relatively quick classifications of all the Gaussian primes, the primes in $\mathbb{Z}[i]$? I found a classification here, but the process comes off as rather tedious. ...
8
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4answers
90 views

$z^n=(z+1)^n=1$, show that $n$ is divisible by $6$.

we are given $z^n=(z+1)^n=1$, $z$ complex number. we want to prove that $n$ is divisible by $6$. I showed that $|z|=|z+1|=1$. Hence $z$ is on the intersection of two unit circles, one centered at ...
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2answers
1k views

Total ordering on complex numbers

Show that there doesn't exist a relation $\succ$ between complex numbers such that (i) For any two complex numbers $z,w$, one and only one of the following is true: $z\succ w,w\succ z,$ or ...
8
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3answers
26k views

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 ...
8
votes
2answers
294 views

The roots of the derivative $P'(z)$ of the polynomial $P(z)\in\mathbb C[x]$ lie in the convex hull of the set of roots of $P(z)$.

Assume $S=\{z_1,z_2,...,z_k\}, z_i\in \mathbb C$$, C(S)$ and define $$C(S):=\{z=a_1z_1+a_2z_2+...+a_kz_k | a_i\ge0 ,a_1+a_2+...+a_k=1\}$$ where $$A:=\{z\in \mathbb C:f(z)=0 ...
8
votes
3answers
148 views

What is the right treatment for $0^i$?

I need to calculate a limit of a complex expression (had it in a physics research) that contains a term $(r-b)^p$ for $r\rightarrow b+$ where $r,b$ are reals, and $p$ is complex, let's suppose for ...
8
votes
1answer
722 views

How to choose a proper contour for a contour integral?

When analyzing real integrals with contour integrals, how does one choose a proper contour integral? Many cases can be solved by integrating around the top half of a circle with radius of infinity ...
8
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2answers
204 views

Expressing a complex function in terms of z

Use the Cauchy-Riemann equations to determine all differentiable functions that satisfy $Re(f(z))=xy$ I think I know how to do this problem. If we let $z=x+iy$, then $f(z)=u(x,y)+iv(x,y)$. We ...
8
votes
2answers
123 views

In a complex vector space, $\langle Tx,x \rangle=0 \implies T = 0$

Suppose $T$ is a linear operator on a complex inner product space. Is it a theorem that if $\langle Tx,x\rangle=0$ for all $x$ in the space then $T=0$. The theorem fails in the real case, as seen for ...
8
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1answer
116 views

Cosh and Sinh analogs

We know that $$\cosh{x}+\sinh{x}=e^x$$ and that his can be expressed as $$\frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}=\frac{(e^x+e^x)+(e^{-x}-e^{-x})}{2}=e^x$$ and this works out nicely because the ...
8
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2answers
356 views

If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear

If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear. I really can't seem to get anywhere on this problem, ...
8
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3answers
196 views

Proving that the limit of a sequence is $> 0$

Let $u$ be the complex sequence defined as follows : $u_0=i$ and $ \forall n \in \mathbb N, u_{n+1}=u_n + \frac {n+1-u_n}{|n+1-u_n|} $ . Consider $w_n$ defined by $\forall n \in \mathbb ...
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0answers
56 views

An Inequality with complex numbers and $1/\pi$ [duplicate]

Let $\displaystyle \{z_1,z_2, \ldots, z_n\}$ be $n$ complex numbers such that: $\displaystyle \sum\limits_{k=1}^n|z_k| = 1$ Then we have to show that, there is a subset $S$ of $\{1,2,\ldots,n\}$, ...
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8answers
2k views

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 ...
7
votes
5answers
902 views

If three complex numbers $z_k$ have modulus $1$, then $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$

Our teacher gave us a hard question (according to her, it is pretty hard for our level): Given that $|z_1| = |z_2|= |z_3|=1,z \in\mathbb{C}$, prove that $|z_1+z_2+z_3| = ...
7
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4answers
786 views

Adding powers of $i$

I've been struggling with figuring out how to add powers of $i$. For example, the result of $i^3 + i^4 + i^5$ is $1$. But how do I get the result of $i^3 + i^4 + ... + i^{50}$? Writing it all down ...
7
votes
7answers
2k views

How to solve $e^{ix} = i$

I am taking an on-line course and the following homework problem was posed: $$e^{ix} = i$$ I have no idea how to solve this problem. I have never dealt with solving equations that have imaginary ...
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8answers
897 views

Most natural intro to Complex Numbers [closed]

This is a soft question but I'm willing to ask. There are few ways to introduce the field of complex numbers, but if You had the opportunity to write an elementary textbook, what would be the most ...
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3answers
227 views

$ \exists a, b \in \mathbb{Z} $ such that $ a^2 + b^2 = 5^k $

I saw this problem recently and found an elegant solution to it, and was curious to see if anybody would think of something else. Nice solutions to nice problems are fun to see! Problem: Prove ...
7
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9answers
2k views

Positive and negative complex numbers?

Can there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number ...
7
votes
2answers
4k views

What does $\mathrm{Re}(x)$ mean?

I see this all the time in Mathematica output as well as in text, such as near the top of the Wikipedia Beta function page.
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3answers
671 views

What is the motivation for complex conjugation?

I have been dealing with complex numbers for few years now. But when I've tried to think about the motivation behind complex conjugation, I was not sure. Let me write what I am working with. For a ...
7
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8answers
746 views

what is$ \sqrt{8i}$

Very simple question with an answer that I cannot understand: I have $\sqrt{8i}$, which, I suppose, is the same as $\sqrt{\sqrt{-64}}$. How come that $2+2i$ is the same as $\sqrt{8i}$? My ...
7
votes
2answers
421 views

How does one find $z\in \mathbb{C}$ such that $\sin z=100?$

I am self-studying Complex Analysis and I am suppose to find $z\in \mathbb{C}$ such that $\sin z=100.$ I know that $$\sin z=\sin x \cosh y+i\cos x\sinh y$$ So I must have $\sin x \cosh y=100.$ I ...
7
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6answers
422 views

Prove if $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} $ is a real number

If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $ i found one link that had a similar problem. Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} ...
7
votes
3answers
443 views

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 ...
7
votes
6answers
245 views

What is $(-1)^{\frac{2}{3}}$?

Following from this question, I came up with another interesting question: What is $(-1)^{\frac{2}{3}}$? Wolfram alpha says it equals to some weird complex number (-0.5 +0.866... i), but when I try ...
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2answers
2k views

Limit of complex function

Im trying to find the limit of: $$ \frac{\operatorname{Re}(z) \operatorname{Im}(z)}{z^2}$$ as z tends to zero.
7
votes
5answers
372 views

Definite integral of even powers of Cosine.

I'm looking for a step-by-step solution to the following integral, in terms of n$$\int_0^{\frac{\pi}{2}} \cos^{2n}(x) \ {dx}$$I actually KNOW that the solution is$${\frac{\pi}{2}} \prod_{k=1}^n ...
7
votes
2answers
363 views

Why is $\left(e^{2\pi i}\right)^i \neq e^{-2 \pi}$?

Here's my (obviously flawed) proof that $1=e^{-2 \pi}$: $$ 1^i=1\\ e^{2 \pi i} = 1\\ \left(e^{2\pi i}\right)^i = 1^i\\ e^{-2 \pi} = 1 $$ What's the issue? I understand that exponentiation is not ...
7
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2answers
147 views

proof for $\frac{1}{i} = -i$?

My physical chemistry textbook seems to be making the implicit assumption that $\cfrac{1}{i} = -i$. I'm not sure how this is valid. Here is the snippet of relevant steps: ...
7
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4answers
3k views

Non-integer powers of negative numbers

Roots behave strangely over complex numbers. Given this, how do non-integer powers behave over negative numbers? More specifically: Can we define fractional powers such as $(-2)^{-1.5}$? Can we ...
7
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4answers
118 views

Deriving an expression for $\cos^4 x + \sin^4 x$

Derive the identity $\cos^4 x + \sin^4 x=\frac{1}{4} \cos (4x) +\frac{3}{4}$ I know $e^{i4x}=\cos (4x) + i \sin (4x)=(\cos x +i \sin x)^4$. Then I use the binomial theorem to expand this fourth ...
7
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2answers
159 views

Invariant under transformation $i\mapsto -i$ implies real?

When one has an expression in terms of $i$, one can send $i$ to $-i$ and, if the expression remains unchanged, one can conclude that the expression is, in fact, real. Analogous statements hold for ...
7
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6answers
111 views

A proper definition of $i$, the imaginary unit [duplicate]

Back when I was in high school, which was a long time ago, I recall my math teacher telling me that the definition of $i$, the imaginary unit, is $\sqrt{-1}$. Knowing little, at the time, I accepted ...
7
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3answers
173 views

How to visualize $f(x) = (-2)^x$

Background I teach Algebra and second year Algebra to middle school students. We are currently studying Exponential, Power, and Logarithmic functions. We study exponential functions (of the form ...
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3answers
650 views

How do I completely solve the equation $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$ where there is a root with the real part of $1$.

I would please like some help with solving the following equation: $$z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$$ All I know about the equation is that there is a root with the real part of $1$. My approach ...
7
votes
5answers
230 views

How to find the center of the circle that contains three given complex numbers?

Suppose $\alpha_1, \alpha_2, \alpha_3 $ are complex numbers which are not collinear. Is it possible to use some geometry to find the center of the circle that contains $\alpha_1, \alpha_2, \alpha_3 $ ...
7
votes
2answers
223 views

An “elementary” approach to complex exponents?

Is there any way to extend the elementary definition of powers to the case of complex numbers? By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots ...
7
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3answers
78 views

Set Theoretic Definition of Complex Numbers: How to Distinguish $\mathbb{C}$ from $\mathbb{R}^2$?

I have spent some time looking for a rigorous, set-theoretic definition of the complex numbers. I have read the book Elements of Set Theory by Herbert Enderton (1977) which does an excellent job of ...
7
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1answer
176 views

Complex Numbers $\stackrel{?}{=} \mathbb{R}^ 2$

Suppose we have a vector field over real numbers $\mathbb R^2$. In additon to vector field proporties define inner product $(x,y) = x_1\cdot y_1 + x_2\cdot y_2$, where $x_1,x_2,y_1,y_2$ are real ...
7
votes
2answers
337 views

How to calculate $i^i$ [duplicate]

I've been struggling with this problem, actually I was doing a program in python and did 1j ** 1j(complex numbers) (In python ...