# Tagged Questions

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### How to teach newbie multiply of complex number

I want to teach a newbie the arithmetic law of complex numbers. the law of add is acceptable psychological. but multiply is not. for example, assume $$z = a+bi, w = c+di$$ He (She) may ask me: why ...
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### What are $a$ and $b$ when the zeropoints of $f(z)=(a+bi)z+2-i=0$ is at $1-i$?

$f(z)=(a+bi)z+2-i$. What are the values of a and b when $1-i$ is the zeropoint of f? $f(z)=(a+bi)z+2-i=0$ $(a+bi)(1-i)+2-i=0$ $a+bi-ai-bi^2+2-i = 0$ $(a+b+2)+(-a+b-1)i=0$ I don't know what the ...
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### How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$
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### the geometric explain of $t = x-\frac{a}{3}$ in the simplify of cubic equation $x^3+ax^2+bx+c=0$

Assume $$f(x) = x^3+ax^2+bx+c$$ we have $$f''(x)=2a+6x$$. we get $x = -\frac{a}{3}$ Magically, If we take the transformation: $$t = x -\left(-\frac{a}{3}\right)$$. we can transform the above ...
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### Making $-{{\pi i}\over n} e^{\alpha i}({{1 - e^{2 n \alpha i}\over{1-e^{2 \alpha i}}}})={\pi \over {n sin(\alpha)}}$; $\alpha={{2m+1}\over{2n}} \pi$

As part of a (much) longer problem in complex analysis, I need to show that the equality mentioned in the title makes sense, but I can't seem to find the right algebra tricks to get from point A to ...
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### Expressing two complex variables with constraints in terms of a single one without

I have two complex variables $u$ and $v$ with the constraints $|u|^2+|v|^2=1\ \ \ \ \ \ \$ and $\ \ \ \ \ \ \ \ uv=|uv|\$. With two constraints there are basically two degrees of freedom, and I ...
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### How could I prove this trigonometric identity?

Show that: $$\left(\frac{1+\tan \theta}{1 - \tan \theta}\right)^n = \frac{1+i\tan n\theta}{1-i\tan n\theta}$$ Original image: http://i.stack.imgur.com/q8Yxj.jpg
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### Finding the Sum of the square of two positive integers.

Write the following equation as the sum of the square of two integers, $a^2 +b^2$. $$(8^2+5^2)(13^2+7^2)$$ I remember that you are supposed to do something with complex numbers or at least that ...
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### Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
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### Let $f(x) = x^5 + 4x^3 + 8x^2 + 32$ be a polynomial. Given that $f(2i) = 0$, express $f(x)$ as a product of linear complex polynomials.

I'm reviewing for my exam and I have a few questions about this solution. The given polynomial $f(x)$ has only real coefficients. Therefore, given $x = 2i$ is a root of $f(x)$, by the Conjugate Roots ...
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### A “complex” complex number problem

$a,b,c$ are cube roots of $p$ ,($p<0$) then for any permissible value of $x,y,z$ which is given by $$\frac{|xa+yb+zc|}{|xb+yc+za|} + (a_1^2-2b_1^2)\omega + \omega^2([x]+[y]+[z]) = 0$$ $\omega$ ...
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### Complex Numbers Question?

I answered the first part of the question. But I'm having a trouble with the second part. I can only find the half-line at $2i$ and $\theta=\pi/6$. Here's the solution guide:
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### Why does this have a complex component?

Why does: $$(-2)^{\frac{2}{3}}$$ have a complex component? I thought it would be equal to: $$((-2)^2)^{\frac{1}{3}}$$ $$= 4^{\frac{1}{3}}$$ which doesn't have a complex component. But Wolfram ...
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### Counterexample for conjugate rules in $\mathbb{Z}[\sqrt[4]{2}]$

We all know that, in a field, such as $\mathbb{Z}[i]$ or $\mathbb{Z}[\sqrt[4]{2}]$, conjugate properties that as $\overline {u \cdot v} = \overline {u} \cdot \overline {v}$ hold. Furthermore, ...
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### Help with rearranging equation to get real and imaginary parts..

I know this is so simple but my algebra is totally failing me.. I have the equation 1/1+2i and I want to extract the real and imaginary parts so I have it in the form.. Re+Im could someone just ...
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### $-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ inside unit circle?

Given $a>0$ I would like to know whether: $\alpha=-ia(1+ \sqrt{1-1/a^2})$ and $\beta =-ia(1- \sqrt{1-1/a^2})$ are inside the unit circle. How can I check that?
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### Find the minimum value of $\operatorname{Im}(z^5)/(\operatorname{Im}(z))^5$ [closed]

If $z$ is a complex number, then find the minimum value of $$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5},$$ where $\operatorname{Im}(z)$ denotes the imaginary part of z.
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### Find all the values of $(1+i)^{(1-i)}$

The question says to find all the values of $(1+i)^{(1-i)}$ I have trouble figuring out firstly, exactly what values are being looked for. I can toy around with the equation a bit to try to make it ...
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### Rewriting $x^3-3xy^2+2xy+i(-y^3+3x^2y-x^2+y^2 )$ in terms of $z$, with $z=x+yi$

How do I write $f=u+iv$ with: $u=x^3-3xy^2+2xy$ and $v=-y^3+3x^2y-x^2+y^2$ in terms of $z$ with $z=x+yi$?
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### Can we simplify $\sqrt{a}*\sqrt{a}$ to $a$ when $a \in \mathbb{R}$ and we do not know whether a is positive or negative?

Can we simplify $\sqrt{a}*\sqrt{a}$ to $a$ when $a \in \mathbb{R}$ and we do not know whether a is positive or negative? (Since $\sqrt{a}$ by itself is undefined in $\mathbb{R}$ when $a$ is negative) ...
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### Solve $x^2-(6+7i)x-4+20i=0$

Solve $x^2-(6+7i)x-4+20i=0$ I probably do too many substitutions, but here is my attempt: https://www.dropbox.com/sh/pqze4iagbo3nics/QhvkBcuZsH
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$(-1+i)^{\frac{1}{3}}$ here, $\tan\theta=-1$ so, $\theta=\tan^{-1}(-1)=\tan^{-1}(\tan(-\frac{\pi}{4}))=\tan^{-1}(\tan(\pi-\frac{\pi}{4}))=\pi-\frac{\pi}{4}$ My question is why can't i write ...
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### Definition of complex number

In many situations (problems as well as solutions) I encounter the complex number $i$ which many times is defined as $i^2=-1$ instead of $i=\sqrt{-1}$, since it is "more preferred". Does anyone know ...
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### Does $z^i=i^z$ have any solutions, beside $z=i$?

Does this equation have any solutions: $$\large{z^i=i^z}$$ Putting polar form of $z$ is better for LHS, But rectangular form is suitable for RHS ! What to do? Thanks!
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### Find the value of $\space\large i^{i^i}$?

Is $\large i^{i^i}$ real ? How to find it? Thank You!
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### Why don't we define division by zero as an arbritrary constant such as $j$? [duplicate]

Why don't we define $\frac 10$ as $j$ , $\frac 20$ as $2j$ , and so on? I know that by following the rules of math this eventually leads to $1=2$ , but we could make an exception and say that $j$ is ...
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### Trigonometric manipulation of complex number, how does this step occur?

I was reading the section about DeMoivre, and my book showed how to derive his formulas. The next part is supposed to be about finding roots of complex and real numbers. Roughly, it says: "Let $z$ be ...
Are there any complex numbers "z" that satisfy this equation? $$z=-\bar z?$$