0
votes
5answers
87 views

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 ...
1
vote
1answer
28 views

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 ...
0
votes
1answer
21 views

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 $(2+6i)^6$
1
vote
2answers
40 views

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 ...
1
vote
1answer
25 views

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 ...
0
votes
1answer
9 views

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 ...
0
votes
2answers
61 views

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
1
vote
1answer
37 views

Confusion about whether $\frac{e^{i 2 \pi} - 1}{e^{-i 2 \pi} - 1}$ is undefined or -1

I'm sure this is a trivial question, so hopefully the response will be quick :) It seems to me that: $$ \frac{e^{i 2 \pi} - 1}{e^{-i 2 \pi} - 1} = \frac{1 - 1}{1 - 1} = \frac{0}{0} \Rightarrow ...
1
vote
1answer
31 views

Conversion of complex numbers to standard form

$$(1/\sqrt{2} - i/\sqrt{2})^8$$ Im not sure where to begin here, should i just expand it out completely and then simplify? $$1/i^{2013}$$ For this one im guessing because $2012$ is basically the same ...
1
vote
1answer
54 views

how to simplify the following expression: $-(4i)^3$

How to simplify the following term $$-(4i)^3$$ I have tried solving it the following way: taking the square root of $-16$ to the third power and taking the negative of that. I am getting an answer ...
1
vote
1answer
49 views

How to work out phase of complex number

This one follows on from the question I just asked about logarithms.. Turns out 1/x questions confuse me (sorry for bombarding your exchange with questions, this isn't homework or anything I am just ...
0
votes
1answer
49 views

Complex De Moivre's theorem question

Express this in terms of multiple angles. $\cos^3x \sin^4x$ I've used the relationships $$\cos(nx) = \frac{z^n+z^{-n}}{2}$$ $$\sin(nx) = \frac{z^n-z^{-n}}{2j}$$ And end up with $$\cos^3x \sin^4x = ...
3
votes
3answers
90 views

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 ...
2
votes
2answers
31 views

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: ...
0
votes
1answer
40 views

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 ...
2
votes
1answer
70 views

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 $$\frac{|xa+yb+zc|}{|xb+yc+za|} + (a_1^2-2b_1^2)\omega + \omega^2([x]+[y]+[z]) = 0 $$ $\omega$ ...
0
votes
1answer
88 views

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:
0
votes
2answers
43 views

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 ...
0
votes
0answers
135 views

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, ...
0
votes
1answer
83 views

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 ...
0
votes
1answer
34 views

$-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ 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?
0
votes
1answer
29 views

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.
1
vote
2answers
170 views

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 ...
0
votes
2answers
142 views

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 $z=x+yi$?
2
votes
3answers
119 views

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) ...
0
votes
3answers
68 views

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
0
votes
1answer
33 views

Quadrant problem

$(-1+i)^{\frac{1}{3}}$ 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 ...
2
votes
1answer
50 views

Algebra problem - length of vector

Algebra - Complex Numbers Calculate the length of the vector given by $\frac{\left( 1+2i \right)\left( 1+\sqrt{3}\cdot i \right)}{\left( 1+i \right)^{3}} $ I start by doing $\left( 1+i ...
2
votes
4answers
63 views

Find the values of $x$, where $x \in \Bbb C$, for which $x^4-1 =0$

Find the values of $x$, where $x \in \Bbb C$, for which $$x^4-1 =0$$ I can see that $x^4-1 = (x^2-1)(x^2+1)=0$ So one set of roots can be taken from $$x^2-1=0$$$$ \Rightarrow x=\pm1$$ However, ...
2
votes
6answers
99 views

Given $ai$ is a root of $x^3-bx^2+a^2x-a^2b=0$, prove it and find the other roots.

Given that $a$ and $b$ are real constants, prove that $ai$ is a root of the equation $$x^3-bx^2+a^2x-a^2b=0$$ Find the other roots of the equation in terms of $a$ and $b$. I realise the conjugate ...
0
votes
2answers
101 views

Given that $z=2-i$ and $z^2=3-4i$ find the roots of the equation $(z+i)^2=3-4i$

Given that $z=2-i$ and $z^2=3-4i$ find the roots of the equation $(z+i)^2=3-4i$ How do you use the given properties to find the roots? I can only obtain them the long way by working through ...
0
votes
3answers
105 views

Deduce that $\tan \frac{3\pi}{8}=1+\sqrt2$

Find the modulus and the argument, in radians in terms of $\pi$, of $$z_1=\frac{1+i}{1-i}, z_2=\frac{\sqrt2}{1-i}, z_3=\left(\frac{1+i}{1-i}\right)^2$$ Plot $z_1, z_2$ and $z_1+z_2$ on an Argand ...
1
vote
2answers
70 views

Given that $T=\frac{x-iy}{x+iy}$, where $x, y, T \in\Bbb R$, show that $\frac{1+T^2}{2T}=\frac{x^2-y^2}{x^2+y^2}$

Given that $T=\frac{x-iy}{x+iy}$, where $x, y, T \in\Bbb R$, show that $$\frac{1+T^2}{2T}=\frac{x^2-y^2}{x^2+y^2}$$ I have got rid of $i$ in the denominator, leaving $$\frac{x^2-y^2-2ixy}{x^2+y^2}$$ ...
5
votes
1answer
72 views

Region in complex plane with $|1-z|\leq M(1-|z|)$

Let $M>0$. Describe the region in the complex plane such that $|1-z|\leq M(1-|z|)$. To start, I take $M=1$. The inequality becomes $|1-z|\leq 1-|z|$. But by triangle inequality, we have ...
1
vote
1answer
62 views

Complex polynomial identity with norm condition

In this question, the following was shown: If $R(z)=\dfrac{P(z)}{Q(z)}$, where $P,Q$ are polynomials in a complex variable $z$, satisfies the condition that $|R(z)|=1$ whenever $|z|=1$, then the ...
9
votes
4answers
481 views

How to show that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ indirectly?

I found this amazingly beautiful identity here. How to prove that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ without directly multiplying the factors? (I've already ...
4
votes
2answers
399 views

Complex Numbers: Finding solutions to $ (z^2-3z+1)^4 = 1 $

I have to find all solutions to $$ (z^2-3z+1)^4 = 1 $$ What I thought could work was $$z^2-3z+1= 1^{1/4} $$ Given that the 4 4th-roots of 1 are $1, i, -i, -1$ my idea was to look at each case ...
2
votes
2answers
97 views

Help with complex numbers $z^4 = i \bar z ^3$

I'm having trouble with complex numbers. For example, I need to find all the solutions to $$ z^4 = i \bar z ^3$$ My attempt was $$z= |z|\ \left(\cos(\alpha)+i\ \sin(\alpha)\right) $$ $$ z^4 = i ...
2
votes
2answers
85 views

Solving equation $z^2 + 6z + 12 - 4i = 0$

How would one solve this complex number equation, I have tried substituting $z$ by $a+ib$ but afterwards I am left with a long equation which i do not think should be like that? $$z^2 + 6z + 12 - 4i ...
0
votes
1answer
84 views

Calculating z/w in polar form

IT HAS BEEN ANSWERED, THANK YOU So we have: z = 7 + 4i w = 8-i I did this: $$7+4i/8-i$$ $$conjagate =\space 8+i$$ $$ (7+4i/8-i) \times (8+i/8+i)$$ $$(56+7i+32i+4i^2)/((8-i)\times(8+i))$$ ...
0
votes
3answers
118 views

Can you get the exact real value of $ \left((-1)^{\frac{1}{180}}\right)^{89}-\left((-1)^{\frac{1}{180}}\right)^{91}$?

By using euler formula,one can obtain: $$ 2\sin\left(\frac{\pi}{180}\right)=\left((-1)^{\frac{1}{180}}\right)^{89}-\left((-1)^{\frac{1}{180}}\right)^{91}. $$ In order to get the exact real value of ...
1
vote
0answers
60 views

Finding $i^{i^i}$. [duplicate]

Express $i^{i^i}$ in the form $a+bi$ where $a,b$ are real. From euler's formula, I get $\ln i=i\pi/2$, which leads to $i^i=e^{-\pi/2}$. Therefore, $\ln i^{i^i}=i^i\ln i=i\pi ...
1
vote
1answer
71 views

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 ...
12
votes
2answers
318 views

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!
25
votes
5answers
1k views
4
votes
3answers
152 views

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 ...
1
vote
1answer
96 views

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 ...
2
votes
1answer
150 views

Complex numbers true or false

Are there any complex numbers "z" that satisfy this equation? $$z=-\bar z?$$
2
votes
2answers
51 views

Problem with operations on exponents

Okay, this is a very basic doubt, and I would like to know where exactly the mistake is. Let's take the equality $1^2=1$ Now I do it this way Step 1: $1^2 = 1^\frac{4}{2}$ Step2: $\implies ...
2
votes
2answers
124 views

trigonometric representation of a complex number.

Let $z=e^{it}+1$ where $0\leq t\leq \pi$, Find the trigonometric representation of $z^2+z+1$. (The trigonometric representation should be in the form of : $r(\cos \theta +i \sin \theta)$, where ...