0
votes
1answer
62 views

Is $\left((-1)^2\right)^\frac12 = (-1)^\left(2\cdot\frac12\right)$? [duplicate]

I'm feeling confused. If I square 1 and -1, the answers should be equal: $1^2 = (-1)^2$ Then I take both sides to the power of $\frac12$: $\left(1^2\right)^\frac12 = \left((-1)^2\right)^\frac12$ ...
0
votes
1answer
45 views

How to show that all roots of $(11+v)q^3-18q^2+9q-2$ have their absolute value less than 1.

The equation is $(11+v)q^3-18q^2+9q-2=0$, where $v>0$ I need to show that either absolute value of all the roots is not greater than one or there exists a root $q: |q|>1$. Using Weierstrass ...
0
votes
0answers
28 views

Ways to compute roots of complex numbers

I know how to use the De Moivre's Formula, but to caculate it one need to use caclulator. Is there any better way to take roots of complex numbers that is more "caclulator-free"? I am particulary ...
1
vote
2answers
61 views

Why are the roots of the polynomial $z^N = a^N$ equal to $z_k = a \ e^{j\frac{2 \pi k}{N}}$?

I am trying to understand equation 3.28 from this image in my book. I get everything that the author is saying, except for when he finds the roots, (zeros), of $z^N = a^N$. Of course, there are ...
0
votes
2answers
128 views

Rational exponents: prove some states

In some rational exponent expressions the solution isn't a real number why? Example (explain what I mean): $$\begin{align} \Big(-x\Big)^{1/n}=\left\{\text{is not a real number}\right\} \end{align}$$ ...
1
vote
1answer
604 views

Multiplying two complex numbers using only three multiplications of real numbers

I have problem given below. Show how two complex numbers $(a+ib)$ and $(c+id)$ may be multiplied using only three multiplications of real numbers, where $i=\sqrt{-1}$. You may use any number of ...
2
votes
4answers
284 views

nth roots of negative numbers

Disclaimer: I know what complex numbers are. Let $x,\space n\in\Bbb R$ What is the complex algebraic solution to $\sqrt[n]{-x}$? Could I have a 'general' formula and a walk through on how to ...
1
vote
1answer
64 views

Proving that: $-\frac{2 \;i\log(i^2)}{2} = \pi$

I'm trying to prove that: $$-\frac{2 \;i\log(i^2)}{2} = \pi$$ This is what I've tried: $$-\frac{2 \;i\log(i^2)}{2} = -i \log(i^2) = -i (i \pi)\implies$$ $$-x\;(x y)=-x\;y\;x\implies$$ $$-i\;(i \pi) ...
2
votes
0answers
168 views

Number of solutions for $x$ of such form that $\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$

Consider $$\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$$ where $y$ is an integer. In relation to solutions for $x$; How could one prove that: $(1)$: There are $y$ solutions for $x$, in total. ...
0
votes
1answer
55 views

Does $i = -\frac{(2\;W({\pi\over2}))}{\pi}$

Let $x = -\frac{(2\;W({\pi\over2}))}{\pi}$, where $W$ denotes the Lambert W-function. As $${\log(i^2)\over i} = \pi$$ and $${\log(x^2)\over x}=\pi$$ Does $x = i$?
2
votes
0answers
129 views

Is there a finite number of solutions to $\mathrm{Re}(a^n)+\mathrm{Im}(a^n)=b^n$, where $a$ is a Gaussian integer and $b \in \Bbb Z$?

Let $$E_n=\{(x,y,b) \in \mathbb{Z}^*\times \mathbb{Z}^*\times \mathbb{Z}^* ~|~ \gcd(x,y,b)=1 ~ \mathrm{and}~\mathrm{Re}((x+iy)^n)+\mathrm{Im}((x+iy)^n)=b^n \}$$ For $n \geq 3$, is $E_n$ finite or not ...
4
votes
5answers
203 views

What is the square root of $i^4$?

What is the $\sqrt{i^4}$? $i^4$ = $(i^2)^2$ So is $\sqrt{i^4}$ = $\sqrt{(i^2)^2}$ = $i^2$ = $-1$? Or is $\sqrt{i^4}$ = $\sqrt{1}$ = $1$? When I plug it into my TI-89 Titanium, I get $1$. Edit: I ...
2
votes
1answer
219 views

Complex division: polar form vs complex conjugate

The original problem In an electricity course which I volunteered to help with, the students solve circuits using phasors. Using phasors requires a good knowledge of complex numbers arithmetics, ...
5
votes
7answers
460 views

Is $0^0=1$ postulate independent of all other axioms of complex numbers?

This question is inspired by the other question which asked for a proof that $i^i$ is a real number. Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he ...
9
votes
3answers
274 views

square root of $1/2 + \sqrt3/2?$

Playing with Maple, I noticed that it gives the square root of $c = 1+\frac{\sqrt3}{2}$ as equal to $a = \frac{1}{2}+\frac{\sqrt3}{2}$. Indeed it checks out. But I got curious: how can I find that ...
0
votes
1answer
104 views

Arithmetic question regarding $\sqrt{1/-1} = \sqrt{-1/1}$ [duplicate]

Possible Duplicate: -1 is not 1, so where is the mistake? $i^2$ why is it $-1$ when you can show it is $1$? Can someone please point out what I'm doing wrong here? $$ \frac{1}{-1} = ...
0
votes
2answers
111 views

Which roots of a negative number can be done?

I'm an Android programmer and am working on a graphing calculator. I have been looking for the limits on which roots can be done. I have a decent understanding of mathematics but can not seem to find ...
0
votes
1answer
92 views

rearrange $z \mapsto z^2 + c$

Mathematics, some of its magic is that a lot is known about how to rearrange its statements (equations). Given the Mandelbrot Set: $z \mapsto z² + c$ (or more precisely) $z_{i+1} = z_i ^2 + c$ ...
5
votes
1answer
65 views

Confused about exponents and imaginary/real answers

I am confused about some exponent behavior. $$(-2)^{7.6} = (-2)^{\frac{76}{10}} = ((-2)^{76})^{\frac{1}{10}} = ((-2)^{\frac{1}{10}})^{76}$$ Is there something wrong in this logic? When I plug the ...
20
votes
6answers
2k views

$-1$ is not $ 1$, so where is the mistake?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} ...