Saltar al contenido principal

# 3.4.E: Problemas en Números Complejos (Ejercicios)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## Ejercicio$$\PageIndex{1'}$$

Verificar que los “puntos reales” en$$C$$ forma de un campo ordenado.

## Ejercicio$$\PageIndex{2}$$

$$z \overline{z}=|z|^{2} .$$Demostrar que Deducir que$$z^{-1}=\overline{z} /|z|^{2}$$ si$$z \neq 0 .^{4}$$

## Ejercicio$$\PageIndex{3}$$

Demostrar que

\ [\ overline {z+z^ {\ prime}} =\ overline {z} +\ overline {z^ {\ prime}}\ text {y}\ overline {z z^ {\ prime}} =\ overline {z}\ cdot\ overline {z^ {\ prime}}
\]
De ahí mostrar por inducción que
$$\overline{z^{n}}=(\overline{z})^{n}, n=1,2, \ldots,$$ y$$\quad \overline{\sum_{k=1}^{n} a_{k} z^{k}}=\sum_{k=1}^{n} \overline{a}_{k} \overline{z}^{k}$$

## Ejercicio$$\PageIndex{4}$$

Definir
\ [
e^ {\ theta i} =\ cos\ theta+i\ sin\ theta.
\]
Describir$$e^{\theta i}$$ geométricamente. Es$$\left|e^{\theta i}\right|=1 ?$$

## Ejercicio$$\PageIndex{5}$$

Calcular
(a)$$\frac{1+2 i}{3-i}$$;
(b)$$(1+2 i)(3-i) ;$$ y
(c)$$\frac{x+1+i}{x+1-i}, x \in E^{1}$$.
Hazlo de dos maneras: (i) usando solo definiciones y la notación$$(x, y)$$ para$$x+y i ;$$ y$$(\text { ii) using all laws valid in a field. }$$

## Ejercicio$$\PageIndex{6}$$

Resolver la ecuación$$(2,-1)(x, y)=(3,2)$$ para$$x$$ y$$y$$ en$$E^{1}$$.

## Ejercicio$$\PageIndex{7}$$

Que
\ [
\ comience {alineado} z &=r (\ cos\ theta+i\ sin\ theta)\\ z^ {\ prime} &=r^ {\ prime}\ left (\ cos\ theta^ {\ prime} +i\ sin\ theta^ {\ prime}\ right),\ text {y}\\ z^ {\ prime\ prime} &=r^ {prime\ prime}\ left (\ cos\ theta^ {\ prime\ prime} +i\ sin\ theta^ {\ prime\ prime}\ derecha)\ end { alineado}
\]
como en Corolario$$2 .$$ Probar que$$z=z^{\prime} z^{\prime \prime}$$ si
\ [
r=|z|=r^ {\ prime} r^ {\ prime\ prime},\ text {es decir,}\ izquierda|z^ {\ prime} z^ {\ prime\ prime}\ derecha|=\ izquierda|z^ {\ prime}\ derecha|\ izquierda|z^ {\ prime\ prime}\ derecha|,\ text {y}\ theta=\ theta^ {\ prime} + \ theta^ {\ prime\ prime}.
\]
Discutir la siguiente afirmación: Multiplicar$$z^{\prime}$$ por$$z^{\prime \prime}$$ medios para rotar en$$\overrightarrow{0 z^{\prime}}$$ sentido antihorario por el ángulo$$\theta^{\prime \prime}$$ y multiplicarlo por el escalar$$r^{\prime \prime}=$$$$\left|z^{\prime \prime}\right| .$$ Considere los casos$$z^{\prime \prime}=i$$ y$$z^{\prime \prime}=-1$$.
[Pista: Eliminar corchetes en
\ [
r (\ cos\ theta+i\ sin\ theta) =r^ {\ prime}\ left (\ cos\ theta^ {\ prime} +i\ sin\ theta^ {\ prime}\ derecha)\ cdot r^ {\ prime\ prime}\ left (\ cos\ theta^ {\ prime\ prime} +i\ sin\ theta^ a^ {\ prime\ prime}\ derecha)
\]
y aplicar las leyes de la trigonometría.]

## Ejercicio$$\PageIndex{8}$$

Por inducción, extienda el Problema 7 a productos de números$$n$$ complejos, y derive la fórmula de Moivre, es decir, si$$z=r(\cos \theta+i \sin \theta),$$ entonces
\ [
z^ {n} =r^ {n} (\ cos (n\ theta) +i\ sin (n\ theta)).
\]
Úselo para encontrar, para$$n=1,2, \ldots$$
\ [
(a) i^ {n};\ hskip 12pt (b) (1 + i) ^ {n};\ hskip 12pt (c)\ frac {1} {(1 + i) ^ {n}}.
\]

## Ejercicio$$\PageIndex{9}$$

De Problem$$8,$$ probar que por cada número complejo$$z \neq 0,$$ hay exactamente números$$n$$ complejos$$w$$ tales que
\ [
w^ {n} =z;
\] se
llaman las raíces$$n$$ th de$$z$$
[Pista: Si
\ [
z=r (\ cos\ theta+i\ sin\ theta)\ text {y} w=r^ {\ prime}\ left (\ cos\ theta^ {\ prime} +i\ sin\ theta^ {\ prime}\ right),
\]
la ecuación$$w^{n}=z$$ rinde, por Problema 8
\ [
\ left (r^ {\ prime}\ derecha) ^ {n} =r\ text {y} n\ theta^ {\ prime} =\ theta,
\]
y a la inversa.
Si bien esto determina de$$r^{\prime}$$ manera única,$$\theta$$ puede ser reemplazado por$$\theta+2 k \pi$$ sin afectar$$z .$$ Así

\ [\ theta^ {\ prime} =\ frac {\ theta+2 k\ pi} {n},\ quad k=1,2,\ ldots
\]$$w$$ Resultado de puntos
distintos sólo de$$k=0,1, \ldots, n-1$$ (entonces repiten cíclicamente).
Así$$w$$ se obtienen$$n$$ valores de.]

## Ejercicio$$\PageIndex{10}$$

Usa el Problema 9 para encontrar en$$C$$
\ [
\ text {(a) todas las raíces cubicas de} 1;\ quad\ text {(b) todas las cuartas raíces de} 1
\]
Describe todas las raíces$$n$$ th de 1 geométricamente.

3.4.E: Problemas en Números Complejos (Ejercicios) is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.