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updated: fixed issue with operations in the extended complex plane
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‎contenido/esfera_de_riemann.html‎

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@@ -123,28 +123,45 @@ <h2>El Punto al infinito</h2>
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infinito.
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</p>
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<p>En contraste con la línea real, para la cual los símbolos $+\infty$ y $-\infty$
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se puede agregar, en el caso de los números complejos $\mathbb C$
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solo necesitamos un símbolo $\infty.$ La razón es que $\mathbb C$
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es un conjunto que no tienen un orden natural como $\mathbb R.$
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Formalmente agregamos el símbolo $\infty$ a $\mathbb C$ para obtener el
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<em>plano complejo extendido</em>, denotado por
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$\mathbb C^*=\mathbb C \cup \{\infty\},$ y definimos las operaciones
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con $\infty$ con las siguientes reglas
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</p><div class="scroll-wrapper">
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\begin{eqnarray*}
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z+\infty&amp;=&amp;\infty\\
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z\cdot \infty&amp;=&amp;\infty \quad \quad \text{provided } z\neq 0\\
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\infty+\infty&amp;=&amp;\infty\\
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\infty\cdot\infty&amp;=&amp; \infty\\
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\frac{z}{\infty}&amp;=&amp;0
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\end{eqnarray*}
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</div>
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para $z\in \mathbb C.$ Notemos que algunas operaciones no están definidas:
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<div class="scroll-wrapper">
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$$\frac{\infty}{\infty}\,,\quad 0\cdot \infty\,,\quad \infty-\infty\,,$$
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</div>
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y además no están definidas por las mismas razones que de su versión de números reales.<p></p>
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<p>A diferencia de la recta real, a la que se pueden añadir
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$+\infty$ y $-\infty$, en $\mathbb{C}$ solo tenemos un
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$\infty.$ La razón es que $\mathbb{C}$ no tiene un
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orden natural como lo tiene $\mathbb{R}.$ Formalmente,
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añadimos un símbolo $\infty$ a $\mathbb{C}$ para obtener
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el <em>plano complejo extendido</em>, denotado por
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$\mathbb{C}^* = \mathbb{C} \cup \{\infty\}.$
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Establecemos operaciones con $\infty$ definiendo:
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<br>
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<div class="scroll-wrapper">
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\begin{eqnarray*}
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z+\infty = \infty + z = \infty
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\end{eqnarray*}
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para todo $z \neq \infty,$ y
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<br>
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\begin{eqnarray*}
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z\cdot \infty = \infty \cdot z = \infty
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\end{eqnarray*}
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para todo $z \neq 0,$ incluyendo $z = \infty.$
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Sin embargo, no es posible definir
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<br>
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\begin{eqnarray*}
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\infty + \infty\,,\, \infty - \infty\,,\, 0\cdot \infty
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\end{eqnarray*}
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</div>
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porque no hay interpretaciones algebraicas ni geométricas
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consistentes para estas operaciones.
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</p>
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<p>
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Por convención especial también definimos:
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<div class="scroll-wrapper">
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\[
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\dfrac{z}{0} = \infty\, \text{ para }\, z \neq 0
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\;\text{ y }\; \dfrac{z}{\infty} = 0\, \text{ para }\, z \neq \infty.
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\]
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</div>
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Los cocientes $\dfrac{0}{0}$ y $\dfrac{\infty}{\infty}$ no están definidos.
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</p>
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<p>
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El plano complejo extendido se puede mapear biyectivamente

‎content/riemann_sphere.html‎

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@@ -113,34 +113,52 @@ <h1>Riemann Sphere</h1>
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<div id="section1">
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<h2>The Point at infinity</h2>
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<p>For some purposes it is convenient to introduce the <em>point at infinity</em>, denoted by $\infty,$
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in
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addition to the points $z\in \mathbb C.$ We must be careful in doing so, because it can lead to
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confusion
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and abuse of the symbol $\infty.$ However, with care it can be useful, if we want
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to be able to talk about infinite limits and limits at infinity.</p>
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<p>In contrast to the real line, to which $+\infty$ and $-\infty$ can be added, we have only one
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<p>For some purposes it is convenient to introduce the
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<em>point at infinity</em>, denoted by $\infty,$
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in addition to the points $z\in \mathbb C.$ We must be
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careful in doing so, because it can lead to
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confusion and abuse of the symbol $\infty.$ However, with care
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it can be useful, if we want
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to be able to talk about infinite
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limits and limits at infinity.</p>
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<p>In contrast to the real line, to which $+\infty$
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and $-\infty$ can be added, we have only one
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$\infty$ for
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$\mathbb C.$ The reason is that $\mathbb C$ has no natural ordering as $\mathbb R$ does. Formally we
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add a
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symbol $\infty$ to $\mathbb C$ to obtain the <em>extended complex plane</em>, denoted by $\mathbb
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C^*=\mathbb C \cup \{\infty\},$ and define operations with $\infty$ by the rules
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</p><div class="scroll-wrapper">
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$\mathbb C.$ The reason is that $\mathbb C$ has no
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natural ordering as $\mathbb R$ does. Formally we
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add a symbol $\infty$ to $\mathbb C$ to obtain the
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<em>extended complex plane</em>, denoted by $\mathbb
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C^*=\mathbb C \cup \{\infty\}.$
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We establish operations with $\infty$ by setting
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\begin{eqnarray*}
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z+\infty = \infty + z =\infty
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\end{eqnarray*}
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for every $z\neq \infty,$ and
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\begin{eqnarray*}
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z+\infty&amp;=&amp;\infty\\
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z\cdot \infty&amp;=&amp;\infty \quad \quad \text{provided } z\neq 0\\
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\infty+\infty&amp;=&amp;\infty\\
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\infty\cdot\infty&amp;=&amp; \infty\\
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\frac{z}{\infty}&amp;=&amp;0
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z\cdot \infty = \infty \cdot z =\infty
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\end{eqnarray*}
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for all $z\neq 0,$ including $z= \infty.$
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However, it is not possible to define
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<div class="scroll-wrapper">
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\begin{eqnarray*}
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\infty + \infty\,,\, \infty - \infty\,,\, 0\cdot \infty
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\end{eqnarray*}
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</div>
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because there is no consistent algebraic
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or geometric interpretations for these operations.
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</p>
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for $z\in \mathbb C.$ Notice that some operations are not defined:
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<div class="scroll-wrapper">
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$$\frac{\infty}{\infty}\,,\quad 0\cdot \infty\,,\quad \infty-\infty\,,$$
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</div>
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and so forth are for the same reasons that they are in the calculus of real numbers.<p></p>
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<p>
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By special convention we also define
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<div class="scroll-wrapper">
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\[
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\dfrac{z}{0}= \infty\, \text{ for }\, z\neq 0
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\;\text{ and }\; \dfrac{z}{\infty} = 0\, \text{ for }\, z\neq \infty.
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\]
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</div>
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The quotients $\dfrac{0}{0}$ and $ \dfrac{\infty}{\infty}$ are undefined.
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</p>
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<p>The extended complex plane can be mapped onto the
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surface of a sphere whose south pole corresponds to the origin and whose north

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