@@ -113,34 +113,52 @@ <h1>Riemann Sphere</h1>
113113
114114 < div id ="section1 ">
115115 < h2 > The Point at infinity</ h2 >
116- < p > For some purposes it is convenient to introduce the < em > point at infinity</ em > , denoted by $\infty,$
117- in
118- addition to the points $z\in \mathbb C.$ We must be careful in doing so, because it can lead to
119- confusion
120- and abuse of the symbol $\infty.$ However, with care it can be useful, if we want
121- to be able to talk about infinite limits and limits at infinity.</ p >
122- 123- < p > In contrast to the real line, to which $+\infty$ and $-\infty$ can be added, we have only one
116+ < p > For some purposes it is convenient to introduce the
117+ < em > point at infinity</ em > , denoted by $\infty,$
118+ in addition to the points $z\in \mathbb C.$ We must be
119+ careful in doing so, because it can lead to
120+ confusion and abuse of the symbol $\infty.$ However, with care
121+ it can be useful, if we want
122+ to be able to talk about infinite
123+ limits and limits at infinity.</ p >
124+ 125+ < p > In contrast to the real line, to which $+\infty$
126+ and $-\infty$ can be added, we have only one
124127 $\infty$ for
125- $\mathbb C.$ The reason is that $\mathbb C$ has no natural ordering as $\mathbb R$ does. Formally we
126- add a
127- symbol $\infty$ to $\mathbb C$ to obtain the < em > extended complex plane</ em > , denoted by $\mathbb
128- C^*=\mathbb C \cup \{\infty\},$ and define operations with $\infty$ by the rules
129- </ p > < div class ="scroll-wrapper ">
128+ $\mathbb C.$ The reason is that $\mathbb C$ has no
129+ natural ordering as $\mathbb R$ does. Formally we
130+ add a symbol $\infty$ to $\mathbb C$ to obtain the
131+ < em > extended complex plane</ em > , denoted by $\mathbb
132+ C^*=\mathbb C \cup \{\infty\}.$
133+ We establish operations with $\infty$ by setting
134+ \begin{eqnarray*}
135+ z+\infty = \infty + z =\infty
136+ \end{eqnarray*}
137+ for every $z\neq \infty,$ and
130138 \begin{eqnarray*}
131- z+\infty&=&\infty\\
132- z\cdot \infty&=&\infty \quad \quad \text{provided } z\neq 0\\
133- \infty+\infty&=&\infty\\
134- \infty\cdot\infty&=& \infty\\
135- \frac{z}{\infty}&=&0
139+ z\cdot \infty = \infty \cdot z =\infty
140+ \end{eqnarray*}
141+ for all $z\neq 0,$ including $z= \infty.$
142+ However, it is not possible to define
143+ < div class ="scroll-wrapper ">
144+ \begin{eqnarray*}
145+ \infty + \infty,,円,円 \infty - \infty,,円,円 0\cdot \infty
136146 \end{eqnarray*}
137147 </ div >
148+ because there is no consistent algebraic
149+ or geometric interpretations for these operations.
150+ </ p >
138151
139- for $z\in \mathbb C.$ Notice that some operations are not defined:
140- < div class ="scroll-wrapper ">
141- $$\frac{\infty}{\infty},,円\quad 0\cdot \infty,,円\quad \infty-\infty,,円$$
142- </ div >
143- and so forth are for the same reasons that they are in the calculus of real numbers.< p > </ p >
152+ < p >
153+ By special convention we also define
154+ < div class ="scroll-wrapper ">
155+ \[
156+ \dfrac{z}{0}= \infty,円 \text{ for },円 z\neq 0
157+ \;\text{ and }\; \dfrac{z}{\infty} = 0,円 \text{ for },円 z\neq \infty.
158+ \]
159+ </ div >
160+ The quotients $\dfrac{0}{0}$ and $ \dfrac{\infty}{\infty}$ are undefined.
161+ </ p >
144162
145163 < p > The extended complex plane can be mapped onto the
146164 surface of a sphere whose south pole corresponds to the origin and whose north
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