\begin{frame} \frametitle{Exponential Functions: Irrational Numbers} \begin{alertblock}{} But what about irrational numbers? What is $2^{\sqrt{3}}$ or $5^\pi$? \end{alertblock} \pause\bigskip Roughly, one can imagine the situation like in this figure: \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,baseline=0cm] \diagram{-2.5}{2.5}{-1}{4}{1} \diagramannotatez \diagramannotatexx{-1,1} \diagramannotatey{1} \begin{scope}[ultra thick] \draw[cgreen,dotted] plot[smooth,domain=-2.5:2,samples=300] (\x,{2^\x}); \end{scope} \end{tikzpicture} } \end{center} \pause We have have defined the function for all rational points, and now want to close the gaps. \pause \bigskip Clearly, the result should be an increasing function\ldots \vspace{10cm} \end{frame}