\begin{definition}[Definition:Real Hyperbolic Tangent/Definition 1]
The '''real hyperbolic tangent''' function is defined on the real numbers as:
:$\tanh: \R \to \R$:
:$\forall x \in \R: \tanh x := \dfrac {e^z - e^{-x} } {e^z + e^{-x} }$
\end{definition}