How can I split the expression?

Question

When I use the command \begin{split} inside the command \begin{itemize} I get an error... Why? Can we not use it there?

The command I wrote is the following:

\begin{description}
\item $\begin{split}+ \ \ : \ \  x_1+x_2 &= a_1+b_1\sqrt{t^2-1}+a_2+b_2\sqrt{t^2-1}=(a_1+a_2)+(b_1+b_2)\sqrt{t^2-1} \\ & \mapsto (a_1+a_2, b_1+b_2)\end{split}$
\item $0\ \  : \ \  0=0+0 \sqrt{t^2-1} \mapsto (0,0)$
\end{description} 

Have I done something wrong?

Answer 1

Not sure what the desired result being sought is, but here

    \documentclass[12pt]{article}
    \usepackage{amsmath}
    \begin{document}

Single line equations:
    \begin{description}
    \item[(a)] $x_{1} + x_{2} = a_{1} + b_{1}\sqrt{t^2-1} + a_{2} + b_{2}\sqrt{t^2-1} = (a_{1} + a_{2} )
    + (b_{1} + b_{2})\sqrt{t^2-1} \mapsto (a_1+a_2, b_1+b_2)$
    \item[(b)] $0 :  0=0+0 \sqrt{t^2-1} \mapsto (0,0)$
    \end{description}

multi-line equation for the first descriptive item:
\begin{description}
\item[(a)] $\begin{aligned}[t] x_{1} + x_{2} &= a_{1} + b_{1}\sqrt{t^2-1} + a_{2} + b_{2}\sqrt{t^2-1} \\
& = (a_{1} + a_{2} ) + (b_{1} + b_{2})\sqrt{t^2-1} \\ & \mapsto (a_1+a_2, b_1+b_2) \end{aligned}$
\item[(b)] $0 + 0 =0+0 \sqrt{t^2-1} \mapsto (0,0)$
\end{description}

    \end{document}

is code that removes the errors mentioned. The output is: