Relativistic doppler shift

Question

When rearranging the doppler shift equation in terms of wavelength to get in terms of $\beta$ we get: $\beta = \cfrac{(\lambda_{source})^2-(\lambda_{obs})^2}{(\lambda_{source})^2 + (\lambda_{obs})^2}$. My question is since $\beta>0$ when the observed wavelength is greater than the source e.g. redshifted, we would get a negative beta so do we have to rearrange the equation another way or would the answer just be the modulus of this equation. Hope this makes sense, thanks!

Answer 1

Note that for the longitudinal Doppler shift \begin{equation} \left. \beta\boldsymbol{=} \begin{cases} \dfrac{v}{c}\boldsymbol{>}0\,,\quad \texttt{if observer moves away from the source}\vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ \dfrac{v}{c}\boldsymbol{<}0\,,\quad \texttt{if observer moves towards the source} \end{cases} \right\} \tag{01}\label{01} \end{equation}