## Derivation of the Approximate Redshift Expression

You may recognise the approximate redshift expression
$z \simeq \frac{v}{c}$
. This equation is easily derived from the exact relationship
$z= \sqrt{\frac{1+v/c}{1-v/c}} -1$
.
Use the binomial expansion and assume
$v \ll c$
so ignore powers of
$v/c$
of 2 and higher.
\begin{aligned} z &= \sqrt{\frac{1+v/c}{1-v/c}} -1 \\ &= (1+v/c)^{1/2}(1-v/c)^{11/2}-1 \\ & \simeq (1+1/2v+...)(1+1/2v/c+...)-1 \\ &= 1+1/2v/c +1/2v/c +...-1 \\ &=v/c \end{aligned}

The exact expression is a form of the Doppler shift.
$\frac{\Delta f}{f} =z = \sqrt{\frac{1+v/c}{1-v/c}} -1 \simeq v/c$
(1)
Using
$c=f \lambda$
,
$0= f \Delta | \lambda + (\Delta f) \lambda \rightarrow \| \frac{\Delta f}{f} \| = \| \frac{\Delta \lambda}{\lambda}$
so we can also write (1) above as
$\frac{\Delta \lambda}{\lambda} =z \simeq v/c$
.