What is the equation for high (cosmological) redshift?

[JArnold]

The equation for cosmological redshift where z << 1 is is commonly given as z = λobs / λemit -1

What is the equation for high-z, accounting for how light surpasses the spatial expansion it leaves behind, but abstracting from gravitational influences? I'm particularly interested in how CMB can be calculated to have a z of only ~1100 if it dates from 380,000 years from the start of cosmological expansion, which would indicate a z of 13,750,000,000 / 380,000 -1, or more than 36,000.

 

[phyzguy]

The equation z = λobs / λemit -1 is valid even at very high redshifts. So the CMB radiation has been shifted by a factor of ~1000. Since the peak wavelength of a blackbody is proportional to 1/T, this means the temperature of the CMB blackbody has also been shifted by ~1000, from ~3000K as emitted to ~3K as we see it today.

Your mistake is in assuming that the redshift is proportional to the lookback time, which isn't the case. It's just more complicated than that. This paper explains the math behind it, and Figure 6 shows a graph of z vs lookback time.