Comparing different magnitudes. Is this statement correct?

Question

I have two methods which allow for a detection of a signal that is at most of magnitude x. If method 2 can detect signal that are half a magnitude larger, i.e. fainter, let's say x.5 mag, is it correct to say that method 2 can detect signals that are about 60% fainter than what method 1 can detect?

Is the 60% correct? I got this number by converting the magnitudes $m_i$ to flux $f_i$, i.e. $$100^{(m_1-m_2)/2.5}=\frac{f_2}{f_1}\approx 0.40$$ $$\Rightarrow f_2=0.40 f_1\sim 40\% f_1$$

Thus, $f_2$ is about $40\% f_1$ which means about $60\%$ fainter. I have never worked with magnitudes before, so is this a valid statement?

Answer 1

Textbooks typically express the flux vs. magnitude relation something like this:

$$m_2 - m_1 = -2.5 \log_{10} \frac{f_2}{f_1}$$

which we can transform into this:

$$\frac{f_2}{f_1} = 10^{(m_1 - m_2) / 2.5}$$

so the original version of your formula was correct.

If m₂ - m₁ = 0.5, then f₂ / f₁ = 0.63. Some readers find comparisons like "x% fainter" confusing, so I would say method 2 is more sensitive than method 1 by a factor of 1.58.