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I was asked today what percentage of the EM spectrum we can see. It looks like a simple question, and yet I don't know how to answer it.

I know that the visible light has wavelengths between $3.8 \times 10^{-7}$ m and $7.4 \times 10^{-7}$ m, but this should amount to 0% compared to the whole, infinite range of possible wavelengths.

Even if we consider the existence of a upper limit for the wavelength, say the width of the observable universe ($1.4 \times 10^{10}$ ly = $1.3 \times 10^{26}$ m), the percentage is so close to zero ($2.7 \times 10^{-31}$%) that I don't feel it's in any way meaningful.

I also thought of using a logarithmic scale, which requires a lower limit as well as an upper one, say the Planck length ($1.6 \times 10^-35$ m). In this case the percentage is 0.48%, which is more meaningful, and yet I'm not sure the idea of using a logarithmic scale is actually a good one or if it's just something I constructed in order to obtain a nicer number.

Can someone help me understand which one of these explanations is valid, or construct something better?

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    $\begingroup$ This is not a meaningful question indeed. $\endgroup$
    – my2cts
    May 6, 2019 at 22:17
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    $\begingroup$ Zero seems like a perfectly reasonable answer. I suspect that the question was asked by someone who doesn’t understand the EM spectrum as well as you do. $\endgroup$
    – G. Smith
    May 7, 2019 at 0:14
  • $\begingroup$ @G.Smith Exactly, and it was inspired by images like this one. $\endgroup$
    – GRB
    May 7, 2019 at 7:41
  • $\begingroup$ In a qualitative sense "very little" would be a perfectly good answer, and is the kind of description I used to give to my students in descriptive astronomy ("Our eyes see very little of the electromagnetic spectrum, which is why we build so many different kinds of telescopes" sort of thing). $\endgroup$ May 7, 2019 at 18:59
  • $\begingroup$ In a quantitative sense you could propose some limits on the width of useful spectrum and take our miserly octave as a fraction of that. Perhaps wavelengths between ten kilometer and a femtometer or attometer for technological limits. Or wavelegnth comparable to the size of the observable universe on one hand and the Planck length on the other for "absolute" limits. $\endgroup$ May 7, 2019 at 19:03

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This

... I don't feel it's in any way meaningful ...

is an intrinsic feature of the problem, because to be able to express the visible range as a fraction of the EM spectrum, you need a measure on the latter, and there is no natural way to do this.

The closest you can get is to assign a measure that is uniform in logarithmic scale, with long- and short-wavelength cutoffs at the size of the universe and at the Planck energy, but that essentially implies that there is an equal amount of interesting phenomena happening at photon energies vastly larger than anything than we've ever had access to, as in the ranges where we do have access, as well as a large almost-bottomless well of wavelengths that don't even fit inside our own galaxy. Physics, of course, doesn't care ─ but we, as humans, do.

All of which is to say: there is no unique answer that comes from physics. If you want an answer that's relevant to humans, then you should base your measure on cutoffs which come from human considerations. I would personally go with a logarithmically uniform measure with cutoffs at about ${\sim}10\:\rm TeV$ (about the highest energies we can reach in particle accelerators, hence about the highest gamma photon energies we can control) and at about ${\sim}1:\rm Hz$ (from the ELF radio band). But, again, that's ultimately a subjective choice.

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