In old quantum theory max Planck derived the energy density of black body radiation , for that purpose he calculated the no. Of modes in the cavity and those are actually standing waves ,different standing waves are different modes
And if length is L and wavelength is λ then L=nλ/2
And from intuition I can say these no. Of modes will be finite and so in integer number (If I say that number of modes is 5 , means there could be 5 different possible of modes or orientation of standing wave but ,There is no meaning of no. Of modes being 2.5 or 1.3 like these fractions ,they are only in inegers)
But Planck calculated no. Of modes is 8πν^2/c^3
Now here I think" 8πν^2/c^3" is not an integer (as π is not integer) Then how can no. Of modes be fraction??