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Would the rainbow look different if we could see the full EM spectrum (or at least a lot more of it)? Would the width of a rainbow be bigger and would all wave lengths of light be in a rainbow?

Jo Shmo
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If we could 'see' a lot more of the EM spectrum, then yes, the rainbow would be wider, and we would have a larger vocabulary to name the 'colors'.

All the wavelengths of the EM would be present, but our eyes would've a limit to its discerning power. For example, $a_0$ nm to $a_1$ nm would be one 'color'. So, we won't see all the wavelengths, but the rainbow would certainly be wider and more 'colorful'

HOWEVER, the nature of water is important. Water would absorb a lot of Infrared part, but not so much of say, the 'green' part. So, the width and which all wavelengths we would be able to see in this rainbow, would depend on the exact nature of behavior of water with different wavelengths of light.