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The metre is defined as the distance light travels in a certain length of time (in a vacuum). Does this assume an infinitely short wavelength (measured in metres!) or is there some sort of quantum uncertainty involved? I'd imagine our definition of the metre should actually have "...plus or minus half the Planck length" after it?
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No, it doesn't assume any wavelength; it doesn't need to. The basic units are all defined assuming a perfect world with no uncertainty; where physical constants need to be involved in this, those are defined to be exact as well. Thus, we get c being exactly 299792458 m/s, and μ0 being exactly 4π*10^-7 N/A2, etc.
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It doesn't have to be defined for a magical world without uncertainty. It could simply be defined as the limit as uncertainty approaches zero.
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Our measurements are much less accurate than the Planck limit. As long as your definition has an error margin smaller than our capacity to measure it, you don't have to care about that error margin.
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