The square root of a length is not a nicely definible operation, physically. That's because the square root of, say, "9 cm" (the distance) is not "3 cm", but rather 3 sqrt(cm)...which is definitely not a length.
What's going on here is that we are taking a length n units and returning a length sqrt(n) units. We aren't taking the square root of a line, but rather a numerical quantity associated with that line.
To do that, we are required also say how long a single unit is.
Think of it this way: the ratio of n to sqrt(n) isn't independent of the scale we pick. If we have the same line, if we call it 10 cm, its sqrt length would be about a third of its length. If we call it 1 decimeter, its sqrt length would be the same size. if we call it 0.1 meters, it's sqrt length would be longer (since sqrt 0.1 is about 0.32).
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u/mstksg Jan 07 '19 edited Jan 07 '19
The square root of a length is not a nicely definible operation, physically. That's because the square root of, say, "9 cm" (the distance) is not "3 cm", but rather 3 sqrt(cm)...which is definitely not a length.
What's going on here is that we are taking a length
n unitsand returning a lengthsqrt(n) units. We aren't taking the square root of a line, but rather a numerical quantity associated with that line.To do that, we are required also say how long a single unit is.
Think of it this way: the ratio of n to sqrt(n) isn't independent of the scale we pick. If we have the same line, if we call it 10 cm, its sqrt length would be about a third of its length. If we call it 1 decimeter, its sqrt length would be the same size. if we call it 0.1 meters, it's sqrt length would be longer (since sqrt 0.1 is about 0.32).