Explain why the sine and logarithm of 1 meter are not well-defined.

Solutions

Expert Solution

If we look at the Taylor series expansion of the function Sin (x) is given by

This implies that x should be dimensionless, otherwise if x is a dimensionfull quantity, e.g. x = 1 meter, then, the above series would be

But, as we can't add two quantities of different dimensions ( similar to the fact that, apples add to apples and apples can't be added with oranges ),
So, 1 meter can't be added with 1 meter^3 and so, on.
This is why, Sine of 1 meter is not well defined.

And in the same way, the Taylor series expansion of the logarithm is given by

And so, x should be dimensionless, otherwise if x is a dimensionfull quantity, e.g. x = 1 meter, then, the above series would be

But, as we can't add two quantities of different dimensions ( similar to the fact that, apples add to apples and apples can't be added with oranges ),
So, 1 meter can't be added with 1 meter^2 and so, on.
This is why, logarithm of 1 meter is not well defined.

genius_generous answered 1 year ago

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