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