Question

In: Physics

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

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.


Related Solutions

Which of the following sets are not well defined? Explain.
Which of the following sets are not well defined? Explain. a. The set of wealthy school teachers b. The set of great books c. The set of natural numbers greater than 100 d. The set of subsets of \(\{1,2,3,4,5,6\}\) e. The set \(\{x \mid x \neq x\) and \(x \in N\}\)  
Define the term "property rights" and explain why the lack of well-defined and enforceable property rights...
Define the term "property rights" and explain why the lack of well-defined and enforceable property rights is detrimental to the smooth functioning of a market system.
The function Sine is defined as: where x is an angle in radians Write a Matlab...
The function Sine is defined as: where x is an angle in radians Write a Matlab script program to compute the sinus value of any angle as the following: - The program should run always until the user enters - 1 to exit from the program. - A sk user to enter: Number of elements (N) that should be included in the sum - C heck that N is a positive integer . [ Hint: use ( round (N) -...
The solution for the finite potential well is a combination of sine and cosine functions a...
The solution for the finite potential well is a combination of sine and cosine functions a combination of exponential functions a combination of exponential functions and sine and/or cosine functions a third degree polynomial
Explain the difference between a set that is well defined and one that is not. Give...
Explain the difference between a set that is well defined and one that is not. Give an example of a well-defined set. Name and describe your well-defined set using roster form and set-builder notation. Give an example of at least 1 subset. NO HANDWRITING PLEASE.
Why is it important for nurses to have a well-defined, straight-forward, unbiased self-concept?
Why is it important for nurses to have a well-defined, straight-forward, unbiased self-concept?
why a well organized "Governance", defined "Role" and adequate "Funding" is so important for a successful...
why a well organized "Governance", defined "Role" and adequate "Funding" is so important for a successful KM system.
Is it more important to speak well, or to listen well? Explain why, and include at...
Is it more important to speak well, or to listen well? Explain why, and include at two strategies or examples to support your response
Explain the four conditions that must hold for the existence of a well-defined property rights system,...
Explain the four conditions that must hold for the existence of a well-defined property rights system, and address why all four matter. Why do economists promote the idea of creating markets for environmental protection? Explain how Pigovian taxes work to control pollution? How do tradable pollution permits work in theory?
Draw and explain how a absolute gravity meter and relative gravity meter works?
Draw and explain how a absolute gravity meter and relative gravity meter works?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT