Why Tailoring Around 1 Makes More Sense

  • Thread starter nhrock3
  • Start date
In summary: So, in this case, the series would be: \sum_{k=1}^{\infty}\frac{(-1)^{k+1}(-Z-1)}{k}=\frac{(-1)^{k+1}(Z-1)}{k+1}=z-1
  • #1
nhrock3
415
0
2qtissi.jpg

why we do tailor around 1 and not 0
?
 
Last edited:
Physics news on Phys.org
  • #2
Hi nhrock3! :smile:

You can "Taylor" around any value where the function is sufficiently continuous and differentiable.

But it isn't at z = 0, so you can't "Taylor" there!

(and I expect they've chosen z = 1 because it's easiest to calculate the derivatives there! :wink:)
 
  • #3
so when x=y we have a problem
how transforming it into a taylor series solves this accuracy problem?
 
  • #4
Why do you think there is a problem when x = y? If x = y, log(x/y) = log(1) = 0.
 
  • #5
the question states how to get over loss of significance problem in here
f=ln x -ln y
x,y>0
and the solution says that we have a problem when x and y are close to each other
how transforming it into a taylor series solves this accuracy problem?
 
  • #6
I think that the idea here is that if x is close to y, then x/y is close to 1. You could have a situation where the actual value of x/y was 1.000000000001253, but because of limitations in computing precision, you might lose the part after the zeros.

By using a Taylor series to approximate log(x) - log(y) = log(z) [itex]\approx[/itex] z - 1, the part after the zeros above is now significant.
 
  • #7
so what if " x/y is close to 1"
"limitations in computing precision"
how does this limitation get solves by this method?
 
  • #8
Try it with a calculator. Enter .00000000000125, and then add 1. Many calculators will display an answer of 1 since they aren't able to maintain enough digits of precision to display the actual answer.

So if you have two numbers that are very different in relative size, (e.g. 1 vs. .00000000000125), adding them causes the loss of digits. If you can strip off the 1, though, there's no problem in storing or computing with the part to the right of all the zeros. In computers, floating point numbers are stored in a way that is similar to scientific notation. Instead of being stored as .00000000000125, it would be stored something like 1.25 X 10-12. Not exactly like that, since the base is 2, not 10, and there are some other differences.

number like 1.00000000000125, where one part is very large
 
  • #9
the question states how to get over loss of significance problem in here
f=ln x -ln y?

f=ln x -ln y=ln z
the tailor series for ln z is:
[tex] \sum_{k=1}^{\infty}\frac{(-1)^{k+1}(Z-1)}{k}[/tex]
but how this expression equals z-1 ?
 
  • #10
The summation doesn't "equal" z - 1. It's "approximately equal" to z - 1. What they have done is discard all terms of degree 2 or larger in the series.
 

What is the significance of tailoring around 1?

Tailoring around 1 means designing or adjusting a system or process to have a baseline or standard value of 1. This helps simplify calculations and comparisons, as well as make the system more intuitive and user-friendly.

Why does tailoring around 1 make more sense?

Tailoring around 1 makes more sense because it allows for easier understanding and interpretation of data. It also eliminates the need for complex conversions or adjustments, as everything is based on a standard value of 1.

How does tailoring around 1 benefit scientific research?

Tailoring around 1 benefits scientific research by providing a common baseline for comparisons and calculations. This helps ensure consistency and accuracy in data analysis, making it easier to draw meaningful conclusions.

Can tailoring around 1 be applied to all scientific disciplines?

Yes, tailoring around 1 can be applied to all scientific disciplines. It is a universal concept that can be used in various fields such as physics, chemistry, biology, and more.

Are there any drawbacks to tailoring around 1?

One potential drawback of tailoring around 1 is that it may not always be the most accurate or precise approach, as it simplifies data and may overlook important nuances. It is important to carefully consider the appropriateness of tailoring around 1 in each specific context.

Similar threads

  • STEM Career Guidance
4
Replies
120
Views
7K
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
19
Views
673
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
272
  • Calculus and Beyond Homework Help
Replies
5
Views
191
  • Classical Physics
Replies
4
Views
213
  • Calculus and Beyond Homework Help
Replies
3
Views
114
  • Calculus and Beyond Homework Help
Replies
9
Views
865
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
Back
Top