Difference between revisions of "2205: Types of Approximation"

Explain xkcd: It's 'cause you're dumb.
Jump to: navigation, search
(Added some additional context for the pi= 1 or 10 approximation, based on what if: Paint the earth.)
m (grammar)
Line 17: Line 17:
  
  
Ponytail offering to use 10 instead of 1 alludes to Randall's preferred style of solving [//en.wikipedia.org/wiki/Fermi_problem Fermi problems], as shown in [https://what-if.xkcd.com/84/ Paint the Earth]. He rounds numbers to the nearest order of magnitude (1, 10, 100, etc.) using on a base 10 logarithmic scale. Numbers between 0.316 and 3.16 are rounded to 1, between 3.16 and 31.6 are rounded to 10, and so on. Pi falls close to this cutoff point, and so by using this form of estimation it doesn't really matter to Ponytail whether pi is approximated to 1 or 10.
+
Ponytail offering to use 10 instead of 1 alludes to Randall's preferred style of solving [//en.wikipedia.org/wiki/Fermi_problem Fermi problems], as shown in [https://what-if.xkcd.com/84/ Paint the Earth]. He rounds numbers to the nearest order of magnitude (1, 10, 100, etc.) using a base 10 logarithmic scale. Numbers between 0.316 and 3.16 are rounded to 1, between 3.16 and 31.6 are rounded to 10, and so on. Pi falls close to this cutoff point, and so by using this form of estimation it doesn't really matter to Ponytail whether pi is approximated to 1 or 10.
  
 
==Transcript==
 
==Transcript==

Revision as of 21:17, 20 September 2019

Types of Approximation
It's not my fault I haven't had a chance to measure the curvature of this particular universe.
Title text: It's not my fault I haven't had a chance to measure the curvature of this particular universe.

Explanation

Ambox notice.png This explanation may be incomplete or incorrect: Created by an APPROXIMATOR. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.
If you can address this issue, please edit the page! Thanks.

In physics and engineering, problem solving typically requires approximations, as physical properties of the universe can be difficult to model. For example, in introductory physics classes, theories are introduced in frictionless environments.

In the comic, Cueball, the physicist, generally dealing with straight math, is introducing a problem with the assumption that the particular curve is a (perfectly) circular arc with a radius represented by R. Megan, the engineer, also assumes that the curve is similar to a circle, with a deviation factor of 1/1000.


The joke arises when Ponytail, the cosmologist, uses the ridiculous approximation of pi equal to 1. In actuality, pi is an irrational number, usually truncated to 3.14. Choosing the value of pi as 1, or 10, as later suggested, completely defeats the purpose of pi for describing a circle. This is a parody of the tendency of cosmology to change the units of measure such that important constants (such as the speed of light or the gravitational constant) are equal to 1, which highly simplifies the formulas without compromising the math. In this case, the number pi is a dimensionless factor, not a directly measured quantity, which means the math will not work.


Ponytail offering to use 10 instead of 1 alludes to Randall's preferred style of solving Fermi problems, as shown in Paint the Earth. He rounds numbers to the nearest order of magnitude (1, 10, 100, etc.) using a base 10 logarithmic scale. Numbers between 0.316 and 3.16 are rounded to 1, between 3.16 and 31.6 are rounded to 10, and so on. Pi falls close to this cutoff point, and so by using this form of estimation it doesn't really matter to Ponytail whether pi is approximated to 1 or 10.

Transcript

Ambox notice.png This transcript is incomplete. Please help editing it! Thanks.

[There are three panels, each labeled on top]

[Physicist Approximations]

Cueball: We'll assume the curve of this rail is a circular arc with radius R.

[Engineer Approximations]

Megan: Let's assume this curve deviates from a circle by no more than 1 part in 1,000.

[Cosmologist Approximations]

Ponytail: Assume pi is one.
Off-panel voice: Pretty sure it's bigger than that.
Ponytail: OK, we can make it ten. Whatever.


comment.png add a comment! ⋅ comment.png add a topic (use sparingly)! ⋅ Icons-mini-action refresh blue.gif refresh comments!

Discussion

The cosmologist is probably using Fermi's a la What-If 84: Paint the EarthOhFFS (talk) 20:34, 20 September 2019 (UTC)

In that What-If, the rounding formula for Fermi problem estimation is given as "Fermi(x) = round10(log10(x))". log10(pi) (Google search, shows calculator) is roughly .4971... so close enough that someone could do a "Fermi rounding" to either 1 or 10 and not really care one way or another. 162.158.142.118 21:19, 20 September 2019 (UTC)

As a physics Phd (though not working in astrophysics), approximating pi to 1 is not all that bad. Especially when the measurable quantities that go into the calculation usually have huge error bars.--172.68.59.120 21:03, 20 September 2019 (UTC)

Using natural units (setting c=hbar=1) is different from setting pi to 1. Using different units is always allowed and not an approximation. Setting pi to 1 on the other hand, is an approximation and is only justifiable if the other quantities in the calculation have huge uncertainty. --172.68.59.120 21:07, 20 September 2019 (UTC)