1047: "Approximations"

[rhomboidal]

Title Text: Two tips: 1) 8675309 is not just prime, it's a twin prime, and 2) if you ever find yourself raising log(anything)^e or taking the pi-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong.

This would make a cool cheat sheet. And an even cooler tattoo.

[bitwiseshiftleft]

Spoiler:

cos(\pi / 7) + cos(3\pi/7) + cos(5\pi/7)
= -\Re(\zeta_7+\zeta_7^2+\zeta_7^3)
= \frac12 - \frac12(1 + \zeta_7 + \zeta_7 ^2 + \zeta_7^3 + \zeta_7^4 + \zeta_7^5 + \zeta_7^6)
= \frac12

[somebody_else]

The error in the first Protip is pretty big though. If only the fractions were added instead of subtracted. I am going to use the radius of the earth one from now on - just to confuse anyone reading my work.

[Shidoshi]

The gas constant is expressed in J/gmol.K and not J/kgmol.K which would be the SI unit.

[Willl]

The stated accuracy of g is about an order of magnitude off - it's accurate to about 1 in 80,000, not 750,000. I don't know about the others, but I was surprised to learn that an international standard was actually defined to that precision, given that it varies by more than 1 in 200, hence I ended up checking it.

[JimsMaher]

Some of the more accurate, yet still elegant ones tempt the idyllic pseudoscientist in me.

G ≈ 1 over {e^(π - 1)^(π + 1)}
... consider {m^3} over {kg*s^2} when off by 1:25000 ... on a galactic scale is still significant. It's the difference between getting a speedy pick-me-up from passing near a black-hole, and winding up with a cyborg's vision of hell set to Disney-music of all things.

[bitwiseshiftleft]

Also, e^(pi * sqrt(163)) ~ 640320^3 + 744. But that one is famous. And was found by a method other than brute force.

[BlitzGirl]

Unfortunately, the only one of these I'm going to remember is the RENT Method.

Present some of the others with heart-wrenching vocals and perhaps I'd retain them.

[hemflit]

𝜋 seconds in a nanocentury, much neater 75^4 in a year.

[NeilUK]

The number of seconds in a year, to better than 1%, is

pi * 10^7

[MacGeb]

Pi seconds in a nanocentury, much neater 75^4 in a year.

My favourite as well. What puzzles me is that sometimes I even use it.

[scooterboo]

My favorite is miles * phi = kilometers
(accurate to .54%)
(phi is about 1.6180339887)

[ptorquemada]

Within about 0.5%, there are pi times ten million seconds in a year.

For added fun when trolling, throw in that Earth's orbit being an ellipse instead of a perfect circle is what throws it off from being exactly right.

[TheoGB]

I don't think I've ever seen accuracy used like that in this sort of context. Does someone want to explain what it means? I had a quick look at the seconds in a year but I't not really any clearer to me. In context of a fluid 1 part in 400 would make sense to me but I've been too long away from Uni to begin to remember this stuff.

[Smatchmo]

2^42.42

So beautiful. Makes me want to jump in the ol' Heart of Gold and go cruising.

Many Kudos, sir.

[ptorquemada]

The gas constant is expressed in J/gmol.K and not J/kgmol.K which would be the SI unit.

I've never seen the abbreviations "gmol" or "kgmol", but the SI units of the gas constant are Joules per mole Kelvin, and moles are defined as if they were based on grams instead of kg. Probably because the chemists threatened insurrection otherwise... grams are a much more convenient laboratory unit.

[JimsMaher]

I don't think I've ever seen accuracy used like that in this sort of context. Does someone want to explain what it means?

Approximations. Notice how everything is rounded off with lots of trailing zeroes.

Consider the variation that the given approximation is from the actual standard.
The size of discrepancy is called one part to the actual measure's Whatever-it-gets-rounded-updown-to.

Demonstrating the first item on the list:
One Light-Year (meters) ≈ 99^8 ... "Accurate to Within: One part in 40" Let's say 1:40 ...
99^8 m = 9 227 446 944 279 201 m
1 ly = 9 460 730 472 580 800 m

1 ly - 99^8 = 233 283 528 301 599 = d
d/ly ≈ 0.0246580... that's just less than 2.5% which can be restated concisely (that's the key here) as, "This approximation is accurate to within one part in Forty".

[Ignitus]

I don't think I've ever seen accuracy used like that in this sort of context. Does someone want to explain what it means? I had a quick look at the seconds in a year but I't not really any clearer to me. In context of a fluid 1 part in 400 would make sense to me but I've been too long away from Uni to begin to remember this stuff.

This took me a while, but it simply means that using this estimation you will be accurate with in one whole number if you don't go beyond this value given

Quick Example
40* 99^8 is a little over 39 light years not 40.

So if you look at the individual estimations they look fairly off since we are used to looking for accuracy up to the 3 or 4th decimal, but in practical terms if your asked does X meters equal a light year X / 99^8th will give a fairly accurate yes or no answer but after 40 light years you start to be off by 1 or 2

[Shidoshi]

The gas constant is expressed in J/gmol.K and not J/kgmol.K which would be the SI unit.

I've never seen the abbreviations "gmol" or "kgmol", but the SI units of the gas constant are Joules per mole Kelvin, and moles are defined as if they were based on grams instead of kg. Probably because the chemists threatened insurrection otherwise... grams are a much more convenient laboratory unit.

I've been recently through ChemEng and now they are starting to use "kilogram moles" and "gram moles", that way everything can be expressed in kg without much problem. Carbon12's molar mass is 12g/gmol or 12kg/kgmol (or even 12lb/lbmol if you swing that way). It's a somewhat new concept yeah, but it sure helps putting everything to SI and making less confusion.

[tetsujin]

Is that an Imperial gallon, a US liquid gallon, or one of the various deprecated or antiquated standards for a gallon?

[bjornart]

The stated accuracy of g is about an order of magnitude off - it's accurate to about 1 in 80,000, not 750,000. I don't know about the others, but I was surprised to learn that an international standard was actually defined to that precision, given that it varies by more than 1 in 200, hence I ended up checking it.

You forgot to divide by g.

[JimsMaher]

Let:
A = actual standard
B = approximation

Expressed as being accurate to within one part in
{ (A - B)/A } rounded up to nearest greater fraction whose reciprocal decimal form has no more than two significant digits.

Check, please.

[Seli]

The gas constant is expressed in J/gmol.K and not J/kgmol.K which would be the SI unit.

I've never seen the abbreviations "gmol" or "kgmol", but the SI units of the gas constant are Joules per mole Kelvin, and moles are defined as if they were based on grams instead of kg. Probably because the chemists threatened insurrection otherwise... grams are a much more convenient laboratory unit.

I've been recently through ChemEng and now they are starting to use "kilogram moles" and "gram moles", that way everything can be expressed in kg without much problem. Carbon12's molar mass is 12g/gmol or 12kg/kgmol (or even 12lb/lbmol if you swing that way). It's a somewhat new concept yeah, but it sure helps putting everything to SI and making less confusion.

That just looks, so wrong.

If one would insist on doing this kilomoles would make some sense and proper use of SI, but kilogram moles... sorry no. Weird engineers.

edit: phew, wiki suggests even engineers adhere to the rules at times and modern practice is using kilomole.

[Willl]

You forgot to divide by g.

How embarrassing. Right you are.

[Coderjoe]

according to Wolfram Alpha, the White House switchboard equation comes out to 0.212456141492...

To get ten places after the decimal, you have to round that nine, giving 2124561415, which is NOT the switchboard number.

ETA: Jenny's constant comes out to 867.53090198..., which is acceptable.

[mrob27]

(I got a tweet about this strip nearly instantly after it went online...)

Today's RIES date:

(((pi+1/2)^(1/7) * pi)^2 * pi)^2 = 2012.0425...

Robert Munafo - mrob.com/ries

[alextrabec]

Randall wins again. P.S. You are all nerds fact checking a web comic. Great Job, make your professors proud! Also, your mom plus me makes a "That what she said joke" - Enjoy

Love the comics and nerds, keep it up bitches!

[Army1987]

The number of seconds in a year, to better than 1%, is

pi * 10^7

Bitch, please. 10_7.5 seconds.

[hthall]

For anyone else still wondering about Jenny's constant:

http://www.youtube.com/watch?v=Dg_YueZ4fi8&t=42s

http://www.snopes.com/music/songs/8675309.asp

http://en.wikipedia.org/wiki/867-5309/Jenny

[peaty]

2^42.42

So beautiful. Makes me want to jump in the ol' Heart of Gold and go cruising.

Many Kudos, sir.

You mean "Much kudos". It's an abstract singular, not a plural.

[Red Hal]

You are all nerds fact checking a web comic.

... and you're a nerd commenting on the nerds, and I'm a nerd commenting on a nerd commenting on the nerds. Oh atheia, we're so meta it burns! Loving the "your mom" joke; it makes you look so edgy. Well done!

[scharb]

if you ever find yourself raising log(anything)^e or taking the pi-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong.

I'll say something has gone horribly wrong! YOU'RE DOING MATH

[jacksonliam91]

9.81 is much easier to remember for g.

[mrob27]

I am Robert Munafo, who made the RIES program. I accidentally re-posted the earlier comment after thinking it was rejected for having a link...

So to comply with the "don't repost, edit instead" rule, and to make this post useful, here are a couple more wacky formulae that didn't make xkcd this time around:

\sqrt{8+\sqrt{2+\sqrt{0+\sqrt{5}}}} = \pi

Start with any X, then calculate the following:

\sqrt{\sqrt{e^{6}/(x+1)}}

then use this as the new X, and repeat. It converges on: 3.1415926825119...

[jutzin]

i believe jenny's constant's last 3 digits are 209, not 902 (http://webcache.googleusercontent.com/search?q=cache:nVZnhecGolcJ:danstheman.com/Jenny.htm).

[jutzin]

... and how cool is Robert Munafo's RIES, by the way!

[Tuinkabouter]

Even now you're too much of a mathematician to dare to say \pi \approx \sqrt{10}. kudos.

[Dobblesworth]

One of the bottom six according to Wolfram is an approximation. The sqrt 5 one. Decimal approximations given by Wolfram for the two terms don't match it seems.
http://www.wolframalpha.com/input/?i=%2 ... 824-4pi%29
http://www.wolframalpha.com/input/?i=sqrt+5

[mistapotta]

My favorite one is π=355/113, just to irritate my students who insist π is irrational.

355/113 is 3.14159292, so much better than 22/7.

[RyanfaeScotland]

I notice Randall uses billions for the world population but millions for the US population. Why is he being dishonest?