1047: "Approximations"

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rhomboidal
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1047: "Approximations"

Postby rhomboidal » Wed Apr 25, 2012 6:31 am UTC

Image

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.

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Re: 1047: "Approximations"

Postby bitwiseshiftleft » Wed Apr 25, 2012 6:41 am UTC

Spoiler:
[math]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[/math]

somebody_else
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Re: 1047: "Approximations"

Postby somebody_else » Wed Apr 25, 2012 6:44 am UTC

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.

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Re: 1047: "Approximations"

Postby Shidoshi » Wed Apr 25, 2012 6:52 am UTC

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

Willl
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Re: 1047: "Approximations"

Postby Willl » Wed Apr 25, 2012 6:53 am UTC

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.

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Re: 1047: "Approximations"

Postby JimsMaher » Wed Apr 25, 2012 7:00 am UTC

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.

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Re: 1047: "Approximations"

Postby bitwiseshiftleft » Wed Apr 25, 2012 7:04 am UTC

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

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Re: 1047: "Approximations"

Postby BlitzGirl » Wed Apr 25, 2012 7:07 am UTC

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.
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Re: 1047: "Approximations"

Postby hemflit » Wed Apr 25, 2012 7:10 am UTC

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

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Re: 1047: "Approximations"

Postby NeilUK » Wed Apr 25, 2012 7:10 am UTC

The number of seconds in a year, to better than 1%, is [math]pi * 10^7[/math]

MacGeb
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Re: 1047: "Approximations"

Postby MacGeb » Wed Apr 25, 2012 7:13 am UTC

hemflit wrote: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
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Re: 1047: "Approximations"

Postby scooterboo » Wed Apr 25, 2012 7:13 am UTC

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

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Re: 1047: "Approximations"

Postby ptorquemada » Wed Apr 25, 2012 7:26 am UTC

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
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Re: 1047: "Approximations"

Postby TheoGB » Wed Apr 25, 2012 7:29 am UTC

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.

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Re: 1047: "Approximations"

Postby Smatchmo » Wed Apr 25, 2012 7:38 am UTC

2^42.42

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

Many Kudos, sir.

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Re: 1047: "Approximations"

Postby ptorquemada » Wed Apr 25, 2012 7:40 am UTC

Shidoshi wrote: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.

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Re: 1047: "Approximations"

Postby JimsMaher » Wed Apr 25, 2012 7:48 am UTC

TheoGB wrote: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".
Last edited by JimsMaher on Wed Apr 25, 2012 7:58 am UTC, edited 1 time in total.

Ignitus
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Re: 1047: "Approximations"

Postby Ignitus » Wed Apr 25, 2012 7:52 am UTC

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

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Re: 1047: "Approximations"

Postby Shidoshi » Wed Apr 25, 2012 7:57 am UTC

ptorquemada wrote:
Shidoshi wrote: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.

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Re: 1047: "Approximations"

Postby tetsujin » Wed Apr 25, 2012 8:00 am UTC

Is that an Imperial gallon, a US liquid gallon, or one of the various deprecated or antiquated standards for a gallon?
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bjornart
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Re: 1047: "Approximations"

Postby bjornart » Wed Apr 25, 2012 8:28 am UTC

Willl wrote: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
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Re: 1047: "Approximations"

Postby JimsMaher » Wed Apr 25, 2012 8:35 am UTC

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
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Re: 1047: "Approximations"

Postby Seli » Wed Apr 25, 2012 8:37 am UTC

Shidoshi wrote:
ptorquemada wrote:
Shidoshi wrote: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.

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Re: 1047: "Approximations"

Postby Willl » Wed Apr 25, 2012 8:44 am UTC

bjornart wrote:You forgot to divide by g.
How embarrassing. Right you are.

Coderjoe
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Re: 1047: "Approximations"

Postby Coderjoe » Wed Apr 25, 2012 8:52 am UTC

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.
Last edited by Coderjoe on Wed Apr 25, 2012 9:19 am UTC, edited 1 time in total.

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Re: 1047: "Approximations"

Postby mrob27 » Wed Apr 25, 2012 8:54 am UTC

(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
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Re: 1047: "Approximations"

Postby alextrabec » Wed Apr 25, 2012 9:15 am UTC

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
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Re: 1047: "Approximations"

Postby Army1987 » Wed Apr 25, 2012 9:23 am UTC

NeilUK wrote:The number of seconds in a year, to better than 1%, is [math]pi * 10^7[/math]

Bitch, please. 107.5 seconds.

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Re: 1047: "Approximations"

Postby hthall » Wed Apr 25, 2012 9:29 am UTC

Look at me, still talking when there's Science to do.

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Re: 1047: "Approximations"

Postby peaty » Wed Apr 25, 2012 9:29 am UTC

Smatchmo wrote: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.

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Re: 1047: "Approximations"

Postby Red Hal » Wed Apr 25, 2012 9:31 am UTC

alextrabec wrote: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!
Lost Greatest Silent Baby X Y Z. "There is no one who loves pain itself, who seeks after it and wants to have it, simply because it is pain..."

scharb
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Re: 1047: "Approximations"

Postby scharb » Wed Apr 25, 2012 9:43 am UTC

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
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Re: 1047: "Approximations"

Postby jacksonliam91 » Wed Apr 25, 2012 9:52 am UTC

9.81 is much easier to remember for g.

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Re: 1047: "Approximations"

Postby mrob27 » Wed Apr 25, 2012 9:56 am UTC

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:

[math]\sqrt{8+\sqrt{2+\sqrt{0+\sqrt{5}}}} = \pi[/math]

Start with any X, then calculate the following:

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

then use this as the new X, and repeat. It converges on: 3.1415926825119...
Last edited by mrob27 on Wed Apr 25, 2012 3:10 pm UTC, edited 2 times in total.

jutzin
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Re: 1047: "Approximations"

Postby jutzin » Wed Apr 25, 2012 10:00 am UTC

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
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Re: 1047: "Approximations"

Postby jutzin » Wed Apr 25, 2012 10:02 am UTC

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

Tuinkabouter
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Re: 1047: "Approximations"

Postby Tuinkabouter » Wed Apr 25, 2012 10:11 am UTC

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

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Re: 1047: "Approximations"

Postby Dobblesworth » Wed Apr 25, 2012 10:22 am UTC

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
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Re: 1047: "Approximations"

Postby mistapotta » Wed Apr 25, 2012 10:27 am UTC

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.

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RyanfaeScotland
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Re: 1047: "Approximations"

Postby RyanfaeScotland » Wed Apr 25, 2012 10:34 am UTC

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


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