1047: "Approximations"

[mrob27]

reminds me of my dad's easy way to do simple multiplications:

7*7? Just take 5*10 and subtract 1.

Curiously, the fact that 2 * 7^2 = 10^2 - 2 is connected to the digit pattern of sqrt(2):

\sqrt{2} = 1.4~14~21~35

Have you ever noticed that the pairs of digits are 2, 2, 3, and 5 times 7? and that the next two digits (62) are almost 9 times 7? The same pattern happens in sqrt(62):

\sqrt{62} = 007.874~007874~011811~019685...

Six digits per group this time, but the same 2, 2, 3, 5 ratio. This one is connected to the fact that 62 * 127^2 = 999998. Both roots get their digit patterns from the Taylor series expansion of sqrt(1/(1-2x)) :

\sqrt{1/(1-2x)} = 1 + x + 3x^{2}/2 + 5x^{3}/2 + 35x^{4}/8 + 63x^{5}/8 + ...

I have written about this on my numbers webpage, but I can't post links yet so you'll have to Google or Bing it.

- Robert Munafo (I do love my numbers)

[mmrevans]

This would make an excellent poster. I'd buy it in a heartbeat. They're just so elegant and cool!

I would as well - I'd hang it in my classroom, and try to find some fun way to incorporate it into my lessons. I'm sure the kids would get a kick out of it.

I think there should be a poster pack that xkcd sells for every beginning science teacher of all his best, including this one. I buy them in bulk and hand them out as door prizes at our Physics meeting; always the first to go!

[keithl]

There is a phone number in New Jersey whose digits (including 1 for the area code) are the starting digits of (9!)! , that is, the factorial of (9 factorial) or 362880! . I invented a puzzler for Clive Maxfield ( http://tinyurl.com/6qc44jv ) involving this calculation. There is a brute force way using calc() and a clever way using a spreadsheet function to calculate this, left as an exercise to the reader.

There are probably ratios of multiplied and nested factorials for expressing all constants to arbitrary precision, though finding them may require factorial amounts of time and very expensive calculations. But if the result can deprive your math/science professor of a night's sleep, it is worth it.

[patzer]

What is jenny's constant? I've tried google, got no results.
I work that approximation out at 867.530902

[Maurits]

The approximation for G is listed as "accurate to one part in 25000." But the actual value of G is only known to one part in 8000.

[anchises868]

What is jenny's constant? I've tried google, got no results.
I work that approximation out at 867.530902

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

[bmonk]

I've long been fond of continued fractions, which also give good approximations, especially just before a large denominator. Which is too bad for the approximation of φ

[Capt.Awesome]

AP Test anyone? I plan on using as many of these as I can think of on the AP Enviromental, Chemistry and Physics tests this year. Hopefully my graders are fans of Randal!

[dga]

Here's one to 1 part in 1400.
300,000,000 m/s
Ratio of space to time (alternately known as 'the speed of light on a vacuum')

[|x-x0|<e]

Light-year:
99^8m?
Usually I use 10^{16}m, which is a lot easier for mental calculations...

[Nyerguds]

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.

I was looking for that one, but it's not in there

[Eternal Density]

My initial reaction: what units is the White House switchboard being measured in?
It took me reading the comments to realise it was just a phone number

[JimsMaher]

The approximation for G is listed as "accurate to one part in 25000." But the actual value of G is only known to one part in 8000.

CODATA Value: Newtonian constant of gravitation
"Relative standard uncertainty = 1.2 x 10^-4"
Closer to ... ±1:8300
Compared to ... within One part in { 6.67384x10^-11 over { (6.67384x10^-11)-(1 over {e^(pi-1)^(pi+1)}) } } ... 1:29000
Comparison of two "known"'s is a bit different than comparison of a "known" to an "unknown".

Anyway, that implies the approximate value is closer to the standard value, than the standard value is expected to be to actual value.
Which means the approximation is potentially actual value, though measurably less likely than the standard.
Of course, these numbers are based on man-made units of distance, mass, and time ...

[da Doctah]

Proof that the metric system is just plain foolish:

Diameter of the sun in miles = number of seconds in a day × 10.

[da Doctah]

reminds me of my dad's easy way to do simple multiplications:

7*7? Just take 5*10 and subtract 1.

Simplifying fractions made easy:

Say you have to simplify 16/64. Just cancel the sixes.

Wait, you say! This doesn't work in general, does it?

Sure it does. Try it with 19/95. See?

If that's not enough to convince you, try 26/65.

Later we'll learn how to conjugate Italian verbs: ami (you love) + amo (I love) = amiamo (we love).

What could be simpler?

[Coderjoe]

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).

That guy was calling the number (867-5309) in every area code. 209 being one where someone was playing with the song. If you wanted to use 209 in the constant, it would be at the left, before the 8.

The actual outcome of Randall's approximation has no end, due to the use of \pi. The result, out to 22 places is 867.5309019854097558275225 (and actually continues with 4961.. instead of that 5).

[ttk]

Yay! Thank you for fixing the sqrt(2) entry, Randall! All is right in the world now.

[loberto]

More accurate (exact I think) for feet in one meter: 1250/381. 1250 feet equals 381 meters, and is the exact height of the Empire State Building.

[mysoxarewhite]

1~m/s = \sqrt 5~mi/hr

to a factor of about 1/2500.

[mnkyman663]

Come on Randall, pi * 10^7 = 31,419,265 not 3,141,592

[Qwert]

Another fun one:

1 AU ~= 499 light-seconds, accurate to 1 in a million.

Or more amusingly coincidental:

1AU ~= 500 ls to within 1 in 500

[atypicaltexan]

One I use a lot (which must make it pretty well-known):

The golden ratio approximates the number of kilometers in a mile (to within 1 part in 185).

[kram2301]

The first post on this one should be edited to include the unique Header text:

Lots of emails mention the physicist favorite, 1 year = pi x 107 seconds.
75^4 is a hair more accurate, but it's hard to top 3,141,592's elegance.

[Biophysiker]

Because he said not to:

\sqrt[\pi]{e} = 1.3748

\frac{11}{8} = 1.375

Accurate to 1 part in 1000

[Florent]

There's a tippo in the approximation of \sqrt{2}, a - instead of a +:

\sqrt{2} = 3/5 + \pi/(7-\pi)

Leading to a less than 0.0001% error.

[phi16180339887]

c = 1 yule gravity Accurate to 1 in 250
yule is Dec 21. actual time to reach the speed of light at constant 1g acceleration Dec 19, 7:45:23 pm

[Popup]

I quite like the approximation from Monty Pythons 'The Galaxy Song', where the speed of light is said to be

twelve million miles a minute

. Admittedly it's about 7% off as well as in non-SI units, but I think that the alliteration makes up for it.

[Beruang]

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

But 22/7 is much easier to remember, and is accurate to 1.264 parts in 1,000 -- better than many of the approximations Russell lists.

[neoliminal]

I found a really cool shortcut for pi.

Spoiler:

http://www.piday.org/million.php

[Dr. Tom]

Aw. My favorite isn't there:

φ ≈ -2sin(666°)

The golden ratio is the doubly negative sine of the beast.

Don't know exactly how accurate it is (or even if it's exact); a cursory calculation or two and comparison shows it's at least 8 digits accurate.

Yeah, that's not an approximation, it's exact. http://oeis.org/A019827

[Erzengel]

Aw. My favorite isn't there:

φ ≈ -2sin(666°)

The golden ratio is the doubly negative sine of the beast.

Don't know exactly how accurate it is (or even if it's exact); a cursory calculation or two and comparison shows it's at least 8 digits accurate.

It's exact.

\sin(666^\circ) = \sin(306^\circ) = \frac{1}{4} (-1 - \sqrt{5})

Multiply that by -2 and you get

-2\sin(666^\circ) = -2\sin(306^\circ) = \frac{1}{2}(1+\sqrt{5}) = \phi

[Maurits]

The approximation for G is listed as "accurate to one part in 25000." But the actual value of G is only known to one part in 8000.

CODATA Value: Newtonian constant of gravitation
"Relative standard uncertainty = 1.2 x 10^-4"
Closer to ... ±1:8300
Compared to ... within One part in { 6.67384x10^-11 over { (6.67384x10^-11)-(1 over {e^(pi-1)^(pi+1)}) } } ... 1:29000
Comparison of two "known"'s is a bit different than comparison of a "known" to an "unknown".

Anyway, that implies the approximate value is closer to the standard value, than the standard value is expected to be to actual value.
Which means the approximation is potentially actual value, though measurably less likely than the standard.
Of course, these numbers are based on man-made units of distance, mass, and time ...

Not sure I follow your logic.

Units are m^3/(kg s^2)
Measured value of G = 6.67384(80)e-11 = 6.67384e-11 +/- 8e-15 = (6.67304e-11 to 6.67464e-11)
XKCD's approximate value of G = e^-((pi - 1)^(pi + 1))
To a high degree of accuracy:

Code: Select all

pi = 3.1415926535897932384626433832795
pi + 1 = 4.1415926535897932384626433832795
e = 2.7182818284590452353602874713527
pi - 1 = 2.1415926535897932384626433832795
(pi - 1)^(pi + 1) = 23.43027491992439
e^(-(pi - 1)^(pi + 1)) = 6.6736110685055e-11


This easily falls within the range which our measurements indicate as probable for the actual value of G, which exists but is currently unknown.

[NeatNit]

This hasn't been mentioned yet

(if you still don't see it, look under the header)

see it for yourself http://xkcd.com/1047/

[abarker]

The original 1791 definition of a meter defines one meter to be 1/10^7 the distance from equator to north pole. So the radius of the earth is R = 2*10^7 / pi meters. That comes out to 6366 km, which is within 0.07% of any other way other measure of the mean Earth radius.

[mrob27]

There is a phone number in New Jersey whose digits (including 1 for the area code) are the starting digits of (9!)! , that is, the factorial of (9 factorial) or 362880! . I invented a puzzler for Clive Maxfield ( he gave a URL here ) involving this calculation. [...]

Using my Hypercalc Perl script (you can Google that), I'm getting 1-609-714-4004. It's cool that the first digit comes out to a 1. Also cool: the next 4 digits are 1001, which could be the person's extension. (My factorial and Gamma functions only give about 30 digits of accuracy though)

(9!)! = 1.60971440041001262110344*10^{1859933}

- Robert Munafo

[Giallo]

If you want something nice, try computing:

\sqrt[3]{1+\sqrt{\frac{28}{27}}} + \sqrt[3]{1 - \sqrt{\frac{28}{27}}}

[Edrees]

I don't get it. How are any of these useful for approximations? Half of them are as hard to calculate or memorize as the actual equations or constants, if not harder. For example I may not know how many seconds in a year, but i can easily calculate it within 10 seconds - 60 x 60 x 60 x 24 x 365...but you couldn't pay me a hundred dollars to remember 75^4 a year from now without putting even more effort in. As for the 2nd point,yYou aren't a very good troll if it takes you more effort to do the trolling than it does for the trollee to calculate the answer. This is one of the worst comics I've seen on xkcd and I'm usually easy to please.

[Seraph]

The first post on this one should be edited to include the unique Header text:

Lots of emails mention the physicist favorite, 1 year = pi x 107 seconds.
75^4 is a hair more accurate, but it's hard to top 3,141,592's elegance.

The header on the comic page bothers me.
First a trivial math error: 3,141,592 != pi*10⁷
Second a formatting issue: Why write 75^4. He clearly knows how to make it look like 75⁴ because he wrote pi x 10⁷.

[Toam]

The fundamental charge one is wrong (and I'm surprised no one has mentioned it?).

3/(14*π^π^π) = 2.6566x10^-6 C

So I guessed maybe a typo and tried:

3/((14π)^π^π) = 1.8139x10^-16 C

Which is closer in comparison, but still no where near.

[Seraph]

Wolfram Alpha disagrees with you:
http://www.wolframalpha.com/input/?i=3% ... pi%5Epi%29