1047: "Approximations"

[uwcsDemo]

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.

You're evaluating 3/(14 (\pi^\pi)^\pi) = 3/(14 \pi^{\pi^2}), not 3/(14 \pi^{\pi^\pi}).

[houseofadams]

For all the chemists out there, Avagadro's Constant is about 25!/26 to 1 part in 106.

[JimsMaher]

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.

We're in agreement then.

[misterajc]

My favorite calculator trick is
Start with the date of the Battle of Hastings (that's 1066 for non Brits)
Divide by 11
Add 1/2
take the square root twice
And you get pi, accurate to about one part in a billion.

As a formula, pi^4 = 2143/22

[ZD3000]

Pretty cool comic. But now I'm desperately longing for a thing I didn't even know I needed before: RIES. However, I'm a total noob and I'm not able to compile the code provided on Rob Munafo's page. (Countless error messages …) The page reads: "ries was first created for Linux1, but is is easily ported to Mac OS X, and nearly any OS with a C compiler." Any OS …
Is there any chance I can get a compiled version that I can run on Windows?

[nickovs]

While much less accurate than pi*10^7 (at only about 1 in 20), the length of a year is pretty close to 2^25 seconds. A handy rule of thumb when you're estimating running times for brute-force searches...

[demerson3]

to convert between miles-per-gallon (the US standard) and liters-per-100km (European):

235/x = y

...works both ways...

[JimmyTMalice]

Has anyone mentioned 22/7 as an approximation for pi? It's good for rough calculations, and it's within 0.002 of the actual value.

[aljohnso]

I'm not sure about this -

My first impression was "the most flagrant example of needing to get a life that I've seen in forever"... sorry Randall.

Then I read the comments, and thought that perhaps getting such nerd hormones flowing had some virtue in itself.

But then again maybe not.

Getting to something with content - if you can make a set of "approximations" that is dense to 1 in 1000, say (that is, if you can create cute expressions for values up to y such that the gap between expressions are less than y/1000), is there any content to the statement that you have found such an "approximation"?

But then again, the cancer stuff (particularly the tattoo thing) make me want to forgive all.

Be relatively happy.

(pick a number at random - congratulations! you are now in a state of statistical sin)

[Widmerpool]

A propos of nothing...the entropy of the Secret Santa distribution is log(\pi +\frac{6}{11})to one part in 80,000

[mrob27]

Pretty cool comic. But now I'm desperately longing for a thing I didn't even know I needed before: RIES. However, I'm a total noob and I'm not able to compile the code provided on Rob Munafo's page. (Countless error messages …) The page reads: "ries was first created for Linux1, but is is easily ported to Mac OS X, and nearly any OS with a C compiler." Any OS …
Is there any chance I can get a compiled version that I can run on Windows?

There is now an online lookup service on the RIES web page. Type in your number and hit the Submit button. I coded it up this afternoon, and yes, /I used regular expressions/.

An earlier reply in this topic gave a link, or type in mrob.com/ries

To address the Windows situation more directly: contact me (contact links on the bottom of my webpage) and we can look at it. RIES only uses basic libraries (malloc, printf, ...) so there probably isn't much needed to get it working.

[Markus__1]

Pretty cool comic. But now I'm desperately longing for a thing I didn't even know I needed before: RIES. However, I'm a total noob and I'm not able to compile the code provided on Rob Munafo's page. (Countless error messages …) The page reads: "ries was first created for Linux1, but is is easily ported to Mac OS X, and nearly any OS with a C compiler." Any OS …
Is there any chance I can get a compiled version that I can run on Windows?

Since you've come as far as "countless error messages", your noobness can't be that bad.
Solution possibilities in order of hardness:
* Install cygwin + gcc, compile it out of the box
* Install mingw, code asprintf yourself (allocate a buffer and sprintf the string concatenation into it)
* Visual studio is hardest, because it's too dumb to grok C99-style variable declarations in the middle of code, and the error message (C2143) does not say that this is the problem. It also doesn't have asprintf.

PS: I've sent patches for all of that to Mr. Munafo.

[Bill Gosper]

"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."
See http://www.tweedledum.com/rwg/idents.htm
--rwg

[klausok]

When I read strip 688, I thought that there was no way it could any nerdier. How wrong I was!

In other words, I really like this one.

[mrob27]

Since you've come as far as "countless error messages", your noobness can't be that bad.
Solution possibilities in order of hardness:
* Install cygwin + gcc, compile it out of the box
* Install mingw, code asprintf yourself (allocate a buffer and sprintf the string concatenation into it)
* Visual studio is hardest, because it's too dumb to grok C99-style variable declarations in the middle of code, and the error message (C2143) does not say that this is the problem. It also doesn't have asprintf.

PS: I've sent patches for all of that to Mr. Munafo.

Thank you Markus__1 for your patches. I have verified that they work on my system at least and they are now part of the source code available at mrob.com/ries (see "RIES Source Code" link). Compatibility with lots of systems is a primary goal so it's great to get rid of that asprintf.

[mrob27]

"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."
See http://www.tweedledum.com/rwg/idents.htm
--rwg

Yeah, I didn't want to challenge Randall directly on this one, (in fact he hasn't answered my initial email about RIES so I think he may be too busy), but...

When RWG and I were working on RIES earlier this year, this issue of wacky expressions like "the pi-th root of 4" came up, and we concluded (along with others in an email discussion) that RIES was more useful if it includes these expressions in its searchspace.

As a long-term TODO item, I hope to add a facility for users to specify patterns that get matched (quasi-regexp-like) against subexpressions in their internal FORTH format. Then such things as a complexity score bias can be attached to matched patterns. The difficulty in implementation comes from the fact that this would totally destroy the pruning efficiency, and thus the performance of the overall RIES alrogithm.

[phenomist]

If you want something nice, try computing:

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

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

1+\sqrt{\frac{28}{27}} +1 - \sqrt{\frac{28}{27}} + 3(\sqrt[3]{1+\sqrt{\frac{28}{27}}} + \sqrt[3]{1 - \sqrt{\frac{28}{27}}})(\sqrt[3]{1+\sqrt{\frac{28}{27}}}\cdot \sqrt[3]{1 - \sqrt{\frac{28}{27}}}) = x^3

2 + 3x(\sqrt[3]{1 - \frac{28}{27}}) = x^3

2 + 3x(-\frac{1}{3}) = x^3

2 - x = x^3

x^3 + x - 2 = 0

(x-1)(x^2+x+2) = 0

Since x^2+x+2 is a quadratic without real roots, x = 1 is the answer.

[drachefly]

For all the chemists out there, Avagadro's Constant is about 25!/26 to 1 part in 106.

I prefer (4!)! It's not as accurate, but much more compact.

[eran_rathan]

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.

39.37 cm/12 in (for US. Survey feet, the only right kind of feet)

0.3048m = 1 international foot (the wrong foot)


-says the surveyor

[J Thomas]

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.

39.37 cm/12 in (for US. Survey feet, the only right kind of feet)

0.3048m = 1 international foot (the wrong foot)


-says the surveyor

3 meters is 10 feet. Off by 1.6%. It isn't cute but it's a far easier approximation than 5/e'th root of pi

[kafkadog]

39.37 cm/12 in (for US. Survey feet, the only right kind of feet)

0.3048m = 1 international foot (the wrong foot)


-says the surveyor

Your conversion implies that 1 US survey foot = 0.3937 m, which is MUCH larger than 1 international foot = 0.3048 m. But the US survey foot is actually only 2 parts per million larger than the international foot. See...

http://www.ngs.noaa.gov/faq.shtml#Feet

I think you meant 1 m = 39.37 in (for the US survey foot, not for the international foot).

[Giallo]

Spoiler:

If you want something nice, try computing:

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

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

1+\sqrt{\frac{28}{27}} +1 - \sqrt{\frac{28}{27}} + 3(\sqrt[3]{1+\sqrt{\frac{28}{27}}} + \sqrt[3]{1 - \sqrt{\frac{28}{27}}})(\sqrt[3]{1+\sqrt{\frac{28}{27}}}\cdot \sqrt[3]{1 - \sqrt{\frac{28}{27}}}) = x^3

2 + 3x(\sqrt[3]{1 - \frac{28}{27}}) = x^3

2 + 3x(-\frac{1}{3}) = x^3

2 - x = x^3

x^3 + x - 2 = 0

(x-1)(x^2+x+2) = 0

Since x^2+x+2 is a quadratic without real roots, x = 1 is the answer.

Exactly
It has been found by del Ferro, if I remember correctly, using the formula for the roots of cubics.

[Not R]

For maximum confusion, I would place the Rent method at 1 in 366, making the other one accurate to within 1 in 300. Also I always thought Jenny's constant was -4442, making -4444 accurate to within 1 in 2221 (sadly not 2222).

[SirH3nry]

Fun fact; when entering "cos(pi/7) + cos(3pi/7) + cos(5pi/7) = 1/2" into WolframAlpha and viewing the maximum number of digits. If you alternate between "fewer digits" and "more digits," it switches between 4.9 repeating and 5.0 repeating randomly.
Exhibit A: http://i.imgur.com/gpAzK.png

[pmackey]

My favorite approximation is probably this one:
π = ln(-1)/i
I hear it's accurate to 1 in ∞!



Patrick

[chewbakken]

Here's a whole bunch more (mostly numerical, not many constants/unit conversions):

http://en.wikipedia.org/wiki/Mathematical_coincidence

[Giallo]

My favorite approximation is probably this one:
π = ln(-1)/i
I hear it's accurate to 1 in ∞!



Patrick

Only if you define well your branch of log... If you use the "classical" ln, the ln(-1) is not even defined. You can also define branches of log where ln(-1) = π(2n + 1) with n any integer.

[flipwook]

A simple one...

\pi^2 \approx 10

(one part in 76)

[ConMan]

My preferred approximation is a bit obscure - in the suburb where I used to live, phone numbers were 8 digits long and (some) started with 9869. So 9869 6044 is, presumably, someone's phone number in that suburb. And 9.8696044 is, to a pretty good approximation, pi^2.

[Greywolf]

Apologies if this one was suggested already, but:

355/113 = π (3ppm)

[fmotoki]

I didn't see anyone mentioning this... g (acceleration of gravity on earth) is approximately pi^2 (in m/s^2)

[PM 2Ring]

Spoiler:

If you want something nice, try computing:

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

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

1+\sqrt{\frac{28}{27}} +1 - \sqrt{\frac{28}{27}} + 3(\sqrt[3]{1+\sqrt{\frac{28}{27}}} + \sqrt[3]{1 - \sqrt{\frac{28}{27}}})(\sqrt[3]{1+\sqrt{\frac{28}{27}}}\cdot \sqrt[3]{1 - \sqrt{\frac{28}{27}}}) = x^3

2 + 3x(\sqrt[3]{1 - \frac{28}{27}}) = x^3

2 + 3x(-\frac{1}{3}) = x^3

2 - x = x^3

x^3 + x - 2 = 0

(x-1)(x^2+x+2) = 0

Since x^2+x+2 is a quadratic without real roots, x = 1 is the answer.

Exactly
It has been found by del Ferro, if I remember correctly, using the formula for the roots of cubics.

Indeed. I guess the expression you posted is of historical significance, but why not give some prettier cubic root expressions that avoid the use of fractions, eg,

(100+sqrt(7803))^(1/3) + (100-sqrt(7803))^(1/3) = 8
and
(9+sqrt(80))^(1/3) + (9-sqrt(80))^(1/3) = 3

[I've used that ugly format so that the expressions can be easily checked in the Google Calculator.]

[sswam]

e = \sqrt[pi]{ \sqrt[i]{-1} }

It's not outside the realm of sensible math.

[chewbakken]

Another one: the temperature of the universe (or at least it's CMB radiation) is e K, accurate to almost 1 in 400.

[PM 2Ring]

On Jenny's Constant:

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).

I guess something weird happened with your copy & paste operation, since it appears that you've lost a couple of digits. I get
867.5309019816854097558275224961431838440297231328116937715658956
1760603903591897835403126064595054279713689805052485005005... etc

[bmonk]

Another approximation: the apparent size of the Sun (from the Earth) equals the apparent size of the Moon (again, from the Earth).