1047: Approximations

2014-01-25T04:21:17Z

173.245.54.78: /* Explanation */ The White House switchboard under Bush, and now under Obama, is (202) 456-1414. It may round to .2024561415, but it's an approximation, and truncation leaves the actual phone number.

{{comic
| number = 1047
| date = April 25, 2012
| title = Approximations
| image = approximations.png
| titletext = 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.
}}

==Explanation==
{{incomplete}}
This comic lists some approximations for numbers, most of them mathematical and physical constants. All of them work astonishingly well. There are reoccurring math jokes along the lines of, “3/5 + π/(7 – π) – √2 = 0, but your calculator is probably not good enough to compute this correctly”, which are mainly used to troll geeks.

Furthermore, there are some useful approximations (which were even more useful in times before calculators) such as “pi is approximately equal to 22/7”.

[[Randall]] makes fun of both of these, using rather strange approximations (honestly: you may handle 22/7, but who can calculate in a sensible way with 99^8, let alone 30^(pi^e)?) to calculate some constants that are easy enough to handle in the decimal system, and stating such “slightly wrong” trick equations, one of which ''is'' actually correct (which may astonish only those who are not familiar with cosines).

There are a few cultural references in this comic:

* 998 and 698 are sexual references.
* “Rent Method” refers to the song “Seasons of Love” from the musical “{{w|Rent (musical)|Rent}}.” The song asks, “How do you measure a year?” One line says “525,600 minutes” while most of the rest of the song suggests the best way to measure a year is moments shared with a loved one.
* (202) 456-1414 is the phone number for the White House switchboard. Truncated, Randall's formula yields 0.2024561414.
* Jenny's constant comes from Tommy Tutone's tune {{w|867-5309/Jenny}}. Randall's formula gives approximately 867.530901981685.
* {{w|42 (number)|42}} is, according to Douglas Adams' ''The Hitchhiker's Guide to the Galaxy'', the Answer to Life, the Universe, and Everything.

And here are some of the mathematical and physical ones, with Wikipedia links.

* Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics.
* The {{w|fine structure constant}} indicates the strength of electromagnetism. It is unitless and around 0.007297, close to 1/137. At one point it was believed to be exactly the reciprocal of 137, and many people have tried to find a simple formula explaining this (with a pinch of {{w|numerology}} thrown in at times), including the infamous {{w|Arthur Eddington|Sir Arthur Adding-One}}.
* In {{w|mathematics}}, the {{w|Euler-Mascheroni constant}} (Euler gamma constant) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
* The {{w|gravitational constant}} relates to, uh, gravity.
* The {{w|gas constant}} relates energy to temperature in physics.
* ϕ is the {{w|golden ratio}}, or (1 + √5)/2. It has many interesting geometrical properties.
* The ruby laser wavelength varies because “ruby” is not clearly defined.
* The {{w|Earth radios#mean radii|mean earth radius}} varies because there is not one single way to make a sphere out of the earth. Randall's value lies within the actual variation of Earth's radius.

The correct equation in the "Pro tip - Not all of these are wrong" section is cos(pi/7) + cos(3pi/7) + cos(5pi/7) = 1/2 as [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2 shown here].

The number 8675309 at the title text refers to the song 867-5309/Jenny as mentioned above, causing a fad of people dialing this number and asking for "Jenny". The number is in fact a {{w|twin prime}} because 8675311 is also a prime. Twin primes have always been a subject of interest, because they are comparatively rare, and because it is not yet known whether there are infinitely many of them.

{{w|Pi}} is a natural constant that arises in describing circles or ellipses. As such, useful as it may be, it doesn't usually occur anywhere in an exponent. When it does, such as with complex numbers, taking the pi-th root is rarely helpful. For example, if we try to derive:

''e''π''i'' + 1 = 0

''e''π''i'' = -1

(''e''''i'')π = -1

''e''''i'' = π√(-1)

We get nowhere.

Same goes for the e-th power: e typically appears in the basis of a power (forming the {{w|exponential function}}), not in the exponent. (This is later referenced in [Lethal Neutrinos http://what-if.xkcd.com/73/].

The software referred to in the comic is [http://mrob.com/pub/ries/ ries], a 'reverse calculator' which forms equations matching a given number.
:{| class="wikitable"
|-
| colspan="2" align="center" | Actual
| colspan="2" align="center" | Approximation
|-
| align="center" | One light year(m)
| align="center" | 9.46x1015
| align="center" | 998
| align="center" | 9.23x1015
|-
| align="center" | Earth Surface(m2)
| align="center" |
| align="center" | 698
| align="center" |
|-
| align="center" | Ocean's volume(m3)
| align="center" |
| align="center" | 919
| align="center" |
|-
| align="center" | Seconds in a year
| align="center" | 31557600
| align="center" | 754
| align="center" | 34640625
|-
| align="center" | Seconds in a year (rent method)
| align="center" | 31557600
| align="center" | 525,600 x 60
| align="center" | 31536000
|-
| align="center" | Age of the universe (seconds)
| align="center" |
| align="center" | 1515
| align="center" | 4.379x1017
|-
| align="center" | Planck's constant
| align="center" | 6.626x10-34
| align="center" | 1/(30πe)
| align="center" | 6.685x10-34
|-
| align="center" | Fine structure constant
| align="center" | 7.297x10-3
| align="center" | 1/140
| align="center" | 7.143x10-3
|-
| align="center" | Fundamental charge
| align="center" | 1.602x10-19
| align="center" | 3/(14 * πππ)
| align="center" | 1.599x10-19
|}

==Transcript==
{{incomplete transcript|this still needs some polishing}}
:'''A table of slightly wrong equations and identities useful for approximations and/or trolling teachers.'''
:(Found using a mix of trial-and-error, ''Mathematica'', and Robert Munafo's ''Ries'' tool.)
: All units are SI MKS unless otherwise noted.

:{| class="wikitable"
|-
| colspan="2" align="center" | Relation:
| align="center" | Accurate to within:
|-
| align="center" | One light year(m)
| align="center" | 998
| align="center" | one part in 40
|-
| align="center" | Earth Surface(m2)
| align="center" | 698
| align="center" | one part in 130
|-
| align="center" | Ocean's volume(m3)
| align="center" | 919
| align="center" | one part in 70
|-
| align="center" | Seconds in a year
| align="center" | 754
| align="center" | one part in 400
|-
| align="center" | Seconds in a year (rent method)
| align="center" | 525,600 x 60
| align="center" | one part in 1400
|-
| align="center" | Age of the universe (seconds)
| align="center" | 1515
| align="center" | one part in 70
|-
| align="center" | Planck's constant
| align="center" | 1/(30πe)
| align="center" | one part in 110
|-
| align="center" | Fine structure constant
| align="center" | 1/140
| align="center" | [I've had enough of this 137 crap]
|-
| align="center" | Fundamental charge
| align="center" | 3/(14 * πππ)
| align="center" | one part in 500
|-
| align="center" | White House Switchboard
| colspan="2" align="center" | 1 / 
π√(e(1 + (e-1)√8)
|-
| align="center" | Jenny's Constant
| colspan="2" align="center" | (7(e/1 - 1/e) - 9) * π2
|-
| colspan="3" align="center" | Intermission: World Population Estimate which should stay current for a decade or two: 
Take the last two digits of the current year

Example: 20[14]

Subtract the number of leap years since hurricane Katrina

Example:14 (minus 2008 and 2012) is 12

Add a decimal point

Example: 1.2

Add 6

Example: 6 + 1.2

7.2 = World population in billions.

Version for US population:

Example: 20[14]

Subtract 10

Example: 4

Multiply by 3

Example: 12

Add 10

Example: 3[22] million

|-
| align="center" | Electron rest energy
| align="center" | e/716 Joules
| align="center" | one part in 1000
|-
| align="center" | Light-year(miles)
| align="center" | 2(42.42)
| align="center" | one part in 1000
|-
| colspan="2" align="center" | sin(60°) = 3√/2 = e/π
| align="center" | one part in 1000
|-
| colspan="2" align="center" | √3 = 2e/π
| align="center" | one part in 1000
|-
| align="center" | gamma(Euler's gamma constant)
| align="center" | 1/√3
| align="center" | One part in 4000
|-
| align="center" | Feet in a meter
| align="center" | 5/(e√π)
| align="center" | one part in 4000
|-
| colspan="2" align="center" | √5 = 2/e + 3/2
| align="center" | one part in 7000
|-
| align="center" | Avogadro's number
| align="center" | 69π√5
| align="center" | one part in 25,000
|-
| align="center" | Gravitational constant G
| align="center" | 1 / e(pi - 1)(pi + 1)
| align="center" | one part in 25,000
|-
| align="center" | R(gas constant)
| align="center" | (e+1) √5
| align="center" | one part in 50,000
|-
| align="center" | Proton-electron mass ratio
| align="center" | 6*π5
| align="center" | one part in 50,000
|-
| align="center" | Liters in a gallon
| align="center" | 3 + π/4
| align="center" | one part in 500,000
|-
| align="center" | g
| align="center" | 6 + ln(45)
| align="center" | one part in 750,000
|-
| align="center" | Proton-electron mass ratio
| align="center" | e8 - 10 / ϕ
| align="center" | one part in 5,000,000
|-
| align="center" | Ruby laser wavelength
| align="center" | 1 / (12002)
| align="center" | [within actual variation]
|-
| align="center" | Mean Earth Radius
| align="center" | (58)*6e
| align="center" | [within actual variation]
|-
| colspan="3" align="center" | Protip - not all of these are wrong:
|-
| colspan="2" align="center" | √2 = 3/5 + π/(7-π)
| align="center" | cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2
|-
| align="center" | γ(Euler's gamma constant) = e/34 + e/5
| align="center" | √5 = 13 + 4π / 24 - 4π
| align="center" | Σ 1/nn = ln(3)e
|}

{{comic discussion}}
[[Category:Charts]]
[[Category:Math]]
[[Category:Physics]]
[[Category:Protip]]

Talk:936: Password Strength

2013-11-11T21:22:04Z

173.245.54.78:

You still have to vary the words with a bit of capitalization, punctuation and numbers a bit, or hackers can just run a dictionary attack against your string of four words. '''[[User:Davidy22|{{Color|purple|David}}y²²]]'''[[User talk:Davidy22|<tt>[talk]</tt>]] 09:12, 9 March 2013 (UTC)

No you don't. Hackers cannot run a dictionary attack against a string of four randomly picked words.
Look at the number of bits displayed in the image: 11 bits for each word.
That means he's assuming a dictionary of 2048 words, from which each word is picked randomly.
The assumption is that the cracker knows your password scheme.
[[Special:Contributions/86.81.151.19|86.81.151.19]] 20:17, 28 April 2013 (UTC)
Willem

Sometimes this is not possible. (I'm looking at you, local banks with 8-12 character passwords and PayPal) If I can, I use a full sentence. A compound sentence for the important stuff. This adds the capitalization, punctuation and possibly the use of numbers while it's even easier to remember then Randall's scheme. I think it might help against the keyloggers too, if your browser/application autofills the username filed, because you password doesn't stand out from the feed with being gibberish. [[Special:Contributions/195.56.58.169|195.56.58.169]] 09:01, 30 August 2013 (UTC)

The basic concept can be adapted to limited-length passwords easily enough: memorize a phrase and use the first letter of each word. It'll require about a dozen words (you're only getting 4.7 bits per letter at best, actually less because first letters of words are not truly random, though they are weakly if at all correlated with their neighbors -- based on the frequencies of first letters of words in English, and assuming no correlation between each first letter and the next, I calculate about 4 bits per character of Shannon entropy). SteveMB 18:35, 30 August 2013 (UTC)

Followup: The results of extracting the first letters of words in sample texts (the {{w|Project_Gutenberg|Project Gutenberg}} texts of ''The Adventures of Huckleberry Finn'', ''The War of the Worlds'', and ''Little Fuzzy'') and applying a {{w|Entropy_(information_theory)|Shannon entropy calculation}} were 4.07 bits per letter (i.e. first letter in word) and 8.08 bits per digraph (i.e. first letters in two consecutive words). These results suggest that first-letter-of-phrase passwords have approximately 4 bits per letter of entropy. --[[User:SteveMB|SteveMB]] ([[User talk:SteveMB|talk]]) 14:21, 4 September 2013 (UTC)

Addendum: The above test was case-insensitive (all letters converted to lowercase before feeding them to the [[http://millikeys.sourceforge.net/freqanalysis.html frequency counter]]). Thus, true-random use of uppercase and lowercase would have 5 bits per letter of entropy, and any variation in case (e.g. preserving the case of the original first letter) would fall between 4 and 5 bits per letter. --[[User:SteveMB|SteveMB]] ([[User talk:SteveMB|talk]]) 14:28, 4 September 2013 (UTC)

I just have RANDOM.ORG print me ten pages of 8-character passwords and tape it to the wall, then highlight some of them and use others (say two down and to the right or similar) for my passwords, maybe a given line a line a little jumbled for more security. [[Special:Contributions/70.24.167.3|70.24.167.3]] 13:27, 30 September 2013 (UTC)
:Remind me to visit your office and secretly replace your wall-lists by a list of very similar looking strings ;) --[[User:Chtz|Chtz]] ([[User talk:Chtz|talk]]) 13:53, 30 September 2013 (UTC)

Simple.com (online banking site) had the following on it’s registration page:

“Passphrase? Yes. Passphrases are easier to remember and more secure than traditional passwords. For example, try a group of words with spaces in between, or a sentence you know you'll remember. "correct horse battery staple" is a better passphrase than r0b0tz26.”

Online security for a banking site has been informed by an online comic. Astounding.
[[Special:Contributions/173.245.54.78|173.245.54.78]] 21:22, 11 November 2013 (UTC)

explain xkcd - User contributions [en]

1047: Approximations

Talk:936: Password Strength