Mathematical approximations and how wrong they are

A great XKCD today: "Approximations: A Slightly Wrong Table of Equations and Identities Useful for Approximations and/or Trolling Teachers."

Approximations

I write books. My latest is a YA science fiction novel called Homeland (it's the sequel to Little Brother). More books: Rapture of the Nerds (a novel, with Charlie Stross); With a Little Help (short stories); and The Great Big Beautiful Tomorrow (novella and nonfic). I speak all over the place and I tweet and tumble, too.

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http://twitter.com/guymac guymac

Pessimist! I see it as “how right they are” ;-)
- Greg Miller
  
  Honestly, in most engineering uses, they’re all good enough. That’s something that’s forgotten a lot, but a problem in engineering is people striving for perfection, when good enough will get the job done sooner and cheaper. There are some things where it’s necessary, but in most cases it isn’t.
  - Paul Renault
    
    I kinda recall that knowing π to a fairly low number of decimal places is plenty accurate for just about anything.
    
    For example, if you memorize π to 31 places (it’s really not that difficult), and you used that number (3.1415926535897932384626433832795) to calculate the circumference of the Milky Way galaxy, you’d be off by about 1/20th of the diameter of a proton.
    
    Just using “How I wish I could enumerate pi” (3.141592) when calculating Earth’s circumference from its diameter, you’d be off by a little more than 7 1/2 meters.
    
    If you used the Katharevousa Greek mnemonic for π, giving you 22 decimal places, for that same calculation, you’d be back down to sub-atomic particle ‘error’ territory:
    Ἀεὶ ὁ Θεὸς ὀ Μέγας γεωμετρεῖ,τὸ κύκλου μῆκος ἵνα ὁρίσῃ διαμέτρῳ,παρήγαγεν ἀριθμὸν ἀπέραντον,καὶ ὅν, φεῦ, οὐδέποτε ὅλον θνητοὶ θὰ εὕρωσι
    - Jorpho
      
      ”How I need a drink, alcoholic of course, after the heavy chapters involving quantum mechanics” will always be my favorite.
davide405

Further evidence that e and pi are part of God’s address.
- Guest
  
  i like pi/e, pi*e, pi+e, and even pi-e. My favorite is pi^e.
  - PaulDavisTheFirst
    
    e^(i * pi) = -1
    
    the universe – it works, bitches!
    - Steve Lord
      
      In its alternate form, e^(i*pi) + 1 = 0, that equation includes five fundamental constants:
      
      The additive identity, 0. [0 + any number = that number]
      The multiplicative identity, 1. [ 1 * any number = that number]
      One of the square roots of -1. [i^2 = -1]
      The base of the natural logarithm.
      The ratio of the circumference of a circle to its diameter.
      - Camp Freddie
        
        I agree with most of the beautiful equation, but Pi is a terrible idea for a fundamental constant. Tau is much more sensible (the ratio of circumference to radius).It makes trigonometry and calculus much more intuitive (tau = 360°, 1/2(tau)r^2 = area, etc.) Also, e^(i×tau)+0=1
        
        Google “Pi is wrong” and have your mind blown. Lots of simple equations based on Pi need an extra factor of 2 in order to balance, since Pi treats diameter as the fundamental property of a circle.
semiotix

As a woodworker, I’m constantly having to switch between metric and English length measurements. 5-over-eth-root-π is so much simpler to remember than 1m≈3′ 3 3/8″. Thanks, XKCD!
- James B
  
  I build a good bit of furniture, and like my combo metric/inch tape measure and combination square. So much easier to find half of 13-9/16″ when it is expressed as 344 mm. And I’m somewhat partial to multiplying inches by 25.4 to get to mm.
  - timquinn
    
    Half of 13 9/16 is 6 1/2 +9/32. The fractional system is built for division by two. You don’t have to add it together and reduce the fractions to find it on the tape measure. If you don’t need 1/32′s say in between 1/4 and 5/16. On the other hand if you were to scribble 344 mms on a piece of paper and then came back later to find you couldn’t tell if it was 394 or 349 or 399 or . . . with a fraction you know when there is a mistake and you have a good idea of what the number should be. If it says 5/10 or 6/16 you know there is problem and what it probably was. Unless, of course, you did not grow up using fractions and just can’t see the advantages at all because you have been propagandized to accept top down solutions to very low level problems.
    - James B
      
      Just saying what works for me. And maybe you caught where I like measuring instruments marked both ways. (Preferably both facing the same direction.) For carpentry work I tend more towards using fractional inch measurements: I intuitively know what 3-1/2″ inches looks like, and most of the work is just measuring and cutting to length. Plus, like you sort of imply, it is less prone to errors. But for laying out mortise and tenons, or when I mark metal for machine work, I still like metric.
jerwin

You’d think that any calculator powerful enough to compute those values would have enough space for “physical constants”.

How precisely is “mean earth radius” known?

I’m at awe at the “proton-electron mass ratio” given that “phi” is (1+sqrt(5))/2
- SamSam
  
  How precisely is “mean earth radius” known?
  
  Unfortunately there seems to be no snarky site called “let me Wikipedia that for you” like there is for Google, so I guess I’ll just be your lazyweb for you instead: http://en.wikipedia.org/wiki/Earth_radius
  - Ambiguity
    
    Perhaps what the Internet needs is a good meta-snark site — a site search site that will find snarky sites (like lmgtfy) for any occasion.
    
    Then we could make ironic spin-offs of that site, and the eschaton would finally come in the form of a meta-singularity!
    - Guest
      
      there is, or will be, an xkcd for that.
      - MarcVader
        
        I like turtles.
  - jerwin
    
    I don’t understand.
    I’ve calculated e*6*8^5 as 6370973.03545089 m
    Your “source” gives the mean radius of the earth as 6,371.009 km, which deviates substantially from Randall’s calculation. Given that Randall never lies, he must be using a more accurate reference for his numbers than some crummy wiki.
    - Ambiguity
      
      The difference between 6370,973 km and 6371.009 km is about 100m. Saying that is falls “within actual variation” may be accurate. The earth is an oblate sphereoid, not a sphere, so there could be some variation in the way it is defined and measured.
      - jerwin
        
        Ah, you are correct. The various methods of calculating the radii give values ranging from 6367.445 m (Meridional Earth radius) to 6,372.797 km (Great Circle radius)– so the actual value is only known to one in a thousand. Not very impressive.
        
        The written form of Randall’s equation is more complicated than the actual value it represents.
    - timquinn
      
      If you read the wiki, not just skim it for numbers, you would see that; #1 radii exist only on spheres, which the earth is not one. #2 Any idea of the mean radius of the earth depends hugely on your needs and is slippery in the best circumstance. So your statement that you do not understand is the most relevant part of your post.
      - timquinn
        
        In other words precision is useless with out accuracy.
Ambiguity

I love the hover-over text on this one: “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.”
nixiebunny

Fortunately, you can replace all occurrences of pi with 355/113 and achieve pretty good results. Because who (especially an XKCD reader) would be bothered to remember the numeric value of pi?
foxtails

3/8 = .38
- http://halfbakedmaker.org Robert Baruch
  
  1 = 2 for sufficient variation in values.
Mark Dow

A better approximation to the fine structure constant is (i^i)^pi, or i^(i*pi). It rolls off the tongue too, “eye to the eye to the pi”.
juan

xln(x)-x = ln(x!)
glittalogik

I’m kind of surprised he didn’t include the Golden Ratio/Fibonacci Sequence for approximate mile-to-km conversion. Each number in miles is close (within .2) to the next number in km.
- invictus
  
  Except for those pesky first members of the set. 1mi. is 1.609km.
Halloween_Jack

Gotta be honest here: this is one of those xkcds that just goes straight over my head, mostly, and I’ve always done well on the math portions of standardized tests. Well enough, anyway.
Jellodyne

Jenny’s Constant for the win . I got it, I got it!
- timquinn
  
  How many 7 digit numbers do you know by heart? Right? and this one is a member of a twin prime pair!
http://garethrees.org/ Gareth Rees

The approximation “kilometres in a mile = φ” is accurate to about one part in 185 and is actually useful.
- Sparrow
  
  I had an old car mileage computer that indicated φmiles rather than kilometres as a metric indication. Now it makes sense.