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An infinite number of thirsty mathematicians

 


Stop me if you've heard this one.  

An infinite number of mathematicians walk into a pub in London.

The first one says to the bartender, "I'll have a pint of your finest lager." 

Second one, "I'll have just half a pint."

Third one, "I'll have a quarter of a pint." 

Bartender raises his hand in a "stop" gesture and says to the whole infinite line, "That's enough-- I'm fetching you the full two pints you're asking for."

Calculus in one lesson.      

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Comments

  1. HenrySeptember 11, 2024 at 8:19 AM

    I could not grasp calculus in college, so forgive this question, but, even if we continue to halve and add up the amounts an infinite number of times, we won't reach a whole pint, will we?

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    Replies
    1. ChristopherSeptember 11, 2024 at 11:40 AM

      The genius contribution of Newton and Leibniz was precisely to treat the halving as a limit, and then to do calculations WITH the limit, as my barkeep does. This was precisely what bothered Zeno of Elea in ancient times. That Achilles the tortoise. Yet if we think in terms of limits we can get to the tortoise and sit on his back for the rest of the race.

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    2. ChristopherSeptember 12, 2024 at 12:11 AM

      I should have said that they treated the "process of halving and addition as a process with a limit, upon which it converges, and then to do calculations...."

      Delete
  2. HemantSeptember 13, 2024 at 5:11 AM

    Addendum: If the first mathematician had said "I'll have a pint," the second: "I'll have half a pint," the third: "one-third of a pint," the fourth: "one-fourth of a pint," and so on... The barkeeper would have replied, "I'm sorry, I don't have enough supply of beer for all of you guys..."

    Unless, of course, he had an infinite quantity of drink available.

    (The sum of the reciprocals of the natural numbers does NOT converge.)

    ReplyDelete
  3. ChristopherSeptember 13, 2024 at 12:02 PM

    True. Indeed, 1/2 plus 1/3 plus 1/4 equals 13/12s, i.e. more than a full pint.

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  4. HenrySeptember 13, 2024 at 3:32 PM

    But if we got rid of the 1/3 and just halved all the way down, would we ever reach a full pint?

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    Replies
    1. ChristopherSeptember 15, 2024 at 12:57 AM

      We would "never reach it" in any finite number of thirsty mathematicians, and presuming that each of the thirsty mathematicians takes some specific amount of time to do his share of the drinking we will all die while the transactions are underway. Still, mathematically the important point is that the halving sets up a series that converges, as a limit, at 2 pints. Think of a Bert and Ernie routine on Sesame Street decades ago, Bert is trying to get to sleep but Ernie keeps him awake because Ernie is excited by what he has learned about shapes. The shapes with the fewest numbers of sides have the sharpest corners, and as shapes get more sides their corners become less pointy. Bert repeatedly tries to get to sleep. Ernie says, "But what about a circle? Is it just one big side or lots of little sides?" Bert thinks about this, but by the time he comes back with, "A circle COULD have some sides Ernie ..." Ernie is snoring. Leibniz and Newton would each have said that the circle (and a planetary orbit, though they are ellipses) have an infinite number of sides of infinitesimal size. Or we might say that the process of doubling the number of sides starting with a triangle results in a circle as a limit. Here is the Sesame Street explanation. (Brilliant). https://www.youtube.com/watch?v=vyuqTYBNs7o

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