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(Alright, how many ben gilo's.)

The Gemara in BM 30b says that each ben gilo visitor to a sick person eliminates 1/60 of their sickness by taking it upon himself. Presumably this means 1/60 of what's left after previous visitors took their portions, not 1/60 of the original. Thus, the disease can't fully be eradicated in this method. But it can get pretty close. How many visitors are needed to get to this point?

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DonielF
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Depends how close you want to get.

(These are the days I wish Mi Yodeya supported MathJax.)

Each person leaves with 1/60 of the guy's sickness. Thus, for x visitors, the choleh is left with (59/60)^x of his disease.

Now, let's say we want to find out how many people need to come to leave him with 10^(-n) of his sickness, for some n (for example, a millionth, 10^(-6)). We can take the log of both sides and solve for x.

log((59/60)^x)=log(10^(-n))

xlog(59/60)=-n

x=137.000588091n

So, for whatever exponent you choose to plug in for n, multiply it by about 137 to figure out how many people need to come to get his sickness down to that point. In our above example of 10^(-6), that would be 6*137=822 ben gilo's.

EDIT: In Vayikra Rabbah (34:1), as the Matnos Kehunah understands it, R' Huna's talmidim originally understood this to mean sixty people should go around curing everybody. R' Huna replied that they remove one-sixtieth of the original if they love the sick person like themselves, but they will remove one-sixtieth of what remains otherwise. According to Maharzu, they remove one-sixtieth of what remains only if they love the person like themselves and they merely provide emotional support but don't remove any of the sickness otherwise. This version makes no mention of a Ben Gilo.

DonielF
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    I've consider this before, too. Of course, this depends on the assumptions that 1/60 is meant literally, and that it's 1/60th of the current illness rather than the original illness. | You may be interested in http://judaism.stackexchange.com/q/22870. – Fred Aug 24 '16 at 04:05
  • @Fred Of course. If it's 1/60, or any fraction, of the original illness, then you need the reciprocal of that fraction to take away all of it, and if it's a different fraction of the current illness, the math still works the same, just with a different number on the left side to take the log of. – DonielF Aug 24 '16 at 04:26
  • Note that 60 is just the Talmuds example of a big number since Babylon used base 60. This genera was presumably never meant to be in any way rigorous. – mevaqesh Aug 24 '16 at 19:54
  • @mevaqesh Yeah, I know that oftentimes that's the case. It's fun to play with the math, though, treating the number literally. – DonielF Aug 24 '16 at 20:22
  • @mevaqesh I guess with this Midrash saying the same concept and R' Huna's discussion with his talmidim it's absolutely meant to be literal. – DonielF Jan 24 '17 at 22:26
  • @mevaqesh I agree that 60 is clearly the "big number" exaggeration in the Talmud. Do you have a source for the assertion that Babylon used base 60? – Chaim Jan 24 '17 at 23:02
  • @Chaim https://en.m.wikipedia.org/wiki/60_(number)#Cultural_number_systems – DonielF Jan 24 '17 at 23:06
  • BTW the passage from Vayikra Rabba is:
    כל מי שמבקר את החולה פוחתים לו אחד מששים בחוליו. Note that this is one of dozens of examples I list in response to your question here in which the divine involvement with Man is presented using the plural.     – mevaqesh Dec 21 '17 at 02:54