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In reference [1], Gelman et al. give an account of one of the first occurrence of Bayesian inference. Laplace used the number of births by gender in Paris from 1745 to 1770 to conclude that the probability that boys have a higher probability to be born than girls is about $1.15\times 10^{-42}$, that is, the data he was using showed with extreme confidence that one is more likely to give birth to a girl than a boy.

Now, $10^{-42}$ is a remarkably small number. Not only for a probability, but also for any physical quantity in SI units (or any unit I know, for that matter). I believe the closest example would be Planck's constant at $10^{-34} [J\cdot s]$, and that is still several orders of magnitude off. And so I wondered, was this, at the time, the smallest non-zero number ever encountered?

In general, my question is: throughout the history of science and mathematics, what were the smallest finite non-zero numbers (in absolute value) encountered as the result of a legitimate calculation?

[1] A. Gelman, J.B. Carlin, H. S. Stern and D. B. Rubin, Bayesian Data Analysis, 2nd edition (2004) pp 34-36

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    $\begingroup$ Closely related: What is the smallest constant that has explicitly appeared in a published paper? $\endgroup$ Jan 29 '19 at 10:45
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    $\begingroup$ Then there's the response given in "The Phantom Tollbooth" :-) $\endgroup$ Jan 29 '19 at 12:20
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    $\begingroup$ In all seriousness, I seem to recall some Statistical Mechanics prof suggesting we calculate the probability that all molecules in a book bounce upwards together (thus levitating the book). Let's see some creative calculations for that! $\endgroup$ Jan 29 '19 at 12:22
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    $\begingroup$ Or $\varepsilon^2$... $\endgroup$
    – KCd
    Jan 30 '19 at 0:59
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    $\begingroup$ @Carl Witthoft: In George Gamow's famous book One Two Three...Infinity, near the beginning of Section 4 (The "Mysterious" Entropy) of Chapter VIII (The Law of Disorder) Gamow discusses and goes through a calculation for the probability that the air molecules in a room all go to the half of the room that you're not in, so that you'd suffocate (not likely, of course, unless they stayed there a few minutes), obtaining $10^{-3 \times 10^{26}}.$ I read this part when I was finishing 6th grade (1971) and found it to be absolutely fascinating. $\endgroup$ Jan 31 '19 at 13:53
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There is a problem with using dimensional magnitudes in this context. The Planck length in meters, the unit of length that light travels in Planck time, beats the Planck constant, it is $1.616229(38)×10^{−35}$ meters. The Planck time beats even the probability in the OP, it is $5.39116(13)×10^{−44}$ seconds. And one can beat this easily by choosing milliseconds or nanoseconds as units. A better, and even smaller, example is the dimensionless gravitational coupling constant, the squared ratio of the electron mass to the Planck mass, $\alpha_G\approx1.7518×10^{−45}$. It measures the strength of gravitational attraction between two electrons in Planck scale terms. Its electrostatic analog is the better known, and much bigger, fine structure constant $\alpha\approx1/137.036$, hence the idea that "gravity is much weaker than electromagnetism".

Probabilities and mathematical constants are also better because they are dimensionless. But "smallest non-zero in absolute value" does not rule out infinitesimals, liberally used by Fermat, Leibniz and Euler in their calculations, and, in recent times, by Robinson and Conway. The Conway's scheme even allows to individuate some of them, like his constant $ε:=\{ 0 | 1, 1/2, 1/4, 1/8,…\}$, and perform calculations on them, similar to the ones in positional systems.

Even if we stick to reals or rationals the precision estimates in the Approximations of $\pi$ easily beat all of the above (other than $ε$). Wikipedia even makes a point of mentioning that

"the circumference of the largest known object, the observable universe, can be calculated from its diameter (93 billion light-years) to a precision of less than one Planck length... using $\pi$ expressed to just 62 decimal places".

Vega calculated first 140 decimal places, of which the first 126 were correct, already back in 1789. That would be to $10^{-126}$ precision. As of November 2016, the record precision is $10^{-22,459,157,718,361 (π^e×10^{12})}$.

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  • $\begingroup$ The last edit specified that the number should be finite. The approximations of $\pi$ however seems to be a very strong contender indeed, and for very large historical segments. $\endgroup$
    – Alexis
    Jan 30 '19 at 9:08
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Below are some examples of possible interest from my 8 April 2002 sci.math post BIG NUMBERS #1. Many of these are things that I had jotted down in personal notes back in the 1970s and early 1980s, and I believe all of them (except the Besicovitch example and the ASCII google groups example) are in a lengthy handwritten manuscript on cardinal numbers, order types, and ordinal numbers that I wrote in Summer 1985.

Something else of possible interest to estimate is discussed here. Also, here I wrote: I suppose we could estimate the odds that someone will live long enough to think of every possible thought while waiting to die of statistical asphyxiation [of the Gamow type]. Seeing as how it's $1$ out of $10^{10^{36}}$ to live at least $1000$ years, this might get us up to around 10^^5 $[ = \sideset{_{}^5}{}{10} = 10^{10^{10^{10^{10}}}}$] for a number that has some remote significance to the physical world.

$5.53 \times 10^{-67}$ -- The gravitational force (in Newtons) between two electrons that are $1$ cm apart.

$6.18 \times 10^{-103}$ -- The gravitational force (in Newtons) between two electrons that are $1$ light year apart.

$10^{-200}$ -- This is the probability that an electron in a 1s orbital of a hydrogen atom is at least $5$ nanometers from the nucleus.

$10^{-1080}$ -- This is the probability of flipping a coin once a second for an hour and getting all heads.

$10^{-2576}$ --- This is a number that appeared in a paper by a 1928 paper by Besicovitch and that I discussed here.

$10^{-42,000}$ -- This is the probability of a monkey typing Hamlet by chance.

$10^{-10^8}$ -- This is the probability of flipping a coin once a second for $100$ years and getting all heads.

$10^{-2.9 \times 10^{12}}$ -- This is the Yukawa nuclear force between two nucleons that are $1$ cm apart.

$10^{-10^{13}}$ -- This is the probability that by randomly typing ASCII characters, a long-lived monkey will type every post, in a pre-selected order, that appears at the Google Groups archive.

$10^{-5.2 \times 10^{18}}$ -- Using non-relativistic quantum mechanics, this is the probability that an electron in a 1s orbital of a hydrogen atom is at least $240,000$ miles from the nucleus (distance to the Moon).

$10^{-3 \times 10^{26}}$ -- Gamow observes (see Chapter 8.4 of [43]) there is a nonzero probability that all of the air molecules in a room will collect together simultaneously on one side of the room, namely $(1/2)^n$ where $n = 10^{27}$ is the number of air molecules in the room.

$10^{-10^{36}}$ -- The probability given by life insurance calculations that someone will live $1000$ years. [This appears at the beginning of Chapter 1 in Volume I of William Feller's text PROBABILITY THEORY AND ITS APPLICATIONS. Feller writes: "We hesitate to admit that man can grow $1000$ years old, and yet current actuarial practice admits no such limit to life. According to formulas on which modern mortality tables are based, the proportion of men surviving $1000$ years is of the order of magnitude of one in $10^{10^{36}}$ -- a number with $10^{27}$ billions of digits."]

$10^{-10^{36}}$ -- Using non-relativistic quantum mechanics, this is also the probability that an electron in a 1s orbital of a hydrogen atom is at least $10$ billion light years away from the nucleus. (Note: The most distant objects that have been observed are a little over $13$ billion light years distant.)

$10^{-2.7 \times 10^{40}}$ -- This is the Yukawa nuclear force between two nucleons that are $10$ billion light years apart.

$10^{-10^{93}}$ -- Using non-relativistic quantum mechanics, this is the probability that every electron in all the hydrogen atoms in the sun are at least $10$ billion light years away.

$10^{-10^{166}}$ --- This is the probability of "randomly typing" the entire space-time universe by selecting the correct particle distribution (out of the $10^{10^{125}}$ total permutations that all elementary particles in the universe (assumed to be distinct) can be distributed into the particle-sized locations in the universe at any given time), for each of the $10^{41}$ instants of time since the Big Bang ("instant" equals the time it takes light to travel the diameter of an atomic nucleus).

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