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What is the most number of digits of a transcendental mathematical constant (for example, $\pi$ or $\mathrm{e}$) that have been necessary for an actual computation?

  • Note that I am asking about the most extreme case, not usual practice. Usual practice is well documented; for example, this article quotes a NASA engineer as saying that they use only 15 digits for their calculations.

  • By "actual computation" I mean a computation that has been performed with some scientific or engineering purpose.

  • By "necessary" I mean that if the value of the fundamental constant were less precise, then the computation could give a result so inaccurate as to be legitimately less useful for the intended scientific or engineering purpose.

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Lamb discusses the issue in How Much Pi Do You Need?:

"I asked a NASA scientist how many digits of pi the agency uses for its calculations. Susan Gomez, manager of the International Space Station Guidance Navigation and Control (GNC) subsystem for NASA, said that calculations involving pi use 15 digits for GNC code and 16 for the Space Integrated Global Positioning System/Inertial Navigation System (SIGI). SIGI is the program that controls and stabilizes spacecraft during missions.

Believe it or not, there is a committee that makes recommendations about the values of these fundamental constants. The Committee on Data for Science and Technology, or CODATA, an interdisciplinary group from the International Council for Science, periodically publishes a set of accepted values of the fundamental physical constants... Peter Mohr, a physicist who works for the Fundamental Constants Data Center at the National Institute for Standards and Technology, which is involved in calculating and disseminating the accepted CODATA values, says that the institute uses 32 significant digits of pi in their computations."

NASA's Marc Rayman explained why one would not need to go far beyond the CODATA precision even in the hypothetical future astronomy:

"The radius of the universe is about 46 billion light years. Now let me ask a different question: How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom (the simplest atom)? The answer is that you would need 39 or 40 decimal places. If you think about how fantastically vast the universe is... and think about how incredibly tiny a single atom is, you can see that we would not need to use many digits of pi to cover the entire range."

One may worry that sensitivity effects in ill-posed problems, such as weather forecasts, would require extra high precision in the input. However, in those cases the imprecision of measured inputs makes high precision in mathematical constants pointless. Still, in 2002 Tucker employed a combination of interval arithmetic and 80-bit IEEE extended arithmetic (17-digit precision) to prove that the Lorenz equations support a “strange” attractor, which solved Smale's problem. Bailey and Borwein describe other examples in High-Precision Arithmetic in Mathematical Physics.

But if "scientific or engineering purpose" includes cryptography then the threshold is raised dramatically. Ciphers based on transcendental numbers have been proposed, and they require exorbitant precision to maintain security. Here is a simple version from Viswanath's Transcendental Numbers and Cryptography:

"The decimal expansion of $\pi$ is known for more than 1000 million places of decimals and is known as the mountain of $\pi$. We use this mountain of $\pi$ in the encryption of messages. One can use any other transcendental number for which the mountain is available! Assume that Bob is encrypting a message with he mountain of $\pi$. In this encryption, the Key $\alpha$ is the position of the decimal place of $\pi$ which is used to begin the encryption. The number at $\alpha$-th place, say $n$ is used to substitute the beginning letter of the plaintext by shifting the alphabet by $n$ units(mod 26). Afterwards the process is continued with the next integer and the next alphabet in the plaintext and so on, till the entire message is encrypted. One can raise the level of security further by using two transcendental numbers one for encryption and the other for decryption and the key is chosen as before."

Actually, digits of $\pi$ are known to 10 trillion places now.

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In cryptography, designs often have "constants": effectively parameters which can use an arbitrary number (or an arbitrary number which doesn't have any simple patterns) but which must use the same number in all implementations. However, it's considered suspicious if the numbers are truly arbitrary: perhaps you calculated them in such a way as to create a "back door". So it is now common to use "nothing up my sleeve" numbers, and elementary transcendental numbers are the most common type of such numbers.

In particular, the Blowfish cipher uses the first 704 bits of the fractional part of pi to initialise its key schedule. That's about 212 decimal digits.

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Of course many mathematical constants can be computed to billions of digits, I suppose the question is about physics: how many digits can be actually tested, what are the most precise measurements.

According to Littlewood (who wrote this in 1948), the most exact computations used in science are time computations in astronomy (up to 15 decimal digits).

"Some of them involve time periods of thousands years which are computed to 0.001 second, and for control one usually keeps one-two more digits".

I don't think this changed since the time of Littlewood.

Reference J. E. Littlewood, A mathematian's Miscellany, London 1957. Chapter "Large Numbers". Originally published in Math Gazette, JUly 1948, vol 32, N 300.

Some computations in quantum electrodynamics (Wikipedia) have 12-13 significant digits which have been tested. So this comes close.

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  • $\begingroup$ I'm not asking about the precision of the output of the calculation, but rather about the precision of the mathematical constants in the input to the calculation necessary for the output to have the necessary precision for some application. $\endgroup$ – ajd Jan 24 '18 at 18:40
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    $\begingroup$ @AlexanderDunlap It'd be about the same. For example, to know $f(\pi)$ to that precision requires approximately the same precision in $\pi$, by the chain rule. $\endgroup$ – J.G. Jan 24 '18 at 22:30
  • $\begingroup$ @J.G. I do not see how, suppose $f(x)=10^{10}x$. Computations with ill-conditioned matrices also require a much higher precision in the input. $\endgroup$ – Conifold Jan 24 '18 at 23:54
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    $\begingroup$ @Conifold: The numbers $10^{10}x$ and $x$ have the same number of SIGNIFICANT digits. $\endgroup$ – Alexandre Eremenko Jan 25 '18 at 1:08
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    $\begingroup$ @AlexandreEremenko Consider $f(x) = \exp\{10^{10}x\}$, or the nonlinear dynamical systems that Conifold mentions in their answer. Not all functions have bounded derivatives! $\endgroup$ – ajd Jan 25 '18 at 6:32

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