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September 6[edit]

Usefulness of a billion digit number?[edit]

Hello to all the "Deskians". In the French scientific magazine La Recherche, May 2019, an insert is devoted to the rapid and exact multiplication of large numbers. They write about numbers of 1 billion digits. I wonder in what area of human knowledge such numbers are useful. A 1 billion digits number it's about 10^1,000,000,000.
Let's not forget that the number of particles in the observable Universe is estimated between 10^60 and 10^90.
I'm thinking of math, for example I read in Wiki: "The largest known prime number is the Mersenne prime number 2 ^ 82 589 933 - 1, which is more than 24 million digits in decimal notation." but to achieve this result needs no large numbers multiplication. I thank you for your cogitations.--Jojodesbatignoles (talk) 07:45, 6 September 2019 (UTC)[reply]

Googolplex has some useful large numbers. Neil Tyson's Welcome to the Universe mentions 10^10¹²⁴ potential universes. Henry ^Flower 09:07, 6 September 2019 (UTC)[reply]

More precisely my question is: I know that mathematicians use numbers greater than 10^1000.000.000 but they don't need ALL THE DIGITS of these numbers; this article reads about getting the exact result (all the digits) of the product of 2 of these billion digit numbers.--Jojodesbatignoles (talk) 09:44, 6 September 2019 (UTC)[reply]

remember that the number of digits is also the precision, that is, 1 billion digits number may just as well be 10^-1,000,000,000. In a number of case, we don't have closed-form solution, but we do have ways to get numerical solutions (Three-body problem fi), but this require the actual value of every digit, and the more you have, the further in time you can figure things (predict in the future/understand in the past). Gem fr (talk) 11:58, 6 September 2019 (UTC)[reply]

All practical primality tests for large prime numbers do multiply numbers at the size of the tested number, and the exact product is needed. The current Mersenne prime record has around 25 million digits but there are already people testing numbers with more than 100 million digits. The usefulness can be debated but thousands of computer years and millions in electricity costs is used to search large primes. Efficient multiplication (not division) is the most important part with the used algortihms. See also Largest known prime number#Prizes and https://www.eff.org/press/releases/eff-offers-cooperative-computing-prizes: "the finder of the first billion-digit prime will receive $250,000". With current computers and algorithms that is a small fraction of the expected electricity cost. Pi#Rapidly convergent series has found not just billions but trillions of digits, but I'm not sure how it's implemented in practice. By the way, I once computed a Smith number above a trillion digits in a few hours on my PC (the stated record is a few million digits!), but I didn't publish it and didn't actually use general multiplication with billions of digits. PrimeHunter (talk) 12:06, 6 September 2019 (UTC)[reply]

Did you store all of the trillion-digit Smith Number in memory? Bubba73 ^{You talkin' to me?} 05:26, 10 September 2019 (UTC)[reply]

Not at the same time. I chose a special form where I could compute the digits in groups, discard them before moving to the next group, and skip long sequences of zeroes. PrimeHunter (talk) 10:32, 10 September 2019 (UTC)[reply]

@Jojodesbatignoles: Nitpick: "Wiki" is not a valid abbreviation of Wikipedia, see WP:DAW.--Jasper Deng (talk) 06:35, 7 September 2019 (UTC)[reply]

There are some ridiculously big numbers that have real meaning, such as Graham's number and TREE(3). Our visible universe does not have enough space and matter to write down power towers that express how many digits those numbers have. Trying to understand how large these numbers are can drive you a little bit insane. Still they are genuine numbers with real meaning. 85.76.74.244 (talk) 19:27, 7 September 2019 (UTC)[reply]

It also isn't just academic, extremely large numbers are useful in cryptography, especially One-way functions used in cryptographic hashes. It is very easy to multiply two large numbers to get a correct answer, but it is extremely hard to factor that larger number to "get back" the two original factors. This is the basis of the P versus NP problem, and has huge implications for cryptography. Currently, the speed of computers limits the size of the numbers they can solve one-way functions by brute force; 40-digit hexadecimal numbers (SHA-1) are the current standard, but it is not out of reason to suspect that in the coming decades, the need for MUCH larger numbers known to the exact ones-digit will be needed. I wouldn't say that someday having to work out a hash with billion-digit numbers will never be useful to that end. --Jayron32 16:44, 10 September 2019 (UTC)[reply]
- Nitpick: integer factorization is suspected to be NP-intermediate and could well be solvable in polynomial time even if P isn't equal to NP.--Jasper Deng (talk) 20:54, 10 September 2019 (UTC)[reply]

Network theory for electrical structures.[edit]

Linear network analysis
Elements

Components

Series and parallel circuits

Impedance transforms

Generator theorems	Network theorems

Network analysis methods

Two-port parameters

	view talk edit

This is from "An Introduction to Information Theory, Symbols, Signals and Noise (Dover Books on Mathematics)" (1980) by John R. Pierce:

"A branch of electrical theory called network theory deals with the electrical properties of electrical circuits, or networks, made by interconnecting three sorts of idealized electrical structures: resistors (devices such as coils of thin, poorly conducting wire or films of metal or carbon, which impede the flow of current), inductors (coils of copper wire, sometimes wound on magnetic cores), and capacitors (thin sheets of metal separated by an insulator or dielectric such as mica or plastic;"

Is the discipline that deals with electrical structures (of three sorts, resistors, inductors, capacitors), still referred as network theory? Is this point of view of any interest at all? It has been a long time since the 80s, so maybe it's a dead field in this form. Anyway, if we are limited to three component like abstract resistors, inductors, and capacitors, can we describe all electrical networks somehow?

Is there any Wikipedia article describing an approach to electrical circuits reducing them to resistors, inductors, capacitors? I imagine these are the nodes, and this is some kind of graph. I don't see this point of view reflected in the article network theory or more specifically Electric network analysis. --C est moi anton (talk) 17:48, 6 September 2019 (UTC)[reply]

This sounds similar to systems theory/systems engineering (those links don't seem to go to what I expected), which attempts to use universal language to describe electrical, mechanical, hydraulic, pneumatic systems, etc. They have to make quite a few compromises to "make the square pegs fit in the round hole", like not mentioning any components that don't fit the mold. SinisterLefty (talk) 21:57, 6 September 2019 (UTC)[reply]

That seems to be in the right direction. Further down the text, the author mentions that:

"In mechanical applications, a spring corresponds to a capacitor, a mass to an inductor, and a dashpot or damper, such as that used in a door closer to keep the door from slamming, corresponds to a resistor"

So, that implies some sort of striving to generalize the principles to all systems.

--C est moi anton (talk) 22:17, 6 September 2019 (UTC)[reply]

Electrical network theory is alive and well. See Network analysis (electrical circuits), Electrical network and Electrical element for a start. In considering networks of linear passive elements, is is usual to allow ideal transformers (and possibly gyrators) in addition to R, L and C. catslash (talk) 22:37, 7 September 2019 (UTC)[reply]

Mechanical–electrical analogies and Hydraulic analogy may also be of interest.catslash (talk) 23:01, 7 September 2019 (UTC)[reply]

Nice answer catslash , thanks! I'll follow the links.C est moi anton (talk) 20:52, 8 September 2019 (UTC)[reply]