Wikipedia:Reference desk/Archives/Mathematics/2020 January 24

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January 24

8 x 10^67 =

Please, eight times ten to the power of 67 equals? I have read that this is the number of combinations that a pack of cards can be shuffled into. Is this equation correct? I would have thought this would have been 52×52×52? Clearly my math skills are lacking, please help. Thanks. Anton 81.131.40.58 (talk) 09:12, 24 January 2020 (UTC)

The number of arrangements of a pack of cards is 52! or factorial 52, i.e. 52×51×50×...×3×2×1. This is because there are 52 ways to choose the first card, then 51 for the second, and so on. 52! is approximately 8 × 10^67, which is 8 followed by 67 zeroes, or more accurately (but not exactly) 8.0658175 × 10^67. See permutations for more explanation. AndrewWTaylor (talk) 09:28, 24 January 2020 (UTC)

... and to give you an idea of scale, this is roughly the same magnitude as the number of protons and neutrons in a galaxy the size of the Milky Way (give or take a few powers of 10). Gandalf61 (talk) 10:00, 24 January 2020 (UTC)

I have replaced letter x with a multiplication symbol × in maths above. --CiaPan (talk) 10:08, 24 January 2020 (UTC)

So that's 80,000, 000,000,000, 000,000,000, 000,000,000, 000,000,000, 000,000,000, 000,000,000, 000,000,000? Thanks. Anton 81.131.40.58 (talk) 11:44, 24 January 2020 (UTC)

Yes, the number you put in the title is precisely that. But 52! (the number of possible orderings of a deck of 52 cards) is about 0.82% more than that. --CiaPan (talk) 11:53, 24 January 2020 (UTC)

See Scientific notation for more info on the notation. --RDBury (talk) 11:54, 24 January 2020 (UTC)

The exact value of 52! is 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. PrimeHunter (talk) 12:26, 24 January 2020 (UTC)

Question regarding sequences

Does the sequence a(n) = a(n-1) + 1/n have a limit?

Thanks, BobsFullBogs (talk) 16:58, 24 January 2020 (UTC)

@BobsFullBogs: Nope. The recursive rule you gave resolves to

$a_{n}=a_{0}+\left({\frac {1}{1}}+{\frac {1}{2}}+\cdots +{\frac {1}{n}}\right)$

The numbers summed in parentheses are terms of a Harmonic progression, so the parentheses' values (those sums) make the Harmonic series – which is known to be divergent, that is: having no limit. --CiaPan (talk) 17:10, 24 January 2020 (UTC)

To be precise, it has no limit in the real numbers. It does have a limit in the extended real numbers, namely +∞. This is sometimes a useful distinction to make; there are sequences that have no limit in the extended real numbers. --Trovatore (talk) 23:43, 25 January 2020 (UTC)

An example is... Georgia guy (talk) 23:53, 25 January 2020 (UTC)

$(0,1,0,1,0,1,\ldots )$ –Deacon Vorbis (carbon • videos) 23:57, 25 January 2020 (UTC)

grr...you beat me to it while I was looking up HTML entities. My example was +1, −1, +1, −1, +1, −1, …. --Trovatore (talk) 00:02, 26 January 2020 (UTC)

Product of primes

Is there a recognised function defined as the product of all primes up to and including a given prime?

Ex: F(13) = 2 x 3 x 5 x 7 x 11 x 13 = 30030.

Thanks. -- Jack of Oz ^{[pleasantries]} 21:47, 24 January 2020 (UTC)

Primorial appears to be what you're looking for. --Wrongfilter (talk) 21:59, 24 January 2020 (UTC)

Ah, thank you. -- Jack of Oz ^{[pleasantries]} 23:03, 25 January 2020 (UTC)

Incidentally, I have a website about prime number records and use the primorial symbol '#' 600 times just on http://primerecords.dk/aprecords.htm. 23 of them are your example 13# = 2 x 3 x 5 x 7 x 11 x 13. I created Primes in arithmetic progression where it's also used many times. PrimeHunter (talk) 17:25, 26 January 2020 (UTC)

I acknowledge your primacy.  :) -- Jack of Oz ^{[pleasantries]} 23:49, 27 January 2020 (UTC)