Wikipedia:Reference desk/Archives/Mathematics/2016 August 14

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August 14

Proof for some bound

Let A be an array (or a tuple) of n positive integers ordered in non-decreasing order. Is that true that ${\frac {\sum A_{i}}{\max _{i}{A_{i}\cdot (n-i)}}}\leq 2$? עברית (talk) 10:09, 14 August 2016 (UTC)

No. I'm assuming that the indexing starts at 0, which makes it easier to hold, but it still doesn't. Take $A=(3,4,6,12)$. Then $\sum _{i}A_{i}=25$, and $\max _{i}A_{i}(n-i)=12$, giving ${\frac {\sum _{i}A_{i}}{\max _{i}A_{i}(n-i)}}>2.$ -- Meni Rosenfeld (talk) 10:24, 14 August 2016 (UTC)

You can actually make the ratio as large as you want; take $A_{i}={\frac {1}{n-i}}$, you'll get $\max _{i}A_{i}(n-i)=1$ and $\sum _{i}A_{i}=\sum _{i=1}^{n}{\frac {1}{i}}$, which is the harmonic series which diverges logarithmically. I'm fairly certain this also gives a bound on the ratio you can get for a particular n. So the correct bound is ${\frac {\sum A_{i}}{\max _{i}{A_{i}\cdot (n-i)}}}\leq \sum _{i=1}^{n}{\frac {1}{i}}=H_{n}\approx \log n+\gamma$. -- Meni Rosenfeld (talk) 10:30, 14 August 2016 (UTC)

Thank you for the counterexample, and for the bound!!

BTW, do you have any idea about how to prove this bound? עברית (talk) 11:55, 14 August 2016 (UTC)

It's straightforward. Since both numerator and denominator are linear (in A as a whole), you can assume wlog that $\max _{i}A_{i}(n-i)=1$. Under this constraint, the sum and the ratio increase with the entries of A. But the constraint gives you $A_{i}\leq {\frac {1}{n-i}}$, so the maximal value is obtained when $A_{i}={\frac {1}{n-i}}$, and then $\sum _{i}A_{i}=\sum _{i=0}^{n-1}{\frac {1}{n-i}}=\sum _{i=1}^{n}{\frac {1}{i}}$. -- Meni Rosenfeld (talk) 13:37, 14 August 2016 (UTC)

Great proof! Thank you! עברית (talk) 16:42, 14 August 2016 (UTC)

You're welcome. -- Meni Rosenfeld (talk) 20:26, 14 August 2016 (UTC)

Why is there 360 degrees in a circle instead of 5040 degrees?

It seems to me that 5040 degrees in a circle would serve humanity much better than 360 degrees. It has much more factors than 360. Another number I can consider is 25200 which has many useful factors. 175.45.116.61 (talk) 23:32, 14 August 2016 (UTC)

See http://mathforum.org/library/drmath/view/59075.html. —Wavelength (talk) 23:58, 14 August 2016 (UTC)

(EC) You're a few thousand years late to the debate. Degree (angle)#History --Tagishsimon (talk) 00:00, 15 August 2016 (UTC)

We can always find a number that has more divisors. Larger numbers are more difficult to work with (especially before the advent of machine computing). We have to find a compromise. The origins of 360° degrees are historical, but I do not know the details. See Degree (angle)#History and highly composite number.

Even when we find that some convention would better server mankind, we often preserve the old one because it is too difficult to change (social inertia) and the benefit is minimal. Such would be the case with your proposed alternatives to degrees. Examples of sub-optimal conventions: In the SI it would have been better (for consistency) to have the base unit of mass be without prefix, as in “grave” instead of “kilogram”. It would have been better to have decimal divisions of time, instead of the quite arbitrary 60 seconds per minute, 60 minutes per hour and 24 hours per day. Attempts to change this have been made and failed.

Mario Castelán Castro (talk) 00:02, 15 August 2016 (UTC).

(edit conflict) The usual explanation is that the 360-degree circle (and also, incidentally, the 60-minute hour) is a hold over from Babylonian mathematics, which had a roughly Sexagesimal numbering system. Having a better idea isn't necessarily a reason to change it, incidentally. We don't do things because they're the best. We often do things because we've always done it that way. Degree (angle) in Wikipedia and This popular science article also discuss the history in more detail. Also, no "rational" number of any size is better than a system divisible by pi. See radian for an alternate way of describing angles of circle. --Jayron32 00:05, 15 August 2016 (UTC)

You seem to forget the fact that each degree has 60 minutes of 60 seconds each. — 79.118.173.2 (talk) 07:11, 15 August 2016 (UTC)

• 360 divides nicely into 2, 3, 4, 5, 6, 8, 9, 10, 12, 18 and 20, and reasonably nicely into 16. These are basically all the numbers that are convenient for mental arithmetic. The only benefit of using 5040 would be that it would also divide by 7 and its multiples. Seven was not especially useful for ancient/medieval counting systems (7 is odd, it's prime and fairly large, which means it doesn't appear that often, and it doesn't have the advantage that 5 had of coincidentally being the number of fingers on a hand) so there just wasn't much advantage to a 5040° circle, especially given. You can see how rare these odd factors were from the bases of imperial measurements and currencies. Foot = 12 inches, yard = 3 feet, pound = 16 ounces, Troy pound = 12 Troy ounces, imperial pint = 20 fluid ounces, US pint = 16 fluid ounces, shilling = 12 pence, pound = 20 shilling, Spanish dollar ("pieces of eight") = 8 real. Similarly Ancient Mesopotamian units of measurement, which the inventors of the 360° convention used, all have bases like 2, 3, 6, 12, 60, 120, 360, etc. The factor of 7 just wasn't important. A couple of minor exceptions: the stone, which is 14 pounds, but a) that was only fixed in the 19th century and b) it's rare that you need to do a sum that involves both mass and angles outside physics; the guinea, which is technically 21 shillings, but is really 1 pound + 1 shilling, and exists to make it easier for people to take a 5% brokers fee; and the days in the week, but these were fixed by religious convention and by the awkward fact that a lunar month is about 28/29 days. Nevertheless, according to our article 5040 (number), Plato (who was a big fan of mathematical perfection) did suggest using 5,040 as the base of society – presumably it didn't catch on precisely because of its unwieldiness. Smurrayinchester 09:39, 15 August 2016 (UTC)

The idea that the Babylonians divided the hour sexagesimally is comprehensively debunked in the relevant Wikipedia articles. Al - Biruni was first to do this at the end of the first millennium. Under his system

1 hour = 60 minutes = 3 600 seconds = 216 000 thirds = 18 960 000 fourths.

In practice, the smallest unit is the second, divided decimally. 86.150.12.166 (talk) 14:18, 15 August 2016 (UTC)

Is your "divides nicely into 16" a typo for 15?? 360 is divisible by 15 (it's 15 times 24.) 352 and 368 are divisible by 16. Georgia guy (talk) 14:55, 15 August 2016 (UTC)

I missed 15. By "reasonably nicely", I mean that 360 / 16 = 22.5. Not an integer, but it terminates, and halves are easy to work with. Smurrayinchester 22:01, 15 August 2016 (UTC)

While thinking about the optimal number of (integer) degrees there should be in a circle, it occurred to me that the numbers of factors is less important that the cost of dividing the number. For example: while the number 25200 has more factors than the number 252, it is just as easy to divide 252 by another integer as it is to divide 25200 by the same integer.

25200 / 100 = 252

252 / 100 = (252 * 100) / 100 / 100 = 252 / 100 = 2.52

That is to say the cost of "25200 / 100" is three symbols {2,5,2} where as the cost of "252 / 100" is four symbols {2,DOT,5,2}

Therefore, what I want to find is an integer number whereby the totalcost by dividing it with "various" integers is the lowest.

Of course there is a problem

252 / 19 = 13.2631578947368...

Here the cost seems to be an infinite number of symbols. The solution is to limit it to a maximum cost of 10 symbols.175.45.116.105 (talk) 03:26, 19 August 2016 (UTC)