Wikipedia:Reference desk/Archives/Mathematics/2014 August 31
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August 31
[edit]Trilateral symmetry
[edit]My question relates to a hypothetical sentient lifeform based on trilateral symmetry. Assume their mathematics to be base-9 (since they have 3 digits on each of their 3 appendages; the only reason humans created the decimal system is that we happened to be created with ten "digits"). —The question is: Are irrational numbers such as π and φ irrational for all base systems –in the sense that they cannot be expressed with a finite set of ordinal digits, (or whatever the proper terminology is)? Does this relate to Commensurability, and would this be applicable to all number-base systems (specifically, base-3 and base-9)? —I might not be expressing myself clearly, but hopefully you get the idea. A second (tangentially related) question might best be asked on the computing or science desk, but I'll give it a try here: is there such a thing as a trinary computer based on (null, +/-); translated as (0,1,2) or base-3 (?) ~:71.20.250.51 (talk) 11:08, 31 August 2014 (UTC)
- Actually, humans developed place-value arithmetic three times, with three different bases. The first place-value system was that of the ancient Babylonians, with base 60. The Mayans used base 20. We use so-called Arabic numerals, which were actually invented in India before being adopted by the Arabs, with base 10. The connection of the arithmetic base with evolutionary anatomy would appear to be sort of random. There are still a few vestiges of Babylonian mathematics, such as 60 minutes to a degree and 60 seconds to a minute, reflecting the use of Babylonian mathematics in astronomy and astrology. Except for that specialized use, Babylonian mathematics did not displace the use of non-place-value systems such as Egyptian, Greek, and Roman numerals. It had the advantage (as do Arabic numerals) of permitting calculations with an arbitrary amount of precision. (That is, you can always carry out a long division to as many decimal places or sexagesimal places as you need, which is important for calculating astronomical events.) It had the disadvantage that it was difficult to memorize the addition and multiplication tables.
- However, the question about rational, irrational, and transcendental (incommensurable) numbers has already been answered, which is that rationality does not depend on the base. The axiomatic formulation of mathematics, with Peano postulates, Dedekind cuts, etc., does not depend on the base. Robert McClenon (talk) 19:21, 31 August 2014 (UTC)
- The definition of irrational is that such a number cannot be expressed as the ratio of two integers. Since being an integer doesn't depend on base, being irrational does not depend on base. The fact that the decimal expansion of irrationals is infinite without repetition is a theorem. If you go through the proof, you'll see that it can be repeated in whatever integer base you like. So yes, π's expansion is infinite without repetition in base 9.
- Since being irrational (and similarly, being rational) does not depend on your base, commensurability does not depend on your base. Otherwise, I don't see much of a way in which it's related.--80.109.106.3 (talk) 12:58, 31 August 2014 (UTC)
- (E.C.) Yes, they are still irrational. An irrational number is one that can't be expressed as a fraction -- or ratio -- of two integers, and this definition is irrespective of base. One consequence of this definition, discussed in Irrational number#Decimal expansions, is that an irrational number cannot be expressed as a terminating or repeating expansion in any natural base (decimal, binary, ternary, whatever), while a rational number can be expressed as a terminating or repeating expansion in every base, although any given rational number may have an infinite but repeating expansion in one base and a terminating one in another. For instance, 1/3 = 0.333333... in base 10 and 0.010101... in base 2, but 0.3 in base 9 and 0.1 in base 3.
- For base 3, see our articles Ternary numeral system, Balanced ternary, and Ternary computer. -- ToE 13:09, 31 August 2014 (UTC)
- Thank you, everyone, for your informative replies and links! ~:71.20.250.51 (talk) 00:16, 1 September 2014 (UTC)
Defining a perfect number
[edit]Go to Perfect number. It says:
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n.
It's a provable theorem that the 2 definitions equate. But what I want to know is why the latter definition is preferred by some modern mathematicians. Georgia guy (talk) 13:42, 31 August 2014 (UTC)
- I can't speak for all of those modern mathematicians but moving out any one exception from a definition looks well worth trading in an additional factor somewhere. 95.112.216.113 (talk) 14:22, 31 August 2014 (UTC)
- While I would not think of a uniform exclusion as an exception, there is a pleasing symmetry between:
- A perfect number is a number for which its positive divisors sum to twice the number, and
- A perfect number is a number for which the reciprocals of its positive divisors sum to 2.
- The second statement becomes rather awkward when the reciprocal of the number itself is omitted. —Quondum 19:15, 1 September 2014 (UTC)
- Probably because mathematicians like to reduce things to other things and annex them as much as possible into theories. So they like to write things with predefined functions, like , which they can define in a natural way by Dirichlet convolution as = Id * 1, and they similarly like Dirichlet convolution because it is related to Dirichlet series. Definitions with - Id might look clunkier.John Z (talk) 20:25, 8 September 2014 (UTC)
- While I would not think of a uniform exclusion as an exception, there is a pleasing symmetry between:
Total degree of elementary symmetric polynomials
[edit]One can think of the Fibonacci numbers as the number of integer solutions to x1, x2, ..., xn ≥ 0, x1+x2, x2+x3, ... xn-1+xn ≤ 1, the number solutions being Fn+2. Define S(n,k) as the number of integer solutions to x1, x2, ..., xn ≥ 0, x1+x2, x2+x3, ... xn-1+xn ≤ k. So S(n,0)=1, S(n,1)=Fn+2. (S(n,k) is the value at k of the Ehrhart polynomial of polytope defined by the first set of inequalities.) I computed S for n and k ≤ 7 and found a matching set of values in OEIS: A050446, but I don't understand the description of the entry "total degree of n-th-order elementary symmetric polynomials in m variables," Also, some insight on how S(n, k) might be related to the degree of an elementary symmetric polynomial would be appreciated. --RDBury (talk) 19:45, 31 August 2014 (UTC)