Comments on: A mathematical fable previsited http://bit-player.org/2007/a-mathematical-fable-revisited An amateur's outlook on computation and mathematics. Fri, 29 Aug 2008 07:49:21 +0000 http://wordpress.org/?v=1.5.2 Comment on A mathematical fable previsited by: Barry Cipra http://bit-player.org/2007/a-mathematical-fable-revisited#comment-1460 Fri, 04 May 2007 19:14:33 +0000 http://bit-player.org/2007/a-mathematical-fable-revisited#comment-1460 Saïd Boutiche writes: "A brief comparison of these 3 methods shows that the Alcuin method is certainly the most elegant since it involves only 3 operations." Actually the Alcuim method involves a fourth, preliminary operation: 4 = int(9/2). This is perhaps more obvious if one asks for a more arbitrary sum, say 1+2+...+1776. Determining the number of pairs (1+1775), (2+1774), etc. clearly requires a nontrivial calculation, at least in our base 10 system -- it would be an easy calculation if we wrote everything base 2! Saïd Boutiche writes:

“A brief comparison of these 3 methods shows that the Alcuin method is certainly the most elegant since it involves only 3 operations.”

Actually the Alcuim method involves a fourth, preliminary operation: 4 = int(9/2). This is perhaps more obvious if one asks for a more arbitrary sum, say 1+2+…+1776. Determining the number of pairs (1+1775), (2+1774), etc. clearly requires a nontrivial calculation, at least in our base 10 system — it would be an easy calculation if we wrote everything base 2!

]]> Comment on A mathematical fable previsited by: Saïd Boutiche http://bit-player.org/2007/a-mathematical-fable-revisited#comment-1459 Fri, 04 May 2007 15:21:06 +0000 http://bit-player.org/2007/a-mathematical-fable-revisited#comment-1459 In mathematics, "elegance" is often associated with the concept of "simplest way" to find a solution for a problem. To illustrate the meaning of "simplest way" in mathematics, I reconsider here the above S calculation for n=10: Let consider first the standard calculation: S = 1+2+3+4+5+6+7+8+9+10 This "standard evaluation" of S, involves a number of 9 operations that is equal to the number of (+) symbols. The calculation of S according to Archimedes (or Heath) starts with 2S = (1+9) + (2+8) + (3+7) + (4+6) + (5+5) + (6+4) + (7+3) + (8+2) + (9+1) + 10 + 10. It involves 4 operations: (3 operations that find 2S= 9x10 +10+10) followed by the one that finds S (S=(10 x 9 +10+10)/2.). While the Alcuin sum S = (1+9) + (2+8) + (3+7) + (4+6) + 5 + 10 = 4 x 10+5+10 involves only 3 operations. A brief comparison of these 3 methods shows that the Alcuin method is certainly the most elegant since it involves only 3 operations. One can ask however, why the number of operations becomes considerably reduced in the Archimedes and Alcuin methods? Both methods use a spatial redistribution of numbers that is "easily accessible" to our perception and sense of evaluation. In mathematics, “elegance” is often associated with the concept of “simplest way” to find a solution for a problem. To illustrate the meaning of “simplest way” in mathematics, I reconsider here the above S calculation for n=10:
Let consider first the standard calculation:
S = 1+2+3+4+5+6+7+8+9+10
This “standard evaluation” of S, involves a number of 9 operations that is equal to the number of (+) symbols.
The calculation of S according to Archimedes (or Heath) starts with 2S = (1+9) + (2+8) + (3+7) + (4+6) + (5+5) + (6+4) + (7+3) + (8+2) + (9+1) + 10 + 10.
It involves 4 operations: (3 operations that find 2S= 9×10 +10+10) followed by the one that finds S (S=(10 x 9 +10+10)/2.).
While the Alcuin sum S = (1+9) + (2+8) + (3+7) + (4+6) + 5 + 10 = 4 x 10+5+10 involves only 3 operations.
A brief comparison of these 3 methods shows that the Alcuin method is certainly the most elegant since it involves only 3 operations.
One can ask however, why the number of operations becomes considerably reduced in the Archimedes and Alcuin methods? Both methods use a spatial redistribution of numbers that is “easily accessible” to our perception and sense of evaluation.

]]> Comment on A mathematical fable previsited by: colm mulcahy http://bit-player.org/2007/a-mathematical-fable-revisited#comment-1458 Fri, 04 May 2007 09:41:02 +0000 http://bit-player.org/2007/a-mathematical-fable-revisited#comment-1458 I'm curious that this surprised y ou: "a Web posting by Saïd Boutiche, dated 2005, also represents 1+2+3+…+100 as (49 × 100) + 50 + 100, and yet there is no evidence at all of influence from Archimedes or Alcuin.)" I find this approach very natural despite the lack of symmetry... the lure of the perfect totals of 100... and I am not alone: I have this problem to a smart 12 year old in my daughter's class as we hiked in the Panama rainforest in March, and he had it within minutes, no paper in sight. At first he didn't realize that he'd overlooked 50 and 100, but he found that method much easier to grasp than pairing numbers to add to 101, which I explained to him later. I’m curious that this surprised y ou:

“a Web posting by Saïd Boutiche, dated 2005, also represents 1+2+3+…+100 as (49 × 100) + 50 + 100, and yet there is no evidence at all of influence from Archimedes or Alcuin.)”

I find this approach very natural despite the lack of symmetry… the lure of the
perfect totals of 100… and I am not alone:

I have this problem to a smart 12 year old in my daughter’s class as we hiked
in the Panama rainforest in March, and he had it within minutes, no paper in
sight. At first he didn’t realize that he’d overlooked 50 and 100, but he found
that method much easier to grasp than pairing numbers to add to 101, which
I explained to him later.

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