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The first example is pretty cool... Not to nitpick, but as he states the problem: Could there be two squares with side [sic] equal to a whole number,

The first example is pretty cool... Not to nitpick, but as he states the problem: Could there be two squares with side [sic] equal to a whole number, n, whose total area is identical to that of a single square with side equal to another whole number, m? Given that he's speaking of whole numbers, the number zero comes to mind, which satisfies this. So his whole proof is shot. -- John Conway's greatest contribution to my life (as opposed to the game of life), and one I use about five times a week is The Doomsday Rule: http://en.wikipedia.org/wiki/Doomsday_rule http://en.wikipedia.org/wiki/Doomsday_rule

dvanduzer 15y ago
It is not non-controversial to include zero in the set of whole numbers. A square with side n, where n=0 would be a point. Except it wouldn't, because we have to define these things differently, or geometry would be incoherent. Incidentally, there is no need for a [sic] there because you only need one side to define a square.
kevinalexbrown 15y ago
> So his whole proof is shot. Not really. Just add the special case of zero. If you want to sound cool say "There are no two nontrivial squares with side ..."
Arjuna 15y ago
"Given that he's speaking of whole numbers, the number zero comes to mind, which satisfies this. So his whole proof is shot." Both Wikipedia [1] and Wolfram [2] indicate that there are varying interpretations regarding which integers are included in the definition of the term whole number. Since Conway is discussing two-dimensional distance in order to determine area, his definition of the term would not include integers that are less than or equal to zero. [1] http://en.wikipedia.org/wiki/Whole_number http://en.wikipedia.org/wiki/Whole_number "Whole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts. Whole numbers may refer to: natural numbers in sense (1, 2, 3, ...) - the positive integers natural numbers in sense (0, 1, 2, 3, ...) - the non-negative integers all integers (..., -3, -2, -1, 0, 1, 2, 3, ...)" [2] http://mathworld.wolfram.com/WholeNumber.html http://mathworld.wolfram.com/WholeNumber.html "0 is sometimes included in the list of "whole" numbers (Bourbaki 1968, Halmos 1974), but there seems to be no general agreement."
- slowpoke 15y ago
  Where I study, it's a general consensus that in Math, N does not include 0 unless explicitly included as N index 0. In CompSci on the other hand, 0 is assumed to be included for practical and technical reasons.
inopinatus 15y ago
No, that is a nitpick, I'm afraid. If you think otherwise may I invite you to draw a square with sides of length zero. Hint: it doesn't exist. Second hint: it is trivial to demonstrate that it doesn't exist. Consider the triangle. Can you define it as a quadrilateral with one side of length zero? What is the angle at the null vertex? Proceed from there. Finally, the problem states "another whole number". Zero is not different from zero. As with any exam, you should read the question carefully first.

dvanduzer 15y ago

It is not non-controversial to include zero in the set of whole numbers. A square with side n, where n=0 would be a point. Except it wouldn't, because we have to define these things differently, or geometry would be incoherent. Incidentally, there is no need for a [sic] there because you only need one side to define a square.

kevinalexbrown 15y ago

> So his whole proof is shot. Not really. Just add the special case of zero. If you want to sound cool say "There are no two nontrivial squares with side ..."

Arjuna 15y ago

"Given that he's speaking of whole numbers, the number zero comes to mind, which satisfies this. So his whole proof is shot." Both Wikipedia [1] and Wolfram [2] indicate that there are varying interpretations regarding which integers are included in the definition of the term whole number. Since Conway is discussing two-dimensional distance in order to determine area, his definition of the term would not include integers that are less than or equal to zero. [1] http://en.wikipedia.org/wiki/Whole_number http://en.wikipedia.org/wiki/Whole_number "Whole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts. Whole numbers may refer to: natural numbers in sense (1, 2, 3, ...) - the positive integers natural numbers in sense (0, 1, 2, 3, ...) - the non-negative integers all integers (..., -3, -2, -1, 0, 1, 2, 3, ...)" [2] http://mathworld.wolfram.com/WholeNumber.html http://mathworld.wolfram.com/WholeNumber.html "0 is sometimes included in the list of "whole" numbers (Bourbaki 1968, Halmos 1974), but there seems to be no general agreement."

slowpoke 15y ago
Where I study, it's a general consensus that in Math, N does not include 0 unless explicitly included as N index 0. In CompSci on the other hand, 0 is assumed to be included for practical and technical reasons.

inopinatus 15y ago

No, that is a nitpick, I'm afraid. If you think otherwise may I invite you to draw a square with sides of length zero. Hint: it doesn't exist. Second hint: it is trivial to demonstrate that it doesn't exist. Consider the triangle. Can you define it as a quadrilateral with one side of length zero? What is the angle at the null vertex? Proceed from there. Finally, the problem states "another whole number". Zero is not different from zero. As with any exam, you should read the question carefully first.