Interesting Numbers Paradox
In mathematics, an interesting number paradox is a paradox that arises when trying to classify numbers as interesting or uninteresting . A number without any special property is called an uninteresting number , all other numbers are interesting numbers .
The interesting number paradox arises from the fact that there is a conclusive, if not entirely serious, proof that shows that no uninteresting numbers can exist and that there are therefore only interesting numbers.
proof
The short and classic-seeming contradiction proof uses the well-ordering of the natural numbers , which says that every non-empty subset of the natural numbers contains a smallest number.
Suppose there is a non-empty set of uninteresting natural numbers. Then there is also a smallest uninteresting natural number because of the well-order of the natural numbers. This smallest uninteresting natural number is particularly distinguished by its minimality property compared to all other uninteresting numbers and is therefore not an uninteresting natural number. But this contradicts our assumption that it is an uninteresting natural number. Thus our assumption of the existence of uninteresting natural numbers is wrong, there are only interesting natural numbers.
Anecdotes
GH Hardy described the number 1729 as "meaningless" according to an anecdote, but was then explained by S. Ramanujan that this "is the smallest natural number that can be expressed in two different ways as the sum of two cubic numbers" ( see S. Ramanujan # anecdotes ).
A number can be called “boring” if it is not explicitly mentioned in the On-Line Encyclopedia of Integer Sequences . For a long period of time, 8,795 was the smallest boring number in that sense.
literature
- Albrecht Beutelspacher : "I was always bad at math ..." , ISBN 978-3-8348-0774-8 , p. 107, (Chapter: What mathematicians (can) laugh about )
- Edward B. Burger, Michael Starbird, Michael P. Starbird: The Heart of Mathematics: An Invitation to Effective Thinking . Springer 2005, ISBN 978-1-931914-41-3 , p. 44 ( limited online version in Google Book Search - USA )
- Francis Casiro: The Paradox of Jules Richard . In: Spectrum of Science - SPECIAL . Special 2/2005. Spectrum , Heidelberg 2007, p. 40-42 .
- Michael Clark: Paradoxes from A to Z . Routledge 2007, ISBN 978-0-415-42083-9 , p. 100 ( limited online version in Google Book Search - USA )
- Martin Gardner : Hexaflexagons and other Mathematical Diversions . University of Chicago Press 1988, ISBN 978-0-226-28254-1 , p. 148 ( limited online version in Google Book Search - USA )