| dc.contributor | Universidade Estadual Paulista (UNESP) | |
| dc.creator | Pegg Jr, Ed | |
| dc.date | 2011-05-26T19:54:14Z | |
| dc.date | 2011-05-26T19:54:14Z | |
| dc.date | 2011-05-26 | |
| dc.date.accessioned | 2017-04-05T16:59:56Z | |
| dc.date.available | 2017-04-05T16:59:56Z | |
| dc.identifier | http://acervodigital.unesp.br/handle/123456789/4270 | |
| dc.identifier | http://objetoseducacionais2.mec.gov.br/handle/mec/9139 | |
| dc.identifier.uri | http://repositorioslatinoamericanos.uchile.cl/handle/2250/831681 | |
| dc.description | Knowledge about algorithms, historical mathematics, integers, number theory, rational numbers, representations of numbers and theorem proving | |
| dc.description | In 1834, Johann Dirichlet noted that if there are five objects in four drawers then there is a drawer with two or more objects. The Schubfachprinzip, or drawer principle, got renamed as the pigeonhole principle, and became a powerful tool in mathematical proofs. Pick a number that ends with 1, 3, 7, or 9. Will it evenly divide a number consisting entirely of ones (a repunit)? Answer: yes. Proof: Suppose 239 was chosen. Take the remainder of 239 dividing 10, 100, 1000, ..., 10^239. The chosen number will not divide evenly into any of those 239 powers of 10, so there are 238 possible remainders, 1 to 238. By the pigeonhole principle, two remainders must be the same, for some 10^a and 10^b. As it turns out, 10^3 and 10^10 both give remainder 44. Subtracting, 9,999,999,000 is the result, which yields 1,111,111 when divided by 9000. When b>a, 10^b-10^a always returns an all-1 number multiplied by 9 and some power of 10, finishing the proof. The reciprocal of the chosen number has a repeating decimal of similar length. Consider: 1,111,111/239 = 4649. 4649 x 9 = 41 841. 1/239 = .00418410041841(...) | |
| dc.description | Componente Curricular::Ensino Fundamental::Séries Finais::Matemática | |
| dc.publisher | Wolfram Demonstrations Project | |
| dc.relation | 238ThePigeonholePrincipleRepunits.nbp | |
| dc.rights | Demonstration freeware using Mathematica Player | |
| dc.subject | Educação Básica::Ensino Fundamental Final::Matemática::Números e operações | |
| dc.subject | Algorithm | |
| dc.subject | Historical mathematics | |
| dc.subject | Number theory | |
| dc.subject | Rational number | |
| dc.subject | Representation of number | |
| dc.subject | Theorem proving | |
| dc.subject | Princípio matemático | |
| dc.subject | Prova matemática | |
| dc.subject | Teoria da prova | |
| dc.title | The pigeonhole principle - repunits | |
| dc.type | Software | |
