Desafio Matemático #6

[Ninguém 0]

Prove, sem uso de falácias, que a dízima abaixo é igual a 1.

0,999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 (ad infinitum =P)

Editado Março 6, 2012 por Davidson Lima

[Carlos de Alcântara 2.765]

Prove, sem uso de falácias, que a dízima abaixo é igual a 1.

0,999999999999999999 (ad infinitum =P)

Vai numa loja e compra algo que custa R$0,99 usando R$1 e observa se alguém te devolve R$0,01. Provado.

[Iago Sousa 49]

hauahuah, de acordo com a lei do arredondamento se o numero anterior for maior ou igual a cinco, esse numero desaparece e o numero que antecede ele recebe mais uma unidade, no caso 0,99...

= 0,9 +0,1 = 1,0 mas isso todo mundo sabe ._.

[Walter Souza 165]

x = 0.999999...

10x = 9.999999... =>

9x = 9 =>

x = 1

Editado Março 6, 2012 por Walter Souza

[Sillas Tsutsui 14]

Vai numa loja e compra algo que custa R$0,99 usando R$1 e observa se alguém te devolve R$0,01. Provado.

EHaiehiahea

Usando Limites. Acho que o Dechichi já provou no outro tópico...

enfim, o negócio do 0,9999...=1 é um limite, uma aproximação, tentar considerar isso como a igualdade entre dois números é bobagem, isso é de fácil entendimento se você estudar limites de funções, é algo até bem intuitivo... é só pensar numa função contínua que é limitada superiormente por um número x qualquer (1,2,3...), sendo ela estritamente crescente, se vc tender a variável independente dela ao infinito vc vai ver que ela tende à x, mas nunca é exatamente x. Se x for 1 por exemplo, as maiores imagens da função vão ser 0,99999..., mas se vc fazer o limite dela vc vê que da 1, é só isso...

[Ninguém 0]

x = 0.999999...

10x = 9.999999... =>

9x = 9 =>

x = 1

Foi a solução que havia pensado. Carlos é muito bobo hahahaha

Embora considerar que o limite de uma função que contenha um ponto, digamos, 1, fora do domínio da mesma, tende a 1 se aproximarmos, por exemplo, pela esquerda (gerando a dízima periódica 0,99...), isso não prova que 1 = 0,99..., consoante o desafio que propus. Como eu disse, ele tende a 1, nunca efetivamente sendo 1.

[... 15]

0.999... é 1. Não é um limite, os símbolos representam a mesma coisa.

[Ninguém 0]

0.999... é 1. Não é um limite, os símbolos representam a mesma coisa.

Não, não pode ser. Seria 1 se fosse 0,999... + 0,00...1. Enfim, não sou tão bom em matemática (ainda). Se essa é a convenção matemática, tudo bem.

[Diniz - Aposentado 468]

Não, não pode ser. Seria 1 se fosse 0,999... + 0,00...1. Enfim, não sou tão bom em matemática (ainda). Se essa é a convenção matemática, tudo bem.

Marcelo está certo.

[Ninguém 0]

Marcelo está certo.

Olha só, vamos pensar, por exemplo, na solução de um polinômio de grau um. Ele só admite um único zero em sua equação.

f(x) = 1 + x

f(-1) = 1 - 1 = 0

Mas se 0,999... = 1 ---> -0,999... = -1, então admitimos também -0,999... como zero desta função. Vamos testar.

f(-0,999...) = 1 - 0,999... = 0,00...1. Não é zero, TENDE a zero.

Explique, por favor. Além disso, poderíamos admitir duas raízes para uma equação do primeiro grau? E se temos dois números no domínio estando, respectivamente, relacionados com um único número na imagem ( x₁ e x₂ ---> y com uma função f: IR ---> IR) deixa de ser uma função.

Temos um paradoxo. Acabou que a solução do Walter não me satisfez =/

Editado Março 10, 2012 por Davidson Lima

[Diniz - Aposentado 468]

Além disso, poderíamos admitir duas raízes para uma equação do primeiro grau?[/b]

Acho que voce nao leu oq marcelo falou, não sao duas raizes pq são o mesmo número oras, só simbologias diferentes.

É a mesma coisa que voce reclamar i^2 é raiz também, então tem um paradoxo.

E toda vez que voce fala

1 - 0,999... = 0,00...1.

Doi meu coração...

Editado Março 10, 2012 por Diniz

[Ninguém 0]

Acho que voce nao leu oq marcelo falou, não sao duas raizes pq são o mesmo número oras, só simbologias diferentes.

É a mesma coisa que voce reclamar i^2 é raiz também, então tem um paradoxo.

E toda vez que voce fala

Doi meu coração...

Tá, entendi hahahahaha

É que parti da ideia:

Se 1 - 0,9 = 0,1

Se 1 - 0,99 = 0,01

Então 1 - 0,999... = 0,00...1

=P