# 1=0.9999...

3 17 182
I always knew I didn't really understand numbers, but I thought I had a pretty good handle on basic arithmetic. So this was a bit of a surprise to me.

It's a given that we can multiply (or add, or subtract, or divide) both sides of an equality by the same factor without violating any of the laws of arithmetic operations. So, for instance:

1/2 = .50
2x(1/2)= 2 x .50
1=1

So, if

1/3 = 0.3333....
and
3 x (1/3) = 3 x 0.3333....
therefore
1 = 0.99999...

(Extracted from Strogatz, S. The Joy of X: A Guided Tour of Math, from One to Infinity (Houghton Mifflin Harcourt 2012)

There's lots of Google stuff under "1 = .9999..." if you're interested in what that really means.

Anúncio Descubra seu nível de inglês em 15 minutos! - Converse grátis por 15 minutos com um professor e verifique como está o seu inglês.

Começar agora!
1 resposta
6 47 1.1k
The problem with Strogatz´ notion (as seen in the above thread) is that it stops somewhere, it doesn´t continues on and on to some place where it converges.

Technically, 0.9999.... equals 1. As proved in Wikipedia as follows:

x = 0.999...
10x = 9.999...
10x-x=9.999... - 0.999...

9x = 9

x = 1
http://en.wikipedia.org/wiki/0.999...

In that page there´s a proof by using limits as well. And other ways.
Ainda precisa de ajuda? Confira algumas opções:
1. Clique no botão "Responder" (abaixo) e faça sua pergunta sobre este assunto;
2. Faça uma nova pergunta;
3. Converse grátis com um professor nativo por 15 minutos: Saiba como!