Is .99999.... equal to 1.00000, or not?

Question

In history class one of the TA was referring to some kind of mathematic history and mentioned that no one solved .999..... = 1.0000...... which are both never ending decimals. Then someone yelled 'wrong they're equal' but the TA disagreed.

I just want to know who is right totally curious.

Can someone answer this question? thanks

score 2 · Accepted Answer · answered Sep 18 '15 at 01:07

2

$\begin{align} x &= 0.999\cdots & (A)\\ 10x &= 9.999\cdots & (B)\\ 9x &= 9.000\cdots & (B-A)\\ \therefore x &= 1.000\cdots \end{align}$

answered Sep 18 '15 at 01:07

how did you get B? – K K Lorence Sep 18 '15 at 01:08
I see it now. crystal clear – K K Lorence Sep 18 '15 at 01:10
Simply multiply $A$ by $10$. – Sep 18 '15 at 01:10

score 0 · Answer 2 · answered Sep 18 '15 at 01:07

0

Yes, they are equal. Proof: $$1/3 = 0.333....$$ Then multiply both sides by 3.

answered Sep 18 '15 at 01:07

Brick

1,345

if you multiply both sides you get 1 = 1? – K K Lorence Sep 18 '15 at 01:09
On the left, you get 1. On the right, you get 0.9999..... – Brick Sep 18 '15 at 01:10
i don't get it .... when you multiply 1/3 by 3 you get 1 which is 1.0000 – K K Lorence Sep 18 '15 at 01:14
Exactly. You wanted to know if 1.000.... = 0.9999...., and that's what you got. On the left side, $3 \times (1/3) = 1.000....$. On the right side you get 0.9999..... – Brick Sep 18 '15 at 01:15
oh i didn't think of it that way, my bad i must be stupid thanks though – K K Lorence Sep 18 '15 at 01:19

Is .99999.... equal to 1.00000, or not?

2 Answers2

$\begin{align} x &= 0.999\cdots & (A)\\ 10x &= 9.999\cdots & (B)\\ 9x &= 9.000\cdots & (B-A)\\ \therefore x &= 1.000\cdots \end{align}$