How do I express 0.999(9) as a fraction?

Question

I'm noob in math. If 0.333(3) is 1/3, 0.666(6) is 2/3, then 0.999(9) is what?

If 3/3 and 0.999(9) is the same, then how can I express one of them without expressing the other?

Answer 1

Well, notice that:

$$\text{n}_1:=0.333\dots=0.\overline{3}=\frac{1}{3}\tag1$$

So, when we muliply $\text{n}_1$ by two, we get:

$$\text{n}_2:=2\cdot0.333\dots=0.666\dots=0.\overline{6}=\frac{2}{3}\tag2$$

So, when we muliply $\text{n}_1$ by three, we get:

$$\text{n}_3:=3\cdot0.333\dots=0.999\dots=0.\overline{9}=\frac{3}{3}=1\tag3$$

Where $\overline{x}$ means a repeating decimal.

Answer 2

$ 0.999 \ldots = 1 $

Proof: Let $x = 0.999\ldots $ . Now consider that $10x = 9.999\ldots$ . So then we have that $9x = 10x -x = 9.99\ldots -0.999\ldots = 9.0$ .

Thus $9x/9 = 9/9 = 1 $.