Are these numbers different from each other?

Question

Let $$x=0.252525252...$$ and $$y=0.25252525...$$

Can we represent these numbers as $x=0.2\dot5\dot2$ and $y=0.\dot2\dot5$ ?

For me they are the same.

But when we think about the last digit $(\infty)^{\text{th}}$ digit of $x$ , it will be $2$. But for $y$ it will be $5$.

What is the difference I can't understand here.

Answer 1

$$x=\dfrac2{10}+\dfrac{52}{990}=\dfrac{198+52}{990}=\dfrac{25}{99}=y$$

Answer 2

Firstly you are correct that $x=0.2\dot5\dot2$. However, $y=0.25255\dot2\dot5\neq 0.\dot2\dot5$ since there is an extra $5$ in the middle.

Thus $$y-x=0.25255\dot2\dot5-0.25252\dot5\dot2=0.0000\dot2\dot73\neq0$$

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Now that you've corrected $y=0.25252525...=0.\dot2\dot5$, of course $x=y$. It's a bit like saying $$\frac19=\overbrace{0.111111...}^{0.\dot1\dot1}=\overbrace{0.11111...}^{0.\dot1}$$

Answer 3

Of course they aren't. They fifth digit after the dot in $x$ is 2 while the fifth digit after the dot in $y$ is 5.

Answer 4

Periodic numbers can be written in many different ways. For example $0.\dot4=0.4\dot4=0.44\dot4\dot4\dot4$. If you try to write them as fractions, they will be all the same when written in their lowest terms.

Namely,

$$0.\dot2\dot5=\frac{25}{99}$$ $$0.2\dot5\dot2=\frac{252-2}{990}=\frac{250}{990}=\frac{25}{99}$$

Nevertheless, the most usual way to write a periodic number is considering that its period starts as "soon" as possible and is as short as possible.