#1**+2 **

what does .99999999999999999999999999999 equel?

hints:

x=.999999999999999999999999999999999

x=1

**It is not so!**

\(x=0.999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999\ \ finite \ decimal \ fraction \\ x=\frac{999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999}{1 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000}\)

\( x=0.\overline{999}..\ periodic \ decimal \ fraction\\x=1\)

!

asinus
Jun 2, 2017

#2**+2 **

I'm pretty sure that you mean what does \(0.\overline{9999}\) equal? I'll use some algebra to show the real value here. This method is well-known, but here it goes anyway:

\(0.\overline{9999}=x\) | I'm going to set this answer equal to a variable. I'll multiply 10 on both sides |

\(9.\overline{9999}=10x\) | This is probably the trickiest step to understand. Subtract \(0.\overline{9999}\) on the left hand side and \(x\) on the right. I can do this because of the first statement I made |

\(9=9x\) | Divide by 9 on both sides |

\(x=1\) | |

Therefore, \(0.\overline{9999}=1=x\).

Now, I have a challenge for you.

\(...9999999=x\)

Using the same algebra I utilized, what does this equal? You should get a bizarre answer

TheXSquaredFactor
Jun 3, 2017

#3**+1 **

I tested the method on another number:

\(0.\overline3=x \\~\\ 3.\overline3=10x \\~\\ 3.\overline3-0.\overline3=10x-0.\overline3 \\~\\ 3.\overline3-0.\overline3=10x-x \\~\\ 3=9x \\~\\ \frac{1}{3}=x\)

It worked! .....Now I will try it on \( \overline9 \) .

\(\overline9=x\)

But what is \( \overline9 \) times 10 ? I need to move the decimal point to the right one place, or add a zero at the "end." But I can't reach the end... it's too far away ! And, if I just try to avoid the issue by writing this...

\(10\,\cdot\,\overline9=10x \\~\\ 10\,\cdot\,\overline9-\overline9=10x-x \\~\\9\cdot\overline9=9x \\~\\ \overline9=x\qquad\text{I get exactly what I started with!}\)

hectictar
Jun 3, 2017