+14915 what does .99999999999999999999999999999 equel?
hints:
x=.999999999999999999999999999999999
x=1
It is not so!
\(x=0.999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999\ \ finite \ decimal \ fraction \\ x=\frac{999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999 \ 999}{1 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000 \ 000}\)
\( x=0.\overline{999}..\ periodic \ decimal \ fraction\\x=1\)
!
+2441 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
+9466 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...
\(10\,\cdot\,\overline9=10x \\~\\ 10\,\cdot\,\overline9-\overline9=10x-x \\~\\9\cdot\overline9=9x \\~\\ \overline9=x\qquad\text{I get exactly what I started with!}\)
