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# what does .99999999999999999999999999999 equel?

0
506
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hints:

• x=.999999999999999999999999999999999

• x=1
Guest Jun 2, 2017
#1
+7493
+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
+2248
+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
+7324
+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
edited by hectictar  Jun 3, 2017