+0

# what does .99999999999999999999999999999 equel?

0
169
3

hints:

• x=.999999999999999999999999999999999

• x=1
Guest Jun 2, 2017
Sort:

#1
+7059
+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
+1221
+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
+4716
+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

### 10 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details