# Road where each section 1/10 of the last section.

1. Aug 4, 2014

### andreasdagen

Imagine having a road thats going straight forward, where each section is 1/10 the size of the last section. The sections would be blue, then red, then blue and so on.

Assuming the size of the first section is one meter, the road would be 1.111... meters repeating.

Each time you drive across a section, there would be another one 1/10 the size so you could never reach the end.

Im just wondering if this has a name and if it would be possible to reach the end of the road.

(im only 15, and im from norway so sorry for my bad english)

2. Aug 4, 2014

3. Aug 4, 2014

### sophiecentaur

Hi and welcome.
Of course you can get to the end of the road as it is always shorter than 1.2m, however many sections you add. (It's shorter than 1.11111112 even). Have you looked at Convergent Series in Maths yet?
Your question is like the Zeno[/PLAIN] [Broken] Paradox (an ancient Greek idea with a flaw that is easy to discover)

Last edited by a moderator: May 6, 2017
4. Aug 4, 2014

5. Aug 4, 2014

### rcgldr

Note, the road would be 10/9ths of a meter long (10/9 = 1.11111...).

6. Aug 4, 2014

### Staff: Mentor

http://mathforum.org/isaac/problems/zeno1.html

Zeno was an ancient Greek philosopher, so this problem has a long history! Basically, in order to resolve it you need to use concepts related to calculus and the summation of infinite series. And yes, the runner does reach the end of the road (and Achilles does catch up with the tortoise, etc.).

http://www.mathcs.org/analysis/reals/history/zeno.html

http://www.iep.utm.edu/zeno-par/

A Google search for "Zeno's paradoxes" turns up many pages. Some of them might be simpler than the two listed above.

(Wow, four people got in ahead of me while I was Googling and writing. I think that's a record. )

7. Aug 4, 2014

### Staff: Mentor

Some fun with algebra (the mathematicians among us may wish to avert their eyes at this point):

Let $x$ be the length of the road. Now we have:
$x=1.111111......$
$10x=11.111111.....$

$10x-x= 11.111111...... -1.111111..... =10$

$9x=10$
$x=10/9$

So the repeating decimal is just a red herring; the road has a perfectly reasonable length and you should be able to traverse it just as if it had any other length. Of course, you still have to deal with Zeno's paradox as the other posters have mentioned; it suggests (incorrectly, of course) that nohing can ever traverse any distance ever.

Last edited: Aug 4, 2014