The Ant And The Rubber Band 


My lastest posts can be found here: Previous blog posts:
Additionally, some earlier writings: 
The Infinitely Patient Ant and

Of course, it's not always an extra 1 cm that the ant gets "for free"
from the stretch of the rubber band. This is the hard bit! Just how
do you work out what's really happening!
If you're struggling, make a model, draw a picture, don't just stare into space and try to visualise it because you won't. Well, probably won't. This is the entire challenge in maths, trying to understand what's happening, and just staring at it won't help. You have to play with the problem. 
Then it walks another cm, so now it's 3 whole centimetres from its starting point. Then the rubber band gets stretched out another metre to a total of three metres long, again, dragging the ant a little "for free". (Tip: Think hard about where it is now)
And so it continues. An extra cm of walking, then the endpoint moved by another metre. Will it ever get there?
More about that next time.
So I thought I'd reply to a few of the comments from my previous post  Irrationals Exist. Someone was checking on what exactly I had shown. and had I shown that an irrational existed in every interval.
Yes. Between any two reals (or algebraics or rationals or integers) there exists an irrational. That's what I proved.
Someone else (Thanks JorgeNuno!) said I'd used the reals to show that the reals existed! What I actually did (and of course he knows it) is to say that if we start with the rationals, we can produce a sequence that should have a limit, but for which no rational can be that limit. After all, we want our number line to be "complete", and if we only have rationals, it isn't.
Another observation is that I never really used any of the properties of the rationals. I'll say more about that next time.
Irrationals Exist  :  Revisiting The Ant ... 
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