What are the ways to represent real numbers, such as π?

[Mehmet Kayaalp]

A good read for your students.

http://qr.ae/Rn7Tya

--mehmet

[kirby urner]

This article reminded me of a recent thread on math-teach (Math Forum) where we brainstormed about the possibility of "divergent numbers" which some suggested were just members of N -- but there was no agreement on that.

Consider "mirror pi"  which has all its digits in front of the decimal point. 

We can say "simply write pi in reverse" to get:

... 951413

The three dots show the digits continue indefinitely to the left. 

But then ...2222222 would be such a number as well.

Given such "numbers" have "infinite digits" to the left of the decimal point, don't they all "diverge to infinity"?

Might we get away with saying "... 11111" < "... 22222" or does that make no sense?

Some might argue these are not "numbers" at all.  But isn't the set N infinite? 

How could N be infinite without numbers allowing its members to have infinite digits?  Wouldn't "mirror pi" be one of those?

We have infinite digits for Convergent Numbers (i.e. pi) so why not for the Divergent numbers?

Anyway, fun wheel spinning.... (maybe).

Kirby

[Paul Libbrecht]

This looks like Non-standard-analysis which had an axiomatic working kind of OK on numbers.
It is basically an axiomatic of the reals with the added assumption that there exists an "infinitely small number".
(inverse that, and you would get a diverging number as you suggest I think).
It did not really take up I believe but proofs about limits were always claimed to be of a better qualuty.

Paul

kirby urner

24 juillet 2015 00:32

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