dividing infinitesimal

[shahar]

Can I divide one infinitesimal by another infinitesimal? Is it a legal operation?

 

[pka]

Can I divide one infinitesimal by another infinitesimal? Is it a legal operation?

In Elementary Calculus: An Infinitesimal Approach​

In Elementary Calculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this. The chapters and whole book is a free down-load at http://www.math.wisc.edu/~keisler/.
Go to Keisler's website and download the first two chapters or even the whole textbook.
Keisler and his students literally wrote the text on this material

 

[Deleted member 4993]

Can I divide one infinitesimal by another infinitesimal? Is it a legal operation?

The calculation of derivative has been legal since Liebniz discovered those monsters.

 

[topsquark]

The calculation of derivative has been legal since Liebniz discovered those monsters.

Well, dy/dx can be treated as if it were a division, but it really isn't one.

-Dan

 

[Deleted member 4993]

Well, dy/dx can be treated as if it were a division, but it really isn't one.

-Dan

I know - but before we take the limit - those are differentials I believe we can do the division then.

 

[topsquark]

I know - but before we take the limit - those are differentials I believe we can do the division then.

Ummmm..... Perhaps I'm wrong then. I thought dx was an example of an infinitesimal. I do know that it took at least a century before Mathematicians could prove that you can use dy and dx as "numbers" in their own right... Addition, multiplication, division etc.

-Dan