October 19th, 2018, 12:19 PM  #1 
Senior Member Joined: Nov 2011 Posts: 239 Thanks: 2  Infinitesimal
Is Infinitesimal is a "number"?

October 19th, 2018, 12:53 PM  #2 
Senior Member Joined: Oct 2009 Posts: 610 Thanks: 187 
Yes, it is (for example) a hyperreal number or a surreal number. More correctly: the hyperreal and surreal numbers contain infinitesimal numbers, plenty of them. The real numbers have no infinitesimals. So no, if you have an infinitesimal it is NOT a real number. (some people allow 0 to be an infinitesimal, I take it to mean nonzero here) 
October 19th, 2018, 02:11 PM  #3 
Senior Member Joined: Aug 2012 Posts: 2,080 Thanks: 595 
I learned something yesterday. Euclid gave a beautiful example of an infinitesimal, a quantity less than any positive quantity. It's the horn angle of a circle on a line. That is, if you draw a circle tangent to a line, the circumference of the circle makes an angle with the line that is less than the angle made by any other line through that point in the direction of the circle. Of course today we'd say the horn angle is zero, since that's the limit. But it's interesting that Euclid recognized the idea of a quantity that was strictly smaller than any other positive quantity. https://mathcs.clarku.edu/~djoyce/el...propIII16.html 
October 19th, 2018, 02:59 PM  #4 
Senior Member Joined: Oct 2009 Posts: 610 Thanks: 187  We wouldn't though. Only recently Euclid's idea about the horn angle was vindicated. In algebraic geometry today, infinitesimals are allowed. And stuff like the Horn angles become easier to study with infinitesimals. The 1800's might have tried to shun the infinitesimal, in the 1900's we tried to get it to come back to us.

October 19th, 2018, 05:22 PM  #5  
Senior Member Joined: Aug 2012 Posts: 2,080 Thanks: 595  Quote:
In basic analysis the angle between two curves is the limit of their tangents, yes? Which is 0 in this case. Appreciate any clarity on this point.  
October 19th, 2018, 11:28 PM  #6 
Senior Member Joined: Oct 2009 Posts: 610 Thanks: 187  Yeah, that's fine. The angle is definitely 0 in basic analysis and differential geometry. I wasn't contradicting you or anything. I was just trying to say that Euclid's idea of horn angles was very genius, because now we came up with similar things in very advanced topics like algebraic geometry.
Last edited by skipjack; October 20th, 2018 at 05:39 AM. 
October 20th, 2018, 06:55 AM  #7 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  
October 20th, 2018, 07:48 AM  #8  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra  Quote:
Nonstandard analysis makes use of infinitesimals to simplify some concepts and is closely related to how the theory was first developed, but infinitesimals are not required and certainly not fundamental.  
October 20th, 2018, 07:53 AM  #9 
Senior Member Joined: Oct 2009 Posts: 610 Thanks: 187 
I'm all for keeping zylo on this forum and discussing with him in HIS threads. But are we like really going to allow him to come on thread by newcomers and confuse them with his nonsense?


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