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November 7th, 2017, 06:02 AM  #1 
Senior Member Joined: Nov 2011 Posts: 116 Thanks: 1  Infinitesimal
Is Infinitesimal a number? [The lowest number that exists or epsilon is the lowest number that exist?]
Last edited by skipjack; November 10th, 2017 at 04:27 AM. 
November 7th, 2017, 06:14 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,034 Thanks: 2342 Math Focus: Mainly analysis and algebra 
I think you mean "smallest" rather than "lowest". If $\epsilon$ were the smallest number, what would $\frac\epsilon2$ be? Infinitesimals don't exist in standard analysis. But this being mathematics, where we set the rules and see what emerges, we can declare the existence of an $\epsilon > 0$ such that $\epsilon < r$ for every positive real number $r$. That results in the Hyperreal numbers which form the basis of (a) nonstandard analysis. Last edited by skipjack; November 10th, 2017 at 04:28 AM. 
November 10th, 2017, 04:12 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,824 Thanks: 752 
Even in nonstandard analysis, where infinitesimals are defined, there is NO "smallest" number. There is no smallest infinitesimal.
Last edited by skipjack; November 10th, 2017 at 04:29 AM. 
November 10th, 2017, 04:24 AM  #4 
Senior Member Joined: Jun 2015 From: England Posts: 697 Thanks: 199 
Infinitesimals find more use in Engineering and Physics, where they are very useful, than in Mathematics, where they are frowned on.

November 10th, 2017, 04:41 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,034 Thanks: 2342 Math Focus: Mainly analysis and algebra  
November 10th, 2017, 06:00 AM  #6 
Senior Member Joined: Jun 2015 From: England Posts: 697 Thanks: 199  
November 10th, 2017, 06:06 AM  #7 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,074 Thanks: 695 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Yeah, they are used in perturbation theory. For example, a density, concentration or temperature perturbation in a fluid element leads to convective mass transport as a natural consequence of adding an infinitesimal change into the equations related to the equation of state of the medium.

November 10th, 2017, 10:14 AM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,034 Thanks: 2342 Math Focus: Mainly analysis and algebra 
I thought perturbation theory used small changes and approximations rather than trying to claim exact results via infinitesimals.

November 10th, 2017, 10:39 AM  #9  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,659 Thanks: 652 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan Last edited by topsquark; November 10th, 2017 at 10:43 AM.  
November 10th, 2017, 10:57 AM  #10 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,034 Thanks: 2342 Math Focus: Mainly analysis and algebra  

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