January 14th, 2016, 04:09 PM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra 
Oh yes, I agree with that. My main problem with the hyperreals is that they don't seem to add any value. The only "benefit" I recall hearing about being that they can make calculus accessible to people who don't understand limits. To which my response is: well teach them limits in a clearer fashion then! Or, don't bother teaching the rigorous foundations at all.

February 23rd, 2016, 04:57 AM  #12 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
It has a historical value When Sir Isaac Newton developed the Calculus, he used 'infinitesmals' without properly defining them. Hyperreal numbers allow one to rigorously define those 'infinitesmals'.

February 23rd, 2016, 06:44 AM  #13 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2665 Math Focus: Mainly analysis and algebra 
Indeed so (and Leibnitz used them too). Infinitesimals show that their intuition that infinitesimals could be used to penetrate the subject was well founded. But unfortunately the modern theory retains the core objection to those infinitesimals by validating the idea that you just ignore them when you are ready to transport your result to the reals. It's all perfectly rigorous but just as unsatisfying now as it was 300 years ago.


Tags 
difference, hyperreal, numbers, real 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Proof difference of two natural numbers  godingly  Calculus  0  November 4th, 2014 10:06 AM 
First proof!  The difference of two numbers is scalable  ryanninjasheep  Applied Math  1  February 28th, 2014 03:55 PM 
Difference between real analytic and holomorph?  ChidoriPOWAA  Complex Analysis  2  April 2nd, 2013 11:58 PM 
Prove the difference between numbers  stevecowall  Algebra  1  March 27th, 2012 06:47 PM 
Find two numbers where difference is 20 & the sum is 4  sharp  Elementary Math  7  October 30th, 2010 08:52 AM 