February 7th, 2015, 10:46 AM  #1 
Newbie Joined: Feb 2015 From: England Posts: 1 Thanks: 0  Trial and Improvement Equation
Hallo, My friend who is from a foreign country never heard of the little GCSE practice we call Trial and Improvement, he thinks its useless and not mathematical. (as it is just guessing and could not provide a correct answer) He claims there would be a different way to answer all trial and improvement equation, Can anyone think of one equation that only can be solved using the trial and improvement method? I am genially now questing the usefulness of this method on the other had would be sweet to prove him wrong. 
February 7th, 2015, 10:56 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,550 Thanks: 2550 Math Focus: Mainly analysis and algebra 
There are other methods and they are generally better, but none is as simple to understand as trial and improvement. There are probably no problems that can only be solved in this way, but it is a good introduction to the concept of approximating solutions. Having said that, the iteration $$a_{n+1} = \frac12 \left(a_n + {b \over a_n}\right)$$ is just about the most efficient approximator for $\sqrt b$ and uses basically the same idea. It was used by the Babylonians around 2000 years ago. Last edited by v8archie; February 7th, 2015 at 11:06 AM. 
February 7th, 2015, 12:05 PM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,780 Thanks: 970 
Archie, help please... I can tell (by quoting your post) that you used this: $$a_{n+1} = \frac12 \left(a_n + {b \over a_n}\right)$$ However, [Math Processing Error] in red ink appears where the result of above should appear; do you have any idea what's wrong with my computer: like, Java not updated or whatever? I see that someone else brought up something similar: Math Error Thanks for any help, suggestions, .... Last edited by Denis; February 7th, 2015 at 12:09 PM. 
February 7th, 2015, 12:10 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,550 Thanks: 2550 Math Focus: Mainly analysis and algebra 
Try refreshing the page. It's all JavaScript, not Java.

February 7th, 2015, 12:23 PM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,780 Thanks: 970 
Refreshing does nothing!! To perhaps clarify further, this is what I "see" when looking at your post: There are other methods and they are generally better, but none is as simple to understand as trial and improvement. There are probably no problems that can only be solved in this way, but it is a good introduction to the concept of approximating solutions. Having said that, the iteration [Math Processing Error]is just about the most efficient approximator for [Math Processing Error] and uses basically the same idea. It was used by the Babylonians around 2000 years ago. Last edited by Denis; February 7th, 2015 at 12:26 PM. 
February 7th, 2015, 12:33 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,550 Thanks: 2550 Math Focus: Mainly analysis and algebra 
Try refreshing with SHIFT+F5 or clearing the browser cache. The page works fine for me. Can you see this? $$a_{n+1} = \frac12 \left( a_n + \frac{b}{a_n} \right)$$ 
February 7th, 2015, 12:44 PM  #7  
Senior Member Joined: Apr 2014 From: Europa Posts: 575 Thanks: 176  Quote:
This is a tedious procedure. Start by estimating the solution (you may be given this estimate). Then substitute this into the equation to determine whether your estimate is too high or too low. Refine your estimate and repeat the process. Example Solve tÂ³ + t = 17 by trial and improvement. Firstly, select a value of t to try in the equation. I have selected t = 2. Put this value into the equation. We are trying to get the answer of 17. If t = 2, then tÂ³ + t = 2Â³ + 2 = 10 . This is lower than 17, so we try a higher value for t. If t = 2.5, tÂ³ + t = 18.125 (too high) If t = 2.4, tÂ³ + t = 16.224 (too low) If t = 2.45, tÂ³ + t = 17.156 (too high) If t = 2.44, tÂ³ + t = 16.966 (too low) If t = 2.445, tÂ³ + t = 17.061 (too high) We know that t is between 2.44 and 2.445. So to 2 decimal places, t = 2.44.  
February 7th, 2015, 12:48 PM  #8 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,780 Thanks: 970 
No, can't see it. And I know about the F5 "refresher"! Plus I keep no history: automatic... Well, I sure ain't alone: http://meta.stats.stackexchange.com/...ocessingerror Last edited by Denis; February 7th, 2015 at 12:54 PM. 
February 7th, 2015, 01:23 PM  #9 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,550 Thanks: 2550 Math Focus: Mainly analysis and algebra 
Well I only ever see that error when I get a slow connection and the JavaScript fails to load properly. I would still clear the cache (that's not the same as the history).

February 7th, 2015, 01:53 PM  #10 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,780 Thanks: 970 
Thanks Archie.


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