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From "equal" signs to "approximate"/"asymptotic" signs...
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HenryDavidT
Aug 28, 2015, 4:12:36 AM8/28/15
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This is the first of 15 lectures....
https://www.youtube.com/watch?v=LYNOGk3ZjFM&list=PL4EF4D3A3D705D560&index=88
Who would have wanted to memorize the dread quadratic formula, or other much more complex methods and tricks, if an easier methods or tricks existed?
The example was X^5 + X = 1
and the trick --- in actuality, it's a method, one that is really sophisticated and extremely powerful (& it was probably why teachers in secondary school don't teach it to their students, lest they over-take the teachers ability to teach within a year or two!) --- is to introduce another variable, called an "epsilon" in this case, to the problem, and then to make powerful yet simplifying mathematical assumptions,.... about ONE TERM of the expression being NEGLIGIBLE and THE OTHER TWO TERMS being ASYMPTOTIC TO ONE ANOTHER (the method is called DOMINANT BALANCE).... and deploying the "epsilon" at the different parts of the equation...
and, then, to go on from there to find an ARBITRARILY CLOSE ANSWER, but not exactly the answer, to the problem.... from simpler problems like the above here (WHICH IN AND OF ITSELF IS REALLY CONSIDERED A "HARD PROBLEM," BECAUSE IT IS A QUINTIC EQUATION, X to the power of 5, and not the X square, or X to the 3, or X to the only 4 power)..... to exceedingly difficult ones that there's no easy answer readily available in books or the Internet...
https://www.youtube.com/watch?v=LYNOGk3ZjFM&list=PL4EF4D3A3D705D560&index=88
Who would have wanted to memorize the dread quadratic formula, or other much more complex methods and tricks, if an easier methods or tricks existed?
The example was X^5 + X = 1
and the trick --- in actuality, it's a method, one that is really sophisticated and extremely powerful (& it was probably why teachers in secondary school don't teach it to their students, lest they over-take the teachers ability to teach within a year or two!) --- is to introduce another variable, called an "epsilon" in this case, to the problem, and then to make powerful yet simplifying mathematical assumptions,.... about ONE TERM of the expression being NEGLIGIBLE and THE OTHER TWO TERMS being ASYMPTOTIC TO ONE ANOTHER (the method is called DOMINANT BALANCE).... and deploying the "epsilon" at the different parts of the equation...
and, then, to go on from there to find an ARBITRARILY CLOSE ANSWER, but not exactly the answer, to the problem.... from simpler problems like the above here (WHICH IN AND OF ITSELF IS REALLY CONSIDERED A "HARD PROBLEM," BECAUSE IT IS A QUINTIC EQUATION, X to the power of 5, and not the X square, or X to the 3, or X to the only 4 power)..... to exceedingly difficult ones that there's no easy answer readily available in books or the Internet...
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