My Math Forum Solving Square Super-Root with Elementary Functions

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 November 29th, 2017, 04:21 PM #1 Newbie   Joined: Nov 2017 From: Boulder, Colorado Posts: 1 Thanks: 0 Solving Square Super-Root with Elementary Functions First, a little background. I joined to ask this question, so apologies if it is in the wrong sub-forum. I am a senior in high school in Calc 1, but I have recently been intrigued with hyper operations. With a little experimenting in the calculator and some logic, I came to the conclusion that I can find the square super-root of any real number on the domain (~.6923, ∞) using a chain (not sure if right word) of elementary functions. Specifically, if we define x^x = y, then: a = y^(1/y) b = y^(1/a) c = y^(1/b) d = y^(1/c) ... and so on. Eventually, the value of y^(1/n) converges to the correct square super-root of y. If x<1, it asymptotes to the square super-root from above. If x>1, it oscillates above and below the correct value and eventually converges. However, computing the square super-root of numbers larger than 15 requires carrying a higher number of digits than I currently can in a computer program. Here is an example. I only wrote down 3 digits each time for brevity, but I carried 10 digits in the calculator. if x^x = 3: a = 3^(1/3) = 1.44 b = 3^(1/1.44) = 2.14 c = 3^(1/2.14) = 1.67 d = 3^(1/1.67) = 1.93 e = 3^(1/1.93) = 1.77 f = 3^(1/1.77) = 1.86 g = 3^(1/1.86) = 1.80 h = 3^(1/1.80) = 1.84 i = 3^(1/1.84) = 1.82 j = 3^(1/1.82) = 1.83 k = 3^(1/1.83) = 1.82 1.82^1.82 ≈ 3 So after 11 iterations, we have 3 correct digits of the 2nd super-root of 3. After infinitely many iterations, we have infinitely many correct digits. Thoughts? Has this been recognized before, and if so, where can I read about it? As far as I have read, square super-roots should not be able to be calculated with elementary functions.
 December 1st, 2017, 03:54 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 Such a function can not be calculated using a finite number of elementary functions. But any function can be calculated as a limit of an infinite sequence of elementary functions.
June 23rd, 2018, 07:47 PM   #3
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Quote:
 Originally Posted by eliboardman First, a little background. I joined to ask this question, so apologies if it is in the wrong sub-forum. I am a senior in high school in Calc 1, but I have recently been intrigued with hyper operations. With a little experimenting in the calculator and some logic, I came to the conclusion that I can find the square super-root of any real number on the domain (~.6923, ∞) using a chain (not sure if right word) of elementary functions. Specifically, if we define x^x = y, then: a = y^(1/y) b = y^(1/a) c = y^(1/b) d = y^(1/c) ... and so on. Eventually, the value of y^(1/n) converges to the correct square super-root of y. If x<1, it asymptotes to the square super-root from above. If x>1, it oscillates above and below the correct value and eventually converges. However, computing the square super-root of numbers larger than 15 requires carrying a higher number of digits than I currently can in a computer program. Here is an example. I only wrote down 3 digits each time for brevity, but I carried 10 digits in the calculator. if x^x = 3: a = 3^(1/3) = 1.44 b = 3^(1/1.44) = 2.14 c = 3^(1/2.14) = 1.67 d = 3^(1/1.67) = 1.93 e = 3^(1/1.93) = 1.77 f = 3^(1/1.77) = 1.86 g = 3^(1/1.86) = 1.80 h = 3^(1/1.80) = 1.84 i = 3^(1/1.84) = 1.82 j = 3^(1/1.82) = 1.83 k = 3^(1/1.83) = 1.82 1.82^1.82 ≈ 3 So after 11 iterations, we have 3 correct digits of the 2nd super-root of 3. After infinitely many iterations, we have infinitely many correct digits. Thoughts? Has this been recognized before, and if so, where can I read about it? As far as I have read, square super-roots should not be able to be calculated with elementary functions.

There are other simpler ways to manage all this stuff on roots and super-roots:
Take a look at this thread:

The Mediant and new high-order root approximating methods.

and this:
https://domingogomezmorin.wordpress.com/

Regards,
Domingo Gomez Morin

 Tags elementary, functions, solving, square, superroot

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