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May 14th, 2018, 09:13 PM   #1
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Equations solvable only by directly providing the answer

Are there equations with solutions, but totally unsolvable mathematically unless by substituting simple variables with given complex answers?
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May 14th, 2018, 10:22 PM   #2
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Are there equations with solutions, but totally unsolvable mathematically unless by substituting simple variables with given complex answers?
Can you explain your question better?

One topic that comes to mind is the solution of polynomials. We know that there is no general algebraic procedure to solve polynomials of fifth degree and higher; but every such polynomial has zeros by the fundamental theorem of algebra. In other words we can prove there are solutions but there is no procedural way to find them. The best we can do is iterative approximation.

Is that what you mean?
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May 14th, 2018, 11:00 PM   #3
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How about the equation x² = -1?
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May 15th, 2018, 09:15 AM   #4
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Originally Posted by Maschke View Post
Can you explain your question better?

One topic that comes to mind is the solution of polynomials. We know that there is no general algebraic procedure to solve polynomials of fifth degree and higher; but every such polynomial has zeros by the fundamental theorem of algebra. In other words we can prove there are solutions but there is no procedural way to find them. The best we can do is iterative approximation.

Is that what you mean?
Yes, that topic is what came to my mind first. It had been proven by Cardano a few hundred years ago, I recall.

Do algebraic functions include the methods for finding most zeros, next to transcendental and "beyond"? Or do exponential and trigonometric functions include just as many possible zeros? Otherwise, would all of these three groups of functions (sans transcendental) follow the fundamental theorem of algebra?
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May 15th, 2018, 10:24 AM   #5
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Yes, that topic is what came to my mind first. It had been proven by Cardano a few hundred years ago, I recall.
Cardano first solved the cubic. Abel was the first to prove the unsolvability of the quintic, in 1824. Galois was the first to work out the general theory of when a polynomial is solvable. Both Abel and Galois died tragically young, Abel at 27 and Galois at 21. So although the examples of Cantor and Gödel show that logic and set theory will drive you crazy, algebra will kill you.

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Do algebraic functions include the methods for finding most zeros, next to transcendental and "beyond"?
Can't understand the question, please be more clear or give a clarifying example please.

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Or do exponential and trigonometric functions include just as many possible zeros?
That doesn't really make sense. The exponential function $e^x$ has no zero over the reals or the complex numbers. The trig functions $\sin$ and $\cos$ each have infinitely many zeros.

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Otherwise, would all of these three groups of functions (sans transcendental) follow the fundamental theorem of algebra?
FTA only applies to polynomials.

Last edited by skipjack; July 21st, 2018 at 09:08 PM.
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May 15th, 2018, 02:18 PM   #6
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Maschke,

Do algebraic functions include the methods for finding most zeros, next to transcendental and "beyond"?

>Is the set greater for the real numbers than that for transcendental numbers plus algebraic numbers?

Or do exponential and trigonometric functions include just as many possible zeros?

>Are polynomial, exponential and trigonometric functions of exclusive yet equivalent sets -- and with what cardinality?

Many mathematicians died before age 40. What's the omen of hearse #40?
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May 16th, 2018, 07:09 AM   #7
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Beer soaked recall follows.
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Cardano first solved the cubic.
Methinks he borrowed the solution more or less from Scipione del Ferro and Niccolò Fontana Tartaglia.
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May 16th, 2018, 11:38 AM   #8
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Beer soaked recall follows.

Methinks he borrowed the solution more or less from Scipione del Ferro and Niccolò Fontana Tartaglia.
....and Pa Kettle....
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May 25th, 2018, 12:59 PM   #9
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Maschke,

Do algebraic functions include the methods for finding most zeros, next to transcendental and "beyond"?
I don't know what you what mean by "methods". It has been proven that there exist polynomials of degree higher than 5 with zeros that cannot be expressed in terms of roots of integers so no formula involving only algebraic operations can give them. But some equations might have solutions that can
be expressed in other ways.

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>Is the set greater for the real numbers than that for transcendental numbers plus algebraic numbers?
If you mean the union of the sets of transcendental and algebraic numbers, no, the transcendental numbers are defined as "all real numbers that are not algebraic numbers". All real numbers are either algebraic or transcendental.

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Or do exponential and trigonometric functions include just as many possible zeros?
The set of all algebraic numbers is "countable". In terms of cardinality, the set of algebraic numbers is just as large as the set of rational numbers. The set of transcendental numbers has the same cardinality as the set of all real numbers. In that sense, the set of all transcendental numbers is far larger than the set of all algebraic numbers.

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>Are polynomial, exponential and trigonometric functions of exclusive yet equivalent sets -- and with what cardinality?
Again, I am not clear what this means. these are all functions of the real numbers.

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Many mathematicians died before age 40. What's the omen of hearse #40?
Do you have any support for that claim? Most of the mathematicians I know are older than 40.
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May 25th, 2018, 03:12 PM   #10
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Many mathematicians died before age 40. What's the omen of hearse #40?
I wonder if you're thinking of the claim that most mathematicians do their best work before age 40. That's not always true either, but I'm sure mathematicians in general don't just drop dead at 40.

Unless they do pioneering work on the quintic. That's generally fatal.
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