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Final Answers
© 2000-2021   Gérard P. Michon, Ph.D.

Irrational  Numbers

He is unworthy of the name of Man who is
ignorant of the fact that the diagonal of a
square is incommensurable with its side.

 Plato  (427-347 BC)
 Michon
 
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There's Plenty of Room at the Bottom  by  Richard P. Feynman (Dec. 1959).
Applications of Forcing in Domain Theory   (Morteza Azad,  2018-06-22).
 
Wikipedia :   Quantum computing   |   Quantum information   |   Qubit   |   Shor's algorithm
 
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Irrational  Numbers


(2018-07-04)   Rational Numbers  &  Real Numbers
A rational number is the ratio of two integers.  An  irrational  one isn't.

The field of rational numbers is the  quotient field  of the integers.

The real numbers are constructed as  equivalence classes of either  Dedekind cuts  (German:  SchnittRichard Dedekind, 1858)  or  Cauchy sequences  of rational numbers  (Bolzano 1817,  Cauchy 1821,  Cantor 1871).

Dedekind's approach is more concise but the latter viewpoint is preferred by most modern authors as it makes basic properties easier to establish.  The real numbers form an  uncountable  complete  field in which the  countable  field of rationals is immersed.

A real number which is not rational is called  irrational.  Almost all are.

Rational number   |   Real numbers


(2018-08-05)   Simple Examples of Irrational Numbers
Surds and logarithmic ratios of coprime integers.

Surds :

Traditionally,  a  surd  is either the square root of any positive integer which isn't a perfect square  (such a thing is sometimes called a  simple surd)  or a linear combination with rational coefficients of several such quantities  (compound surds).  The word  surd  is short for  absurd,  which is how the ancient  Pythagorean cult  perceived them at a time when they didn't recognize numbers besides ordinary fractions  (i.e., ratios of integers).

Proving that a simple surd is irrational  (theorem of Theodorus, c.399 BC)  is easily accomplished by contradiction by considering the factorizations of the numerator and denominator of a purported rational simple surd.

There's also a brilliant modern  proof by infinite descent  which doesn't use divisibility at all.

Ratio of the logarithms of two coprime integers:

With two  coprime integers  a  and  b,  we consider the ratio:

x   =   Log a  /  Log b

If  x  was a rational  p/q  then we would have:

p  Log b   =   q  Log a     Therefore:     b p   =   a q

As the latter isn't possible when  a  and  and  b  are coprime,  we deduce that  x  must be irrational.   QED

An irrational power of an irrational base can be rational :

The above shows that   x  =  Log 9 / Log 2   and   y  =  2½ are irrational.  Yet:

y x   =   2 (Log 3 / Log 2)   =   3   QED


(2018-07-04)   Constructible Numbers
What can be built with straightedge and compass  (or compass alone).

This was of considerable interest to the  ancient Greeks  and it remained so for two millenia or so,  when Euclid's Elements  were still dominating mathematical teaching.  Three infamous classical problems which actually call for the  (impossible)  constructions  of some particular numbers:

  • Squaring the circle  (constructing pi).
  • Duplicating the cube  (constructing the Delian constant).
  • Trisecting the angle  (constructing a solution to a  cubic equation).

It turns out that rule-and-compass constructions can built  (from a given segment of unit length)  precisely for those numbers which can be obtained with finitely many additions, subtractions, multiplications, divisions and square roots.  Such  constructible  numbers are enough to solve any quadratic equation with constructible coefficients but some cubic equations don't have constructible roots  (so that the trisection of the angle is impossible classically).

All constructible numbers are  algebraic  (they are roots of polynomials with integer coefficients,  but the converse isn't true  (the Delian constant is an example of an algebraic number which isn't constructible).  Therefore,  transcendental  numbers  (i.e., non-algebraic ones)  are not constructible.  Squaring the circle is not possible because  p  is  transcendental,  as proved by  Ferdinand von Lindemann (1852-1939)  in 1882,  using little more than the method devised in 1873 by  Charles Hermite (1822-1901)  to prove the transcendence of  e.

The notion of  constructible number  is now only of  historical  or  cultural  interest.  Akin to  Egyptian fractions,  which were also of tremendous importance for thousands of years but are now all but forgotten,  except for  recreational mathematics.


(2018-07-04)   Algebraic Numbers
An  algebraic number  is a root of a polynomial with integer coefficients.

All  rationals  are algebraic  (as each one obeys an equation of degree 1).  So are all  constructible numbers.

 Come back later, we're
 still working on this one...


 Gerard Michon (2018-07-04)   Subreal Numbers   =   Computable Numbers
They're countable,  because  Turing machines  are too.

 Come back later, we're
 still working on this one...

Numericana :   subreal numbers   |   Countable sets


(2018-07-04)   Transcendental Numbers
By definition,  those are the  non-algebraic  real numbers.

Some of them are  subrealalmost all  of them are not.

 Come back later, we're
 still working on this one...

Transcendental numbers
 
Transcendental numbers powered by Cantor's infinities (17:18)  by  Burkard Polster   (Mathologer, 2017-04-22).


(2018-07-04)   Irrationality Measure  (or irrationality exponent)

 Come back later, we're
 still working on this one...

Irrationality measure   |   Roth's theorem   |   Klaus Roth (1921-2015; Fields Medal 1958)
Joseph Liouville (1809-1882)   |   Axel Thue (1863-1922)   |   Carl L. Siegel (1896-1981)   |   Freeman Dyson (1923-2020)


(2018-07-04)   Liouville Numbers
Numbers whose  irrationality measure  is infinite.

 Come back later, we're
 still working on this one...


(2018-07-04)   Pickover's  Flint Hills  series   (C. A. Pickover,  2002)
A series whose  convergence  depends on the  irrationality measure  of p.

Flint Hills  (Kansas)  was given its name,  in 1806,  by means of an entry in the diary of the explorer  Zebulon Pike (1779-1813).

US counties  are named after  Zebulon Pike  in ten states:  AlabamaArkansasGeorgiaIllinoisIndianaKentuckyMississippiMissouriOhio  and  Pennsylvania.  Where did the epic westward journey of  Sweet Betsy  start from?  Well,  she crossed the wide mountains with her lover Ike, two yoke of oxen, a big yeller dog, a tall Shanghai rooster and one spotted hog.  East of the Appalachians?  That's Pennsylvania.

The series whose  n-th  term is  1 / (n3 sin2 n)  was given the name of that place  by  Clifford A. Pickover (1957-)  as he introduced it in his book  The Mathematics of Oz:  Mental Gymnastics from Beyond the Edge  (Vol. 2, Ch. 25, pp. 57-59 & 265-268.  Cambridge University Press,  2002).

The quantity   sin2 n   decreases to zero as the index  n  goes through the successive numerators of the  convergents  of  p :

1, 3, 22, 333, 355, 103993, 104348, 208341, 312689, 833719...  (A046947)

Max Alekseyev  has generalized the above to any series whose  n-th  term is:

1 / ( nu sink n)       (for given real  u  and integer  k)

He found that

 Come back later, we're
 still working on this one...

Flint Hills Series   |   Cookson Hills Series
 
On the convergence of the Flint Hills series  by  Max A. Alekseyev  (2011-04-27).


(2018-07-04)   Transcendence over a subfield  K
Algebraically independent sets  &  transcendence degree.

 Come back later, we're
 still working on this one...

An extension  L  of a field  K  is said to be  algebraic  when every element of  L  is a root of some polynomial whose coefficients are in  K.  Otherwise,  L  is called a  transcendental extension  of  K.

Galois theory  deals exclusively with algebraic extensions  (Galois called it  [algebraic] "ambiguity theory").  In the letter he wrote to his friend Auguste Chevalier the night before his fateful duel  (his celebrated scientific  testamentEvariste Galois (1811-1832)  said:

Mes principales méditations,  depuis quelques temps,  étaient dirigées sur l'application à l'analyse transcendante de la théorie de l'ambiguité.  Il s'agissait de voir a priori,  dans une relation entre des quantités ou fonctions transcendantes,  quels échanges on pouvait faire,  quelles quantités on pouvait substituer aux quantités données,  sans que la relation put cesser d'avoir lieu.  Cela fait reconnaître de suite l'impossibilité de beaucoup d'expressions que l'on pourrait chercher.  Mais je n'ai pas le temps, et mes idées ne sont pas encore bien développées sur ce terrain, qui est immense.

The next morning  (Wednesday, May 30)  Galois was mortally wounded in the gut and he died from peritonitis one day later  (May 31, 1832).  How Galois would have  developped  for transcendental relations what he had done for algebraic equations is something we're only beginning to guess...

Algebraic independence   |   Transcendence degree
Schanuel's conjecture (cf. Grothendieck period conjecture)   |   Stephen H. Schanuel (1933-2014)
Mumford-Tate conjecture   |   David Mumford (1937-)   |   John Tate (1925-2019)

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