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(2018-07-04)  
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 classesof either Dedekind cuts (German: Schnitt. Richard 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 mostmodern authors as it makes basic properties easier to establish. The real numbers form an uncountable complete fieldin which the countable  field of rationals is immersed.

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

(2018-08-05)  
Surds and logarithmic ratios of coprime integers.

Surds :

Traditionally,  a surd  is either the square rootof any positive integer which isn't a perfect square  (such a thing is sometimescalled a simple surd)  or a linear combination with rationalcoefficients 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 besidesordinary fractions  (i.e., ratios of integers).

Proving that a simple surd is irrational (theorem of Theodorus, c.399 BC) is easily accomplished by contradictionby considering the factorizations of the numerator and denominatorof 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   =   Loga  /  Logb

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

p  Logb   =   q  Loga    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. 

An irrational power of an irrational base can be rational :

The above shows that   x  =  Log 9 / Log 2   and  y  =  2^½ are irrational.  Yet:

y ^x   =   2 ^{(Log 3 / Log 2)}   =   3  

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

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

• Squaring the circle  (constructingpi).
• Duplicating the cube  (constructing theDelian 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 ofunit 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 quadraticequation 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 ofpolynomials with integer coefficients,  but the converse isn't true (theDelian constant is an example of an algebraic numberwhich isn't constructible).  Therefore, transcendental numbers  (i.e., non-algebraic ones)  are not constructible. Squaring the circle is not possible because   is transcendental,  as proved by Ferdinand von Lindemann (1852-1939) in 1882,  using little more than the method devised in 1873by Charles Hermite (1822-1901) to prove the transcendence of e.

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

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

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

(2018-07-04)  
They're countable,  because Turing machines  are too.

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

Some of them are subreal, almost all  of them are not.

(2018-07-04)  

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

(2018-07-04)  
A series whose convergence depends on the irrationality measure  of .

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

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   :

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 / ( n^u sin^k n)       (for given real  u and integer  k)

He found that

(2018-07-04)  
Algebraically independent sets  & transcendence degree.

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

Galois theory  deals exclusively withalgebraic 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 testament) Evariste Galois (1811-1832)  said:

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