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 uncountablecompletefield
in which the countable field of rationals is immersed.
A real number which is not rational is called irrational.
Almost all are.
(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.
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 = 2½ are irrational. Yet:
y x = 2 (Log 3 / Log 2) = 3
(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:
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.
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 :
(2018-07-04) Transcendence over a subfield K Algebraically independent sets & transcendence degree.
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 testament)
Evariste 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...
