this is a - standard - field extension: basis: 1, pi.

Dear Anton Schober,

Thank you for the reply. I was not aware of transcendental extension. It is countable, because simple extension of a countable field is countable.

One has the field of rational functions with rational coefficients: Q(X). Here X is an abstract transcendental element. Well, Q(pi) is also a field, and is isomorphic with Q(X). The isomorphism just does f(pi) = X and preserves all rational numbers. The only difference is that pi is not an abstract transcendental quantity, but a concrete one. The reason of Q(X) to be countable is that (Q U QxQ U QxQxQ U ...)^2 is a countable set.

Dear Mihai,

Thank you. I got your point.

@Mihai Prunescu: "...The only difference is that pi is not an abstract transcendental quantity, but a concrete one" . Here is my question: consider the number seven: is 7 now an abstract integer quantity or a concrete one? In case you don't like this, take sqrt7.

I differ, pi is not a concrete one quantity, it is irrational and not rational number so it has no limit, now the question if Q (pi) is countable, there are infinite Q (pi), so it is not countable. It is not tue that  f(pi) = X and preserves all rational numbers, becauese it is imposible to represent an irrational number equal to rational number.

Jose William Porras:

1) "pi is not a concrete one quantity". pi is the area of a unit circle, 2pi is its circumference.

2) "because it is imposible to represent an irrational number equal to rational number": sqrt2 is the diagonal of the unit sqare.

This was geometrical. There a noumerous further representations of pi in series or definite integrals. Resume: As far as "quality" is concerned, the integers and the reals are abstract mathematical objects, the number 1 does'nt exist in nature.

Anton Schober, but the area of unit circle or area of any circle is not concrete one quantity.....always it is an approximationis  give one concrete one quantity (value). The irrational number have infinite mantissa......geometrically, you can represent an irrational number, but not with "concrete one quantity".......

Q(pi) is the field of fractions of Q[pi] which is isomorphic (really only need a bijection) with the ordinary polynomial ring Q[x]. Now Q[x] is equal to the Union of the countable collection QuQxuQx^2uQx^3u... That is, it's a countable Union of countable sets (each Qx^n is in one to one correspondence with Q, and hence countable) and so Q[x] is countable. Now, if an integral domain is countable, it is not difficult to prove that it's field of fractions is countable. Hence Q(pi) is countable.

Thank you Jerry Rosen, Jose William.

Dear Jose William and Anton,

What is the definition for concrete quantity and abstract quantity?

Are they well-defined?

@Jose William Porras: this "concrete one quantity" exists eventually in your mind. I ask you: is an open set, subset of a topological space a "concrete one quantity"? Is the Lagrangian of the electromagnetic field a "concrete one quantity"? AND AGAIN: is the number 1 a "concrete one quantity"? Eventually you answer exactly these three questions.

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in the binary system there exist only zeroes(0) and ones (1). Now "concrete"?

next: also 1/3 = .333... is not "concrete" in your system

Anton Schober: In my mind not. Concrete quantity is a physical object or a collection of such objects. According with Definition in: http://www.finedictionary.com/Concrete%20quantity.html

Panchatcharam Mariappan:

Concrete quantity is a physical object or a collection of such objects. According with Definition in: http://www.finedictionary.com/Concrete%20quantity.html

abstract: Considered apart from concrete existence.

In Mathematics. Abstrct Quantity:

a number that does not designate the quantity of any particular kind of thing.

References: http://dictionary.reference.com/browse/abstract-number

I am not sure whether this wiki page is correct. https://en.wikipedia.org/wiki/Concrete_number

It says the following:

In mathematics the term "number" is usually taken to mean an abstract number. A denominate number is a type of concrete number with a unit of measure attached with it. For example, 5 inches is a denominate number because it has the unit inches after it.

If it is correct, then pi is a concrete quantity. A circle with unit metre has pi metre^2 as area. So, it should be concrete number. If we consider a thread whose length is same as circumference of a circle of radius 1 inch, then half of its length is pi inches.

@Panchatcharam Mariappan: “Concrete quantity is a physical object or a collection of such objects” is like discrete topology (trivial = worthless). In categories we have objects and morphisms. These objects are the “abstract quantities”, and the morphisms are well defined mappings.

I agree with Anton Schobers. In addition quantity is not the same as number.

Dear Anton and Jose

Thanks for clarifications.

Of course, Anton Schober's answer for Q(pi) is OK.

Q(pi) is certainly countable, otherwise any real is a rational fraction in pi.

dear Dr. Gabriel Thomas, "any real is a rational fraction in pi." could you please demonstrate this by sqrt2 and pi?

Q(pi) is included in R (real numbers).

If Q(pi) is not countable, then it is equipotent to R (continuum hypothesis).

Thus there is a bijection between R and Q(pi).

It is the same argument for sqrt(2).

However it is possible to show that the real number e is not an algebraic number so it is not in Q(sqrt(2)). 

For Q(pi) : using an argument from Cantor on the countability of the polynomials with integral coefficients, you get the conclusion.

@dear Dr. Gabriel Thomas, "Q(pi) is included in R", sure, bur sqrt2 is not icluded in Q(pi) 

"For Q(pi) : using an argument from Cantor on the countability of the polynomials with integral coefficients, you get the conclusion" I would like to see this polynomial! This should be an easy task! 

no integer polynomial in pi has sqrt2 as a root, 

no integer polynomial in sqrt2  has pi as a root,

That makes sense, since actually Q(pi) isn't equal to R, nor equipotent to it.