The above is the sum of fractions which can, by standard methods, be expressed as a single fraction; i.e., a rational number.

Algebraic Numbers

Algebraic numbers are numbers which are solutions to polynomial equations with integer coefficients, such as √2, which is a solution to x²-2 = 0. Integers and rational numbers are special cases of algebraic numbers. Although the set of algebraic numbers includes many irrational numbers like √2, it does not contain all irrational numbers. For example, the constant π is not an algebraic number. It is called a transcendental number; i.e., a real number which is not an algebraic number. There are only a few familiar transcendental numbers. The base of the natural logarithms, e=2.71828182859045..., is another familiar one.

It might seem that there are many more algebraic numbers than transcendental numbers, but actually the reverse is the case. There is a higher order of infinity more transcendental numbers than algebraic numbers. The cardinality of the algebraic numbers is ₀, the same as the natural numbers (nonnegative integers), integers and rational numbers. This means that the cardinality of the set of transcendental numbers is the same as that of the set of real numbers, the order of the continuum.

To prove the above assertion let us first consider the set of all polynomials with integer coefficients. This is just as general as considering polynomials with rational number coefficients because one can multiply by the denominators of rational coefficients to get integer coefficients. The set of all integer coefficient polynomials is the union of the set of all such linear equations, quadratic equations, cubic equations and so on.

The cardinality of the set of all integer coefficient linear equations is (Z-{0})×Z. The leading coefficient of a polynomial should not be zero. The cardinality of the set of all linear equations is
₀×₀=₀

Likewise the cardinality of the set of all integer-coefficient quadratic equations is ₀×₀×₀ = ₀. Similarly the cardinality of all n-th degree polynomials is ₀.

Any n-th degree polynomial has at most n distinct roots. Therefore the cardinality of the set of roots of all n-th degree polynomials is at most n×₀=₀. Thus the cardinality of the set of roots to all integer-coefficient polynomials is at most

₀+₀+₀+... =
₀×₀ = ₀
 

Remarkably the set of all such roots is countable.

If the coefficients are allowed to be roots of integer-coefficient polynomials the cardinality of the set of roots is still ₀. Thus by expanding the set of coefficients we never get a set that has a cardinality greater than ₀

Although algebraic numbers are usually defined as the roots of integer coefficient polynomials the concept can obviously be generalized through a recursive definition. Let S₁ be the roots of all integer coefficient polynomials. Let S_m be the set of roots of all polynomials which have coefficients belonging to the set S_m-1. For each m it is true that S_m-1 ⊂ S_m. The generalized algebraic numbers are members of the set which is the limit of S_m as m goes to infinity. Let us denote that set as S_∞.

The denumerable cardinality of the generalized algebraic numbers is established by a process similar to that used in establishing the denumerability of the rational numbers. Let A_m = S_m - S_m-1 and A₁ = S₁.

The cardinality of S₁ has previously been established as ₀, and likewise the roots of polynomials with coefficients from a denumerable set is denumerable. The set A_m, being a subset of a denumerable set, is also denumerable. Each A_m can be put in an order sequence, say:

(A_1,1, A_1,2, A_1,3...)
(A_2,1, A_2,2, A_2,3...)
(A_3,1, A_3,2, A_3,3...)
...........................
...........................
...........................

 

Clearly the Axiom of Choice is needed in this construction.

Now consider the ordering

(A_1,1, A_1,2, A_1,3, A_3,1, A_2,2, A_3,1, ....)
 

Every element of the sets is picked up in this sequence and thus ∪A_m = S_∞ is put into a one-to-one correspondence with the natural numbers. The proof must allow for the possibility that some A_m might be finite or even empty. This constitutes no problem because the sequence can pick up the next existing element in the pattern.

Note that the generalized algebraic number must be defined recursively. A circular definition that a generalized algebraic number is a root of a poynomial with generalized algebraic number coefficients would lead to the argument that π is a generalized algebraic number because it is the solution to the linear equation x - π = 0. Such a definition leads only to the real numbers.

Let us now consider a more detailed analysis of the above construction. Let the set of coefficients be C and its cardinality #C. The cardinality of the set of n-th degree polynomials is less than or equal to the caridinality of {C-{0}}×C×...×C; i.e., (#C-1)(#C)^n-1. The cardinality of the set of roots of n-th degree polynomials is less than or equal to n(#C-1)(#C)^n-1. Thus the cardinality of the set of roots to all finite polynomials of degree less than or equal to N is:

Σ_n=1^N[n(#C-1)(#C)^n-1] =
(#C-1)[N(#C)^N-1/(#C-1) - (#C)^N/(#C-1)²]
 

The Axiomatic Definition of Real Numbers

This approach to the real numbers uses eleven axioms to define a complete ordered field, the real numbers. Let (S, +, *, <) be a set and two function, + and *, called addition and multiplication, respectively, and an order relation <, which is a boolean function. The functions are usually called operations and represented in infix notation, but they are nothing more than special binary functions. The first eight axioms define a field:

• Closedness of Addition and Multiplication: If a and b belong to S then so does a+b and a*b.
• Associativity:
  If a, b, and c belong to S then
  a + (b + c) = (a + b) + c
  a*(b*c) = (a*b)*c
• Commutativity of Addition and Multiplication:
  a + b = b + a
  a*b = b*a
• Distributivity of Multiplication Over Addition:
  If a, b, and c belong to S then a*(b + c) = a*b + a*c.
• Additive and Multiplicative Identities
  There exits two members of S, 0 and 1, such that
  for all a belonging to S, a + 0 = a
  and a*1 = a.
• Additive Inverses:
  For each a belonging to S there exists an element of S, say b, such that a + b = 0. The element b is usually denoted -a.
• Multiplicative Inverses:
  For each a belonging to S other than the additive identity 0 there exists an element of S, say b, such that a*b = 1. The element b is usually denoted a^-1.
• Trichotomy:
  If a and b belong to S either a < b or b < a or a=b.
• Transitivity of the Order Relation:
  If a < b and b < c then a < c.
• Isotony of the Order Relation:
  If a < b then for all c belonging to S a + c < b +c.
  If a < b and 0 < c then a*c < b*c.
• Completion:
  If T is a subset of S with an upper bound b belonging to S such that for all a belonging to T, a < b, then there exist a least upper bound to T; i.e., an element c belonging to S such that c is less than or equal to every upper bound of T.

It must be shown that that the set of infinite sequences of digits satisfies these axioms. This is not trivial. Note again that for most rational numbers there are two infinite sequences that represent the same real number. For example, the multiplicative identity 1 can be represented as 1.000... and 0.999....

Consider how to represent the sum of the two sequences 0.a₁a₂a₃... and 0.b₁b₂b₃.... If the sequences ended then the sum could defined in terms of starting with the right-most digits and expressing the sum in terms of the carry operation. For an unending sequences this procedure is not available. Thus since we cannot write out √2 or √3 it is even less possible to write out √2+√3. The defining of a*b would be even harder.

From the above one can see why the formal definition in mathematics of the reals is in terms of what are called Dedekind Cuts. Another definition is in terms of what are called Cauchy sequences.

In terms of Cauchy sequences a real number is a convergence sequence. Now the definition of sum and product is easy. If X={x_i: i=0,1,2,...} and Y={y_i: i=0,1,2,...} are two convergent sequences then the sum and product are simply the sequences X+Y={(x_i+y_i): i=0,1,2,...} and X*Y={(x_i*y_i): i=0,1,2,...}, which can be shown to be convergent. For more on this see Real Numbers as Cauchy Sequences.

Although more complex these approaches yield nothing other than the relatively naive approach of infinite sequences of digits previously presented. The gain of the other definitions is in rigor and the provability of propositions.

The Set of Transcendental Numbers

It is exceedingly difficult to prove that a specific number, such as π, is transcendental. Indeed, virtually all the transcendental numbers cannot even be specified. There is however one useful result which was was discovered independently by the Russian mathematician Aleksandr Gelfond and the German mathematician Theodor Schneider in 1934. This result, called the Gelfond-Schneider Theorem, says that

If α is an algebraic number other than 0 or 1, and β is an irrational algebraic number then α^β is transcendental; i.e., non-algebraic.

Thus the Hilbert Problem of whether or not 2^√2 is transcendental is solved by the Gelfond-Schneider Theorem. The number 2 is algebraic and √2 is an irrational algebraic number. Thus 2^√2 is transcendental. Likewise 2^√3 is transcendental, as is also (√2)^√2.

The Gelfond-Schneider Theorem applies for complex numbers as well, but that is a different story.