What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)
CHAPTER II. THE NUMBER SYSTEM OF MATHEMATICS
*§6. ALGEBRAIC AND TRANSCENDENTAL NUMBERS
1. Definition and Existence
An algebraic number is any number x, real or complex, that satisfies some algebraic equation of the form
(1) anxn + an-1xn-1 + · · · + a1x + a0 = 0 (n ≥ 1, an ≠ 0)
where the ak are integers. For example, is an algebraic number, since it satisfies the equation
x2 – 2 = 0.
Similarly, any root of an equation with integer coefficients of third, fourth, fifth, or any higher degree, is an algebraic number, whether or not the roots can be expressed in terms of radicals. The concept of algebraic number is a natural generalization of rational number, which constitutes the special case when n = 1.
Not every real number is algebraic. This may be shown by a proof, due to Cantor, that the totality of all algebraic numbers is denumerable. Since the set of all real numbers is non-denumerable, there must exist real numbers which are not algebraic.
A method for denumerating the set of algebraic numbers is as follows:
To each equation of the form (1) the positive integer
h = |an| + |an-1| + · · · + | a1 | + | a0| + n
is assigned as its “height.” For any fixed value of h there are only a finite number of equations (1) with height h. Each of these equations can have at most n different roots. Therefore there can be but a finite number of algebraic numbers whose equations are of height h, and we can arrange all the algebraic numbers in a sequence by starting with those of height 1, then taking those of height 2, and so on.
This proof that the set of algebraic numbers is denumerable assures the existence of real numbers which are not algebraic; such numbers are called transcendental, for, as Euler said, they “transcend the power of algebraic methods.”
Cantor’s proof of the existence of transcendental numbers can hardly be called constructive. Theoretically, one could construct a transcendental number by applying Cantor’s diagonal process to a denumerated table of decimal expressions for the roots of algebraic equations, but this procedure would be quite impractical and would not lead to any number whose expression in the decimal or any other system could actually be written down. Moreover, the most interesting problems concerning transcendental numbers lie in proving that certain definite numbers such as πand e (these numbers will be defined on pages 297 and 299) are actually transcendental.
**2. Liouville’s Theorem and the Construction of Transcendental Numbers
A proof for the existence of transcendental numbers which antedates Cantor’s was given by J. Liouville (1809-1882). Liouville’s proof actually permits the construction of examples of such numbers. It is somewhat more difficult than Cantor’s proof, as are most constructions when compared with mere existence proofs. The proof is included here for the more advanced reader only, though it requires no more than high school mathematics.
Liouville showed that irrational algebraic numbers are those which cannot be approximated by rational numbers with a very high degree of accuracy unless the denominators of the approximating fractions are quite large.
Suppose the number z satisfies the algebraic equation with integer coefficients
(2) f(x) = ao + a1x + a2x2 + · · · + anxn = 0 (an ≠ 0),
but no such equation of lower degree. Then z is said to be an algebraic number of degree n. For example, is an algebraic number of degree 2, since it satisfies the equation x2 – 2 = 0 but no equation of the first degree; is of the third degree because it satisfies the equationx2 – 2 = 0 and, as we shall see in Chapter III, no equation of lower degree. An algebraic number of degree n > 1 cannot be rational, since a rational number z = p/q satisfies the equation qx – p = 0 of degree 1. Now each irrational number z can be approximated to any desired degree of accuracy by a rational number; this means that we can find a sequence
of rational numbers with larger and larger denominators such that
Liouville’s theorem asserts: For any algebraic number z of degree n > 1 such an approximation must be less accurate than 1/qn+1; i.e., the inequality
(3)
must hold for sufficiently large denominators q.
We shall prove this theorem presently, but first we shall show how it permits the construction of transcendental numbers. Let us take the number (see p. 17 for the definition of the symbol n!)
z = a1. 10–1! + a2. 10–2! + a3. 10–3! + · · · + am. 10–m! + a(m+1). 10–(m+1)! + · · ·
= 0.a1a2000a300000000000000000a40000000.,
where the ai are arbitrary digits from 1 to 9 (we could, for example, choose all the ai equal to 1). Such a number is characterized by rapidly increasing stretches of 0’s, interrupted by single non-zero digits. Let us denote by zm the finite decimal fraction formed by taking only the terms of z up to and including am. 10–m!. Then
(4) | z – zm | < 10.10–(m+1)!.
Suppose that z were algebraic of degree n. Then in (3) let us set p/q = zm = p/10m!, obtaining
for sufficiently large m. Combining this with (4), we should have
so that (n + 1)m! > (m + 1)! − 1 for all sufficiently large m. But this is false for any value of m greater than n (the reader should give a detailed proof of this statement), which gives a contradiction. Hence z is transcendental.
It remains to prove Liouville’s theorem. Suppose z is an algebraic number of degree n > 1 which satisfies (1), so that
(5) f(z) = 0.
Let zm = pm/qm be a sequence of rational numbers with zm → z. Then Dividing both sides of this equation by zm – z, and using the algebraic formula
we obtain
(6)
Since zm tends to z as a limit, it will differ from z by less than 1 for sufficiently large m. We can therefore write the following rough estimate for sufficiently large m:
(7)
which is a fixed number, since z is fixed in our reasoning. If now we choose m so large that in the denominator qm is larger than M, then
(8)
For brevity let us denote pm by p and qm by q. Then
(9)
Now the rational number zm = p/q cannot be a root of f(x) = 0, for if it were we could factor out (x – zm) from f(x), and z would satisfy an equation of degree less than n. Hence f (zm) ≠ 0. But the numerator of the right hand side of (9) is an integer, so it must be at least equal to 1. Hence from (8) and (9) we have
(10)
which proves the theorem.
During the last few decades, investigations into the possibility of approximating algebraic numbers by rational numbers have been carried much farther. For example, the Norwegian mathematician A. Thue (1863-1922) proved that in Liouville’s inequality (3) the exponent n + 1 may be replaced by (n/2) + 1. C. L. Siegel later showed that the even sharper statement (sharper for large n) with the exponent holds.
The subject of transcendental numbers has always fascinated mathematicians. But until recently, very few examples of numbers interesting in themselves were known which could be shown to be transcendental. (In Chapter III we shall discuss the transcendental character of π, from which follows the impossibility of squaring the circle with ruler and compass.) In a famous address to the international congress of mathematicians at Paris in 1900, David Hilbert proposed thirty mathematical problems which were easy to formulate, some of them in elementary and popular language, but none of which had been solved nor seemed immediately accessible to the mathematical technique then existing. These “Hilbert problems” stood as a challenge to the subsequent period of mathematical development. Almost all have been solved in the meantime, and often the solution meant definite progress in mathematical insight and general methods One of the problems that seemed most hopeless was to prove that
is a transcendental, or even that it is an irrational number. For almost three decades there was not the slightest suggestion of a promising line of attack on this problem. Finally Siegel and, independently, the young Russian, A. Gelfond, discovered new methods for proving the transcendental character of many numbers significant in mathematics, including the Hilbert number and, more generally, any number ab where a is an algebraic number ≠ 0 or 1 and b is any irrational algebraic number.