The Discovery of Transcendental Numbers Opens Up the World of Numbers

Real numbers can be visually represented using a "number line," but the number line is a mixture of different kinds of numbers. For example, there are numbers that can be expressed as fractions and those that cannot. The former are called rational numbers, and the latter are called irrational numbers. For instance, 2/3 is a rational number, while √2 and pi π are irrational numbers. However, if we think about it differently, 2/3 and √2 are the same type of number, while π is a different type. First, multiplying 2/3 by 3 gives 2. In other words, 2/3 is a number that becomes 0 when substituted into the polynomial 3x −2. Next, √2 is a number that becomes 2 when squared. In other words, it is a number that becomes 0 when substituted into the polynomial x2−2. Numbers that become 0 when substituted into a polynomial f(x) with integer coefficients are called algebraic numbers. On the other hand, it is known that π is not an algebraic number. This means that no matter how complex the polynomial f(x) with integer coefficients is, substituting π will never result in f(π) =   0. Numbers that do not become 0 when substituted into any polynomial f(x) with integer coefficients are called transcendental numbers (see Figure 1).  

To represent a transcendental number with specific digits rather than a symbol like π, an infinite series is required. The most familiar form of an infinite series is a decimal representation. √2 and π, when expressed as decimals and infinite series, respectively, are as follows.  

The sequence of digits appearing in each place of these decimals is random, and no regularity can be found. As is clear from the cases of √2 and π, it is extremely difficult to distinguish between algebraic numbers and transcendental numbers from their decimal representations (which lack regularity).

becomes a transcendental number. Here, the positions where 1 appears are the 1, 2, 3, 5, 8, 13, 21, 34, 55, ... , which is the famous Fibonacci sequence (a sequence where adding two adjacent terms gives the next term).

By the way, once one transcendental number is obtained, different transcendental numbers can be created one after another by substituting it into polynomials, taking its reciprocal or square root, and so on. An interesting fact is that the transcendental numbers newly created from a single transcendental number through such operations are insignificant when viewed from the entirety of all transcendental numbers. To explain this, the following concepts exist. For example, substituting √π and π into the two-variable polynomial x2−y results in a value of 0. Numbers that are related in this way, linked by a polynomial with multiple variables, are said to be algebraically dependent. Conversely, numbers that cannot be linked no matter what polynomial is used are said to be algebraically independent. For example, using the theory handled in our laboratory, it can be shown that the aforementioned θ1 and the number θ2, represented by the following decimal and infinite series, are algebraically independent.  

Here, the positions where 1 appears are the sequence 2, 4, 6, 10, 16, 26, 42, 68, ... , which is obtained by doubling each term of the Fibonacci sequence. Furthermore, if we consider θ1, θ2, and the number θ3, represented by the following decimal and infinite series, it can be similarly shown that these three numbers are algebraically independent.

Here, the positions where 1 appears are the sequence 3, 6, 9, 15, 24, 39, 63, ... , which is obtained by tripling each term of the Fibonacci sequence. The fact that θ1, θ2, and θ3 are algebraically independent means that θ3 cannot be obtained by repeatedly performing calculations such as the four arithmetic operations or taking square roots from θ1 and θ2. Similarly, if we create numbers up to θn, it can be shown that θ1, θ2,…, θn are also algebraically independent (in the decimal representation of θn, 1 appears only in the positions corresponding to the sequence obtained by multiplying each term of the Fibonacci sequence by n).

Algebraically independent transcendental numbers act, so to speak, as guideposts or lighthouses in the largely unknown world of transcendental numbers. In that sense, I will explain how the above research is useful. Consider the polynomials with integer coefficients P1 ( X1 , X2 ,…, Xn ), P2 ( X1 , X2  ,…, Xn  ),…, Pm ( X1 , X2 ,…, Xn ) of the variables X1 , X2 ,…, Xn. The key point here is that there can be any number of variables X1 , X2 ,…, Xn. Also, these polynomials need only be different from each other, i.e., for all different numbers i and j, we assume Pi ( X1 , X2 ,…, Xn ) − Pj ( X1 , X2 ,…, Xn ) ≠ 0. The values P1 ( θ1 , θ2  ,…, θn ), P2 ( θ1 , θ2  ,…, θn  ),…, Pm ( θ1 , θ2  ,…, θn ) obtained by substituting θ1 , θ2 ,…, θn into these polynomials are all different transcendental numbers, no matter how large n and m are, allowing us to generate a large number of examples of transcendental numbers simultaneously.  

In fact, the following "mechanism" is hidden behind the fact that θ1, θ2,…, θn are algebraically independent.

Starting with this example, one of the themes of our laboratory is to clarify the "mechanisms" that allow for the efficient generation of algebraically independent transcendental numbers. Elucidating the mechanism by which many algebraically independent transcendental numbers can be obtained from a single sequence like the Fibonacci sequence provides a foothold for exploring the world of numbers.