We have learned to solve equations in school, and the most common ones we solve are so-called algebraic equations, such as 3x -1 = 0, x 2 -2x -8 = 0, etc. One of the main characteristics of these equations is that every term containing an unknown variable only includes positive integer powers of the unknown variable. In addition, these equations have two very important features: the number of terms is finite, and the coefficients of the unknown variables are all integers. Now, we want to answer the following question: Are all real numbers roots of integer-coefficient algebraic equations? For convenience, we will simply refer to integer-coefficient algebraic equations as algebraic equations. First of all, it is obvious that all rational numbers q =m /n are roots of the algebraic equation nx m =0, where m and n are integers (n ≠0). Secondly, anyone who has studied the quadratic equation knows that the roots of an algebraic equation may not be rational. For example, the two roots of x 2 -2 = 0 are irrational numbers. Therefore, the roots of integer-coefficient algebraic equations can be either rational or irrational, thus at least superficially having the potential to represent all real numbers.
However, having the potential does not necessarily mean it can be achieved. The first person to provide a negative answer and proof to the question "Are all real numbers roots of integer-coefficient algebraic equations?" was the French mathematician Liouville. He not only proved that some real numbers are not roots of any integer-coefficient algebraic equation, but also specifically constructed such real numbers, thus providing the most convincing negative answer. The non-algebraic roots of equations constructed by Liouville were represented by continued fractions, and later he also constructed such real numbers using decimal fractions. In memory of Liouville's contributions, the numbers constructed by him are called Liouville numbers.
Now we know that many important real numbers, such as the base of natural logarithms e and the pi, are not roots of integer-coefficient algebraic equations. To make the expression more convenient, mathematicians call numbers that can be represented by the roots of integer-coefficient algebraic equations algebraic numbers, and call numbers that cannot be represented by the roots of integer-coefficient algebraic equations transcendental numbers. Real numbers include both algebraic numbers and transcendental numbers. Rational numbers and π are examples of algebraic numbers; e and π are examples of transcendental numbers. Our question can be rephrased in this new terminology: Are all real numbers algebraic numbers? The answer, as mentioned above, is negative.
However, even though the answer has been revealed, proving that a specific number is transcendental is often not an easy task. For example, it took mathematicians a lot of effort to prove that the two most important constants e and π are transcendental. The transcendence of the number e was proved by the French mathematician Hermite in 1873. Judging whether π is transcendental is even more difficult, and it was not until 1882 that the German mathematician Lindemann proved that π is also transcendental. As for whether simple combinations such as e+π and e-π are transcendental, it remains a mystery to this day.
Next, we can ask another question: Are there more algebraic numbers or transcendental numbers? Judging from the difficulty of constructing and proving transcendental numbers, many readers may guess that only a small number of numbers are transcendental. In fact, the opposite is true. In 1874, the German mathematician Cantor proved that transcendental numbers far outnumber algebraic numbers (this involves the comparison of the number of elements in infinite sets). In fact, he proved that almost all real numbers are transcendental - in other words, if we randomly pick a real number from the set of real numbers, the probability that it is an algebraic number is 0!
The existence of transcendental numbers not only has abstract classification significance, but can also solve some specific mathematical problems. For example, it has been proven that the length of the line segments that can be constructed by the ruler-and-compass method in geometry can only be algebraic numbers (given a unit length). Therefore, the seemingly only having abstract classification significance of the transcendence of π directly proves thatOne of the "Three Geometric Problems of Compass and Straightedge" that have puzzled mathematicians for more than 2,300 years, namely "squaring the circle", is impossible to achieve.