For numbers, we are used to the classification of imaginary numbers, real numbers, irrational numbers and rational numbers. ...
But we can also divide it according to the solution of algebraic equation, and call the number that can satisfy the algebraic equation with integral coefficient as algebraic number; Numbers that do not satisfy any algebraic equation with integral coefficients are called transcendental numbers.
Real transcendental number is a special case of irrational number.
The three famous transcendental numbers I know are irrational numbers. They are:
Pi π = 3.14154380.10000000005 ...
The base of natural logarithm e = 2.71828182459045. ...
Tensile constant γ = 0.5772156649015328.
Historical origin of transcendental number
The concept of transcendental number first appeared in the book Introduction to Infinite Analysis published by Euler in 1748. In the sixth chapter of the first volume of the book, he asserted without proof: "If the number B is not a power based on A, its logarithm is no longer an irrational number. In fact, if both A and B are rational numbers, this equation cannot be established, so the logarithm of this number B is not a power based on A, and should be properly named as a transcendental number. " The French mathematician joseph liouville (J. Joseph Liouville, 1809 ~ 1882) proved the existence of transcendental numbers for the first time in history. He constructed a number in 185 1: this infinite decimal was later called "Louisville number". Joseph liouville successfully proved that this number is a transcendental number. Since a complex set contains both algebraic numbers and transcendental numbers, how many are there? More than twenty years after the "Louisville number" was constructed, the mathematician Cantor proved that all sets of algebraic numbers are countable, that is, there are as many algebraic numbers as there are natural numbers! On this basis, Cantor learned that the complex number set is uncountable according to another conclusion in his set theory-the real number set is uncountable, and further came to the conclusion that there must be complex numbers that are not algebraic numbers, so there must be transcendental numbers! This is the first unstructured proof of the existence of transcendental numbers. In other words, Cantor proved their existence without constructing a specific transcendental number! Many proofs in mathematics are realized by unstructured methods. Joseph liouville's method is constructive, that is, actually producing an object and proving it. These two methods are commonly used in mathematical proof. Generally speaking, it is much easier for us to consider a concrete object than an abstract object, but in mathematics, it is sometimes the opposite: it is far more difficult and complicated to prove that a concrete number is a transcendental number than to prove the existence of a transcendental number in an unstructured way. After joseph liouville, mathematicians made various efforts to prove the transcendence of certain numbers: 1873, French mathematician C. Hermite, 1822 ~ L90 1) proved that the base e of natural logarithm is 2.7182818 ...1882. German mathematician Lin Deman (1852 ~/kloc-). 1900, among the 23 questions raised by Hilbert at the International Congress of Mathematicians, the tenth one was about transcendental numbers. Hilbert speculated that numbers like this are transcendental numbers. In 1929, it is proved that it is a transcendental number. 1930 is also proved to be a transcendental number. It is of great significance to prove that some numbers are transcendental. For example, the transcendental proof of π completely solved the problem of turning a circle into a square in the three major drawing problems in ancient Greece, that is, it was impossible to turn a circle into a square. It is too difficult to judge whether some given numbers exceed the number. Mathematicians have worked hard for more than a century to obtain the above results. Even so, this field is still foggy. For example, people still can't tell whether numbers like e+π sum are algebraic numbers or transcendental numbers. Transcendental numbers and algebraic numbers are obviously different, and even the arithmetic rules are different. For example, the addition and multiplication elimination methods that are suitable for algebraic numbers are not suitable for transcendental numbers. For example, if the following formula holds for three transcendental numbers A, B and C: A+B = A+C but B = C is not necessarily true. Similarly, for these three numbers, if the following formula holds: a× b = a× c but b = c is not necessarily true. Even more surprising, according to Cantor's conclusion, although there are infinite algebraic numbers and transcendental numbers, algebraic numbers are countable and transcendental numbers are uncountable. In other words, the number of transcendental numbers that people know little about is much more than algebraic numbers! Mathematics is indeed a vast ocean, even in the field of "number" research, there are many unknown mysteries waiting for people to explore the answer!