Monday, December 11, 2023

# Famous unknowns | The game of science

Over the past two weeks we have prepared a list of very interesting topics (it is not a majestic plural: several readers have actively participated in the preparation), which, from smallest to largest, are the following:

0, 1, i, √2, Φ, 2, e, π, 5, 8, 9, 10, 113, 6174

(The position of i in the list is arbitrary because it is an imaginary number.)

Note that all digits except 3, 4 and 6 are present. Therefore, it would be a comparative offense not to include them:

#### 3

He is the first cousin and first cousin of Fermat and also the first cousin of Mersenne. It is a Lucas and Fibonacci number… Can you think of other characteristics of the number 3?

#### 4

It is the first composite number and the square of the first prime number. They are a flawed and raised number as well as a pastel and padovan number.

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#### 6

It is the first perfect number (because it is equal to the sum of its divisors: 6 = 1 + 2 + 3) and the product of the first two prime numbers. It’s a practical, elongated number…

Our regular commentators Bretos Bursó and Salva Fuster rightly believe that the demonstration of the irrationality of the square root of 2 (of which we saw the most common version last week) can be expressed more succinctly, as long as we take it for granted that every number will be on unique way factored as a product of prime numbers: It cannot be a² = 2 b², since 2 occurs even often when factoring a² (twice as often as when factoring a) and a odd frequency in 2 b² (twice as often as in b, plus 1). For the same reason, the square root of any prime number generally turns out to be irrational. Furthermore, the square root of a natural number is only rational if it is a perfect square.

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And in addition, Bursó and Fuster propose other interesting numbers: 0.5 = 1/2, since certainly the concept of half was the starting point of rational numbers; the cube root of 2 (1.2599…), due to its importance in the development of algebra; and the Euler constant (or Euler-Mascheroni constant), which is represented by the Greek letter gamma and whose value is 0.5772…

#### Gamma, Omega, Aleph…

Euler’s constant is the limit of 1 + 1/2 + 1/3… + 1/n – log n as n approaches infinity. It is a much less well-known number than π, but even more mysterious: just as we know millions of decimal places for π, only a few thousand could be calculated for Euler’s constant, and it is not even known whether it is rational or irrational.

And when you talk about gamma, Euler’s constant, you inevitably think of other very important numbers, but very little known to the general public and difficult to understand even for laypeople. First, let’s look at a few that are on most mathematicians’ favorite lists:

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##### Chaitin constant

Represented by the Greek letter Omega, it is an irrational number that indicates the probability that a series of instructions will stop a universal Turing machine. It is named in honor of the American mathematician and later Argentine citizen Gregory J. Chaitin, who formulated it in the 1960s.

##### Alef 0

The first of Cantor’s transfinite numbers, which corresponds to the infinity of natural numbers, which is a lower order infinity than the irrational numbers, which are uncountable (i.e. they cannot be brought into a one-to-one correspondence with the irrational numbers ). natural numbers), as Georg Cantor demonstrated in 1873. But that’s another article.

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