The Brilliant Unknowns | Science game

In the past two weeks, we have prepared a list of very interesting cases (it is not a grandiose plural: many readers have actively participated in its preparation), which is, from smallest to largest, as follows:

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

(The location of i in the list is random, because it is an imaginary number.)

Note that all numbers are present except 3, 4 and 6, so it would be a relative crime not to include them:

3

He is Gharib’s first cousin and Verma’s first cousin, and also Mersenne’s first cousin; It is the Lucas and Fibonacci number.. Can you think of other properties of the number 3?

4

It is the first complex number and the square of the first prime number. It is a flawed and sublime number, as are those of Bastille and Padovan.

6

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

Our regular commentators Britus Borsu and Silva Foster rightly argue that the proof of the irrationality of the square root of 2 (the most popular version of which we saw last week) could be expressed more succinctly, since we take it for granted. Each number is factored as a product of prime numbers in a unique way: a² cannot be = 2 b², because 2 appears an even number of times in the factorization of a² (twice as many times as it appears in the factorization of a) and a is an odd number of times in 2 b² (twice as many The times it appears in b, in addition to 1). In general, for the same reason, it turns out that the square root of any prime number is irrational. Furthermore, the square root of a natural number is rational only if it is a perfect square.

In addition, Bourseau and Foster suggest other interesting numbers: 0.5 = 1/2, since the concept of half was certainly the starting point for rational numbers; The cube root of 2 (1.2599…), due to its importance in the development of algebra; The Euler constant (or Euler-Mascheroni), which is represented by the Greek letter gamma, has a value of 0.5772…

Gamma, Omega, Alpha…

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 much more mysterious: just as we know millions of decimal places for π, only a few thousand have been able to calculate Euler’s constant, and it is not even known whether it is rational or irrational.

When we talk about gamma, Euler’s constant, it is inevitable to think about other numbers that are very important but little known to the general public, and even difficult to understand for non-specialists. Let’s now see a couple that appear on most mathematicians’ favorite lists:

Chaitin fixed

It is symbolized by the Greek letter omega, which is an irrational number that represents the probability that a set of instructions will stop the Universal Turing Machine. It was named after the American mathematician turned Argentine citizen Gregory J. Chaitin, who formulated it in the 1960s.

thousand 0

The first of Cantor’s infinite numbers, which correspond to the infinity of natural numbers, is an infinity of a lower order than irrational numbers, which cannot be counted (i.e., cannot be placed in one-to-one correspondence with natural numbers), as Georg Cantor demonstrated in 1873. But This is another article.

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