#rational numbers

Rational Numbers A rational number is a number which can be written in the form p/q where p and q are integers and q should be greater than zero. P and q must not have a common factor between then except 1. Examples of rational numbers is like 1/5, 3/7, 4/11, etc. Irrational numbers Irrational are numbers are numbers that cannot be written in the form p/q. Irrational numbers cannot be…

Cantor (1845-1918) conceived of "set theory" and used this concept to cut into "infinity". This is a field that other mathematicians have not dabbled in because of its monstrous appearance.

The weapon is "one-on-one correspondence". For example, natural numbers 1, 2, 3, 4... and even numbers 2, 4, 6...

Which one is more? Suppose you asked me. It is normal to think that "the even numbers are the natural numbers minus the odd numbers, so there are more natural numbers", but Cantor's answer is different. An even number 2 is the first, an even number 4 is the second, and so on. Since all natural numbers and even numbers can be in one-to-one correspondence, it means that there are the same number of natural numbers and even numbers. In this way, we can conclude that there are as many fractions (= rational numbers) as there are natural numbers.

"Then what about irrational numbers?" --Here is a very interesting proof. The goal is to reach a conclusion based on the law of contradiction that "there are more irrational numbers than natural numbers."

If it is possible to have a one-to-one correspondence with the natural numbers, the irrational numbers will be expressed as decimals (rational numbers will be converted to recurring decimals) and all of them will be arranged to create a list. Now, take a number that differs from the first number in the list in the first decimal place, and from the number in the second list in the second digit, and so on. The number obtained by the procedure is not in the list ! ・・・ I was able to derive a contradiction due to contradiction. Irrational numbers are infinitely more than natural numbers. In an elegant proof, this is called the diagonal argument.

Kronecker (1823-1891) raises an objection. This person was quite eccentric and professed that the natural numbers were created by God, and that all other numbers were created by humans. Kronecker seems to have had the ambition to rebuild all mathematics on the basis of natural numbers. That's why he couldn't help but hate Cantor, the guy who came to the conclusion that there are more irrational numbers than natural numbers. Kronecker persecutes Cantor at every opportunity. As a result, he suffered from mental illness and spent the last half of his life in and out of mental hospitals.