Summary of the history of Mathematics

Three crises in the history of Mathematics

Mathematics originated from the counting and measurement of people in life . In the long history of the development of mathematics, there have been many great men , Pythagoras is one of them .

Pythagoras (Pythagoras, B.c. 580 year - B.c. 500 year ) Is an ancient Greek mathematician 、 philosopher . Pythagoras' development of mathematics is mainly in two aspects ： One is Pythagorean theorem , The other is the golden section .

Pythagoras pioneered pure rational Mathematics . And proved the Pythagorean theorem （ Pythagorean theorem ）, But due to the limitations of his time , He denied the existence of irrational numbers , It is called the place where later generations criticized him .

The first crisis

In the age of Pythagoras , People realize that numbers are limited to rational numbers , That is what we usually call scores , They all have $\frac{p}{q}$ Form like this , among $p$ and $q$ Are integers. , such as $\frac{2}{3}$. Of course, integer itself is also a special rational number , Their denominators are 1. Rational numbers have a very good property , Add any two rational numbers 、 reduce 、 ride 、 After division （0 Except for the denominator ）, The result is still a rational number , It's perfect . So integers and fractions are called Rational number (rational number).

It was after the emergence of Pythagoras' theorem , When people re-examine the numbers, they find terrible problems , It destroys the perfection of numbers .

If the length of both right angles of a right triangle is 1, Then how much is the beveled edge ?

According to Pythagorean Theorem , The square of the hypotenuse should be equal to the sum of the two right angle sides , Beveled edge $c^2=2$, And this $c$ Is not a rational number . This is the first mathematical crisis .

Yes The first mathematical crisis is summarized Come on , All numbers are rational numbers , But based on Pythagoras' theorem, it is found that $\sqrt{2}$ Not a rational number . The following proof $\sqrt{2}$ Not a rational number .

prove : Use counter evidence

hypothesis $\sqrt{2}$ It's a reasonable number , Then it can be expressed as $\frac{p}{q}$ In the form of , And meet :

(1) $p,q$- Are integers. ;

(2) $p,q$ Coprime .

(3)$\frac{p}{q}$ The square of is equal to 2.

from (3) have to : $(\frac{p}{q}{)}^2=2\Rightarrow p^2=2q^2$

First of all, we found that $p$ I must be , Because the square of an odd number cannot be even . Then we might as well set $p=2s$, Bring it up to

$$2s^2=q^2$$

The same can be , $q$ It must also be an even number .

since $p,q$ Is an even number , Then they cannot be mutually prime . That explains it $\sqrt{2}$ Not a rational number , But an irrational number (irrational number).

With the discovery and further understanding of irrational numbers , It can be regarded as solving the first mathematical crisis .