First, the origin of irrational numbers:

1, the length of the diagonal of the square is unmeasurable (if the side length of the square is 1, the length of the diagonal is the root number 2, not a rational number);

2. In a circle, the ratio of circumference to diameter is called pi, and pi is "?" It is also an unpredictable number, not a rational number; This unfathomability is quite different from the Pythagorean school's philosophy of "everything is a number" (referring to rational numbers).

Second, the history of irrational numbers:

Pythagoras (about 580 BC to 500 BC) was a great mathematician in ancient Greece. He proved many important theorems, including Pythagorean theorem named after him, that is, the sum of the areas of two right sides of a right triangle is equal to the area of a square with the hypotenuse as the side.

In 500 BC, hippasus, a disciple of Pythagoras School, discovered an amazing fact: the diagonal of a square is incommensurable with the length of one side (if the side length of a square is 1, the length of the diagonal is not a rational number), which is quite different from Pythagoras School's philosophy of "everything is a number" (referring to a rational number).

By the second half of19th century. 1872, the German mathematician Dai Dejin started from the requirement of continuity, defined irrational numbers through the division of rational numbers, and established the theory of real numbers on a strict scientific basis, thus ending the era when irrational numbers were regarded as "irrational numbers" and the first great crisis in the history of mathematics that lasted for more than two thousand years.

The concept and basic introduction of irrational number;

First, the concept of irrational numbers:

Irrational number, also known as infinite acyclic decimal, cannot be written as the ratio of two integers. If written in decimal form, there are infinitely many digits after the decimal point, which will not cycle. Common irrational numbers include the square root, π and E (the latter two are transcendental numbers) of incomplete square numbers.

Another feature of irrational numbers is the expression of infinite connected fractions. Irrational numbers were first discovered by a disciple of Pythagoras.

Second, the basic introduction:

Irrational Numbers In mathematics, irrational numbers are all real numbers that are not rational numbers, and the latter is a number composed of the ratio (or fraction) of integers. When the length of two line segments is irrational, the line segments are also described as incomparable, that is, they cannot be "measured", that is, they have no length ("measured").