1, rational number is one of the important contents in the field of "number and algebra", which is widely used in real life and is the basis for continuing to learn real numbers, algebraic expressions, equations, inequalities, rectangular coordinate systems, functions, statistics and other mathematical contents and related disciplines. Mathematically, a rational number is the ratio of an integer a to a positive integer b, such as 3/8. The general rule is a/b, and 0 is also a rational number.

2. Irrational numbers, also known as infinite acyclic decimals, 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.

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First, the naming origin of rational numbers

The name "rational number" is puzzling, and rational numbers are no more "reasonable" than other numbers. In fact, this seems to be a mistake in translation. The word rational number comes from the west and is rational in English. Rational usually means "rational".

China translated western scientific works in modern times into "rational numbers" according to Japanese translation methods. However, this word comes from ancient Greece, and its English root is ratio, which means ratio (the root here is English and the Greek meaning is the same).

So the meaning of this word is also very clear, that is, the "ratio" of integers. In contrast, "irrational number" is a number that cannot be accurately expressed as the ratio of two integers, but it is not unreasonable.

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.

After Pythagoras skillfully used mathematical knowledge, he felt that he could not be satisfied with solving problems, so he tried to expand from the field of mathematics to the field of philosophy and explain the world from the perspective of numbers.

After some hard training, he put forward the view that "everything is number": the element of number is the element of everything, the world is made up of numbers, and everything in the world can't be expressed by numbers, and numbers themselves are the order of the world.

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