Irrational number

[wú lǐ shù]

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Irrational numbers, also known as infinite non recurring decimal numbers, cannot be written as two integer Ratio. If it is written as decimal Form, there are unlimited numbers after the decimal point, and they will not loop 。 Common irrational numbers are Incomplete square Of square root 、 π and e (The latter two are both Transcendental number ）Etc. Another characteristic of irrational numbers is infinite Continued fraction expression. Irrational numbers were first created by Pythagorean school disciple Hebersos Discovery.^[1]

Chinese name: Irrational number
Foreign name: Irrational number
Alias: Infinite acyclic decimal
Presenter: Hebersos

Applied discipline: mathematics
Nature: It cannot be expressed by scores
Corresponding concept: Rational number
Scope: real number

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In mathematics, irrational numbers are all real numbers that are not rational numbers, and the latter are numbers composed of the ratio (or fraction) of integers. When the length ratio of two line segments is irrational, the line segments are also described as non comparable, which means that they cannot be "measured", that is, there is no length ("measured").

Common irrational numbers are: the ratio of circumference to diameter π, Euler number e, golden ratio φ, etc.

It can be seen that the representation of irrational numbers in the positional numeral system (for example, in decimal numbers or any other natural basis) does not terminate or repeat, that is, it does not contain a subsequence of numbers. For example, the decimal representation of the number π starts from 3.141592653589793, but there is no finite number of digits that can accurately represent π and do not repeat. The evidence of the decimal extension of a rational number that must be terminated or repeated is different from the evidence of the decimal extension of a rational number that must be terminated or repeated. Although it is basic and not lengthy, both proofs need some work. Mathematicians usually do not take "termination or repetition" as the definition of rational number.

Irrational numbers can also be processed by non terminating consecutive fractions.

An irrational number is a number that cannot be expressed as the ratio of two integers within the range of real numbers. In simple terms, irrational numbers are infinite non cyclic decimals in decimal system 10, such as pi

Etc.

The rational number is composed of all fractions and integers, and can always be written as integer , finite decimal or infinite Recurring decimal , and can always be written as the ratio of two integers, such as 3/4, etc.^[2]

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Infinite acyclic decimal Π

Pythagoras (Pythagoras, from about 580 BC to 500 BC) was a mathematician in ancient Greece. He proved many important theorems, including those later named after him Pythagoras theorem （ Pythagorean theorem ）, i.e right triangle Two right angled sides are long square The sum of the areas of is equal to hypotenuse Is the area of a square with long sides. When Pythagoras applied mathematical knowledge skillfully, he felt that he could not only use it to solve problems, so he tried to expand from the field of mathematics to the field of philosophy, explaining the world from the perspective of numbers. After a lot of hard practice, he put forward the view that "all things are numbers": the elements of numbers are the elements of all things, the world is composed of numbers, everything in the world can not be expressed by numbers, and numbers themselves are the order of the world.

Irrational number

In 500 BC, Pythagorean school 's disciple Hebersos (Hippasus) discovered an amazing fact that the length of the diagonal of a square and its side is incommensurable (if the side length of the square is 1, the length of the diagonal √ 2 is not a rational number), which is quite different from the philosophy of "all things are numbers" (referring to rational numbers) of the Bishop School. This discovery terrified the leaders of the school, believing that it would shake their dominance in the academic world, so they tried to block the spread of the truth, and Hebrews was forced to exile. Unfortunately, Hebrews still met the disciples of Bishop on a sea boat, and they were brutally thrown into the water to kill them. The history of science has just begun, but it is a tragedy.

Irrational number

Hebrews' discovery revealed the defects of the rational number system for the first time, proving that it cannot be treated equally with continuous infinite straight lines, and rational numbers are not full Number axis The point on the number axis has "pores" that cannot be expressed by rational numbers. This kind of "pore" has been proved by later generations to be "countless". Therefore, the ancient Greeks regarded rational numbers as the kind of continuous connection arithmetic Continuum The idea of ". Uncommeasurable discovery together with Zeno paradox Known as The first mathematical crisis Has had a profound impact on the development of mathematics for more than 2000 years, prompting people to turn from relying on intuition and experience to relying on proof, and promoting axiom The development of geometry and logic Calculus Thought sprouts.

What is the essence of irreducibility? For a long time, there have been different opinions, and no correct explanation can be obtained. The two incommensurable ratios have also been considered as unreasonable numbers. Famous Italian painter in the 15th century Da Vinci Called "irrational number", German astronomer in the 17th century Kepler It is called "nameless" number.

However, truth cannot be concealed after all, and it is "unreasonable" for Bi School to obliterate truth. People named incommensurable quantities "irrational numbers" in memory of Hebrews, a respectable scholar who dedicated himself to the truth - that is the origin of irrational numbers.

Fraction=finite decimal+infinite cyclic decimal, infinite non cyclic decimal is irrational

Caused by irrational numbers Mathematical crisis It lasted until the second half of the 19th century. 1872, German mathematician Dedekind from Continuity Based on the requirements of, the "division" of rational numbers is used to define irrational numbers, and Real number theory It is based on strict science, thus ending the era when irrational numbers are considered "irrational", and ending the History of mathematics The first big crisis on.