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.
Irrational number is a concept opposite to rational number, which refers to a real number that cannot be expressed by the ratio of two integers. Simply put, an irrational number is a real number that cannot be expressed by fractions or decimals.
Irrational numbers were first discovered in Greece in the 5th century BC. At that time, scholars found that some short-term segments could not be composed of integers. This discovery broke the belief of ancient Greeks and Euclid that everything can be composed of integers, and inspired more work to explore irrational numbers.
There are many differences between irrational numbers and rational numbers in nature. First of all, irrational numbers are infinite cyclic decimals, while rational numbers can be finite cyclic decimals or infinite cyclic decimals. For example, π and √2 are irrational numbers, and their values cannot be accurately expressed whether expressed in fractions or decimals.
In addition, irrational numbers have no expression, that is, there is no clear derivation, and only after further financial mathematics and geometric calculation can they be accurately defined.
Irrational numbers have a very important application in mathematics. In geometry, irrational numbers are widely used in some important theorems and formulas, such as Pythagorean theorem.
For example, in Pythagorean Theorem A? +b？ =c？ If a and b are positive integers, then c is irrational number 2. In physics, irrational numbers are often used to describe the quantity of physical quantities. For example, when measuring pi, the higher the precision of irrational numbers, the more accurate the calculated values will be.
In a word, irrational numbers explain the transcendence of a number, break through the limitations of integers and expand our understanding of logarithm and mathematics. Historically, the process of discovering irrational numbers also contains human's thirst for knowledge and profound observation of the world, constantly opening up the boundaries of human thinking and creating a broader space for later mathematical science.
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.
Basic introduction
Chinese name: irrational number
Mbth: irrational number
Nickname: infinite acyclic decimal
Speaker: Hibersos
Applied subject: mathematics
Property: cannot be expressed in fractions.
Corresponding concept: rational number
Range: real number
Keywords definition, history, proof method, extension, example,
definition
In mathematics, irrational numbers are all real numbers of irrational numbers, which are numbers 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"). Common irrational numbers are: the ratio of circumference to diameter, Euler number e, golden ratio φ and so on. It can be seen that the representation of irrational numbers in the positional number system (for example, in decimal numbers or any other natural basis) will not be terminated or repeated, that is, it does not contain subsequences of numbers. For example, the decimal representation of the number π starts from 3.141592653589793, but there is no finite number that can accurately represent π, so it is not repeated. The evidence that the decimal extension of rational numbers must be terminated or repeated is different from the evidence that rational numbers must be terminated or repeated. Although this is basic and not lengthy, both proofs need some work. Mathematicians usually don't define "termination or repetition" as the concept of rational numbers. Irrational numbers can also be treated with non-terminating continued fractions. Irrational number refers to a number that cannot be expressed as the ratio of two integers within the real number range. Simply put, an irrational number is an infinite cyclic decimal with 10 as the base, such as pi and so on. Rational numbers are composed of all fractions and integers, which can always be written as integers, finite decimals or infinite cyclic decimals, and can always be written as the ratio of two integers, such as 2 1/7.
history
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. 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). This discovery frightened the leaders of the school, thinking that it would shake their dominant position in the academic world, so they tried their best to stop the spread of this truth, and Herbesos was forced into exile. Unfortunately, he met his disciples on a seagoing ship. Was brutally thrown into the water by Pisces disciples and killed. Thus began the history of science, but it was a tragedy. The discovery of Herbesos revealed the defects of rational number system for the first time, and proved that it could not be treated as a continuous infinite line. Rational numbers are not covered by points on the number axis, and there are "holes" on the number axis that rational numbers cannot express. And this kind of "pore" was proved to be "countless" by later generations. In this way, the ancient Greeks' idea that rational numbers were regarded as a continuous arithmetic continuum was completely shattered. The discovery of incommensurable measure, together with Zeno's paradox, is called the first mathematical crisis in the history of mathematics, which has had a far-reaching impact on the development of mathematics for more than 2000 years, prompting people to rely on proof instead of intuition and experience, promoting the development of axiomatic geometry and logic, and gestating the bud of calculus thought. What is the nature of irreducibility? There have been different opinions for a long time, and there is no correct explanation. The ratio of two incommensurable degrees has always been considered unreasonable. /kloc-Leonardo da Vinci, a famous Italian painter in the 0/5th century, called it an "irrational number", and Kepler, a German astronomer in the 0/7th century, called it an "indescribable number". However, after all, the truth cannot be submerged, and it is "unreasonable" for the main Sect to obliterate the truth. People named this incommensurable quantity "irrational number" in memory of this respectable scholar Ebersus who devoted himself to truth-this is the origin of irrational number. The mathematical crisis caused by irrational numbers lasted until 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. ? Fraction = finite decimal+infinite cyclic decimal, which is irrational.
Proof method
In Euclid's Elements of Geometry, a classical method to prove irrational numbers is put forward: prove that √2 is irrational, assuming that it is not irrational ∴ it is rational, so that the squares of both sides of (,coprime sum,) can be obtained by shifting terms, and it is obtained that ∴ must be even ∴ must be even ∴ if simplified, it must be.
develop
Proved to be an irrational number (integer), coprime. If there is an assumption, then a is an even number. If there is a positive integer, then b is even. If there is a contradiction with the minimum integer whose condition (,) is coprime, then the assumption is not true, then it must be an irrational number.
example
If the positive integer n is not a complete square number, it is not a rational number (it is an irrational number). Proof: Assuming rational numbers, we assume that both P and Q are positive integers (not necessarily coprime). If the integer part of p and q is assumed to be a, there is inequality. When both sides are multiplied by q, P, Q and A are integers, and p-aq is also a positive integer. Multiply on both sides of the above inequality and get: obviously, qN-ap is also a positive integer. So we found two new positive integer sums, satisfaction, that is, sum and have. By repeating the above steps, you can find a series of methods to make peace. Because this step can be repeated indefinitely, it means that it can be reduced indefinitely, but this contradicts that the minimum positive integer is 1. So the assumption is wrong, not rational.