With a segment of unit length, ancient mathematicians already knew: they knew, for example, the incommensurability of the diagonal and the side of the square, which is equivalent to the irrationality of the number.
Irrational are:
Irrationality Proof Examples
Root of 2
Assume the contrary: it is rational, that is, it is represented as an irreducible fraction, where and are integers. Let's square the supposed equality:
.From this it follows that even, therefore, even and . Let where the whole. Then
Therefore, even, therefore, even and . We have obtained that and are even, which contradicts the irreducibility of the fraction . Hence, the original assumption was wrong, and is an irrational number.
Binary logarithm of the number 3
Assume the contrary: it is rational, that is, it is represented as a fraction, where and are integers. Since , and can be taken positive. Then
But it's clear, it's odd. We get a contradiction.
e
Story
The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manawa (c. 750 BC - c. 690 BC) found that the square roots of some natural numbers, such as 2 and 61 cannot be explicitly expressed.
The first proof of the existence of irrational numbers is usually attributed to Hippasus of Metapontus (c. 500 BC), a Pythagorean who found this proof by studying the lengths of the sides of a pentagram. In the time of the Pythagoreans, it was believed that there is a single unit of length, sufficiently small and indivisible, which is an integer number of times included in any segment. However, Hippasus argued that there is no single unit of length, since the assumption of its existence leads to a contradiction. He showed that if the hypotenuse of an isosceles right triangle contains an integer number of unit segments, then this number must be both even and odd at the same time. The proof looked like this:
- The ratio of the length of the hypotenuse to the length of the leg of an isosceles right triangle can be expressed as a:b, where a and b selected as the smallest possible.
- According to the Pythagorean theorem: a² = 2 b².
- As a² even, a must be even (since the square of an odd number would be odd).
- Insofar as a:b irreducible b must be odd.
- As a even, denote a = 2y.
- Then a² = 4 y² = 2 b².
- b² = 2 y², therefore b is even, then b even.
- However, it has been proven that b odd. Contradiction.
Greek mathematicians called this ratio of incommensurable quantities alogos(inexpressible), but according to the legends, Hippasus was not paid due respect. There is a legend that Hippasus made the discovery while on a sea voyage and was thrown overboard by other Pythagoreans "for creating an element of the universe, which denies the doctrine that all entities in the universe can be reduced to whole numbers and their ratios." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the assumption underlying the whole theory that numbers and geometric objects are one and inseparable.
see also
Notes
| Numerical systems | |
|---|---|
| Counting sets |
Integers () |
We have already shown earlier that $1\frac25$ is close to $\sqrt2$. If it were exactly equal to $\sqrt2$, . Then the ratio - $\frac(1\frac25)(1)$, which can be turned into a ratio of integers $\frac75$ by multiplying the upper and lower parts of the fraction by 5, would be the desired value.
But, unfortunately, $1\frac25$ is not the exact value of $\sqrt2$. A more precise answer $1\frac(41)(100)$ is given by the relation $\frac(141)(100)$. We achieve even greater accuracy when we equate $\sqrt2$ to $1\frac(207)(500)$. In this case, the ratio in integers will be equal to $\frac(707)(500)$. But $1\frac(207)(500)$ is not the exact value of the square root of 2 either. Greek mathematicians spent a lot of time and effort to calculate the exact value of $\sqrt2$, but they never succeeded. They failed to represent the ratio $\frac(\sqrt2)(1)$ as a ratio of integers.
Finally, the great Greek mathematician Euclid proved that no matter how the accuracy of calculations increases, it is impossible to get the exact value of $\sqrt2$. There is no such fraction that, when squared, will result in 2. It is said that Pythagoras was the first to come to this conclusion, but this inexplicable fact impressed the scientist so much that he swore himself and took an oath from his students to keep this discovery a secret . However, this information may not be true.
But if the number $\frac(\sqrt2)(1)$ cannot be represented as a ratio of integers, then no number containing $\sqrt2$, for example $\frac(\sqrt2)(2)$ or $\frac (4)(\sqrt2)$ also cannot be represented as a ratio of integers, since all such fractions can be converted to $\frac(\sqrt2)(1)$ multiplied by some number. So $\frac(\sqrt2)(2)=\frac(\sqrt2)(1) \times \frac12$. Or $\frac(\sqrt2)(1) \times 2=2\frac(\sqrt2)(1)$, which can be converted by multiplying the top and bottom by $\sqrt2$ to get $\frac(4) (\sqrt2)$. (We should not forget that no matter what the number $\sqrt2$ is, if we multiply it by $\sqrt2$ we get 2.)
Since the number $\sqrt2$ cannot be represented as a ratio of integers, it is called irrational number. On the other hand, all numbers that can be represented as a ratio of integers are called rational.
All integer and fractional numbers, both positive and negative, are rational.
As it turns out, most square roots are irrational numbers. Rational square roots are only for numbers included in a series of square numbers. These numbers are also called perfect squares. Rational numbers are also fractions made up of these perfect squares. For example, $\sqrt(1\frac79)$ is a rational number because $\sqrt(1\frac79)=\frac(\sqrt16)(\sqrt9)=\frac43$ or $1\frac13$ (4 is the root square of 16, and 3 is the square root of 9).
The material of this article is the initial information about irrational numbers . First, we will give a definition of irrational numbers and explain it. Here are some examples of irrational numbers. Finally, let's look at some approaches to finding out if a given number is irrational or not.
Page navigation.
Definition and examples of irrational numbers
In the study of decimal fractions, we separately considered infinite non-periodic decimal fractions. Such fractions arise in the decimal measurement of the lengths of segments that are incommensurable with a single segment. We also noted that infinite non-recurring decimals cannot be converted to common fractions(see the translation of ordinary fractions to decimals and vice versa), therefore, these numbers are not rational numbers, they represent the so-called irrational numbers.
So we came to definition of irrational numbers.
Definition.
Numbers that in decimal notation represent infinite non-recurring decimal fractions are called irrational numbers.
The sounded definition allows to bring examples of irrational numbers. For example, the infinite non-periodic decimal fraction 4.10110011100011110000… (the number of ones and zeros increases by one each time) is an irrational number. Let's give another example of an irrational number: −22.353335333335 ... (the number of triples separating eights increases by two each time).
It should be noted that irrational numbers are quite rare in the form of infinite non-periodic decimal fractions. Usually they are found in the form , etc., as well as in the form of specially introduced letters. The most famous examples of irrational numbers in such a notation are arithmetic Square root out of two, the number “pi” π=3.141592…, the number e=2.718281… and golden number.
Irrational numbers can also be defined in terms of real numbers, which combine rational and irrational numbers.
Definition.
Irrational numbers- This real numbers, which are not rational.
Is this number irrational?
When a number is given not as a decimal fraction, but as a certain root, logarithm, etc., then in many cases it is quite difficult to answer the question of whether it is irrational.
Undoubtedly, in answering the question posed, it is very useful to know which numbers are not irrational. It follows from the definition of irrational numbers that rational numbers are not irrational numbers. Thus, irrational numbers are NOT:
- finite and infinite periodic decimal fractions.
Also, any composition of rational numbers connected by signs is not an irrational number. arithmetic operations(+, −, ·, :). This is because the sum, difference, product and quotient of two rational numbers is a rational number. For example, the values of the expressions and are rational numbers. Here we note that if in such expressions among rational numbers there is one single irrational number, then the value of the entire expression will be an irrational number. For example, in the expression, the number is irrational, and the rest of the numbers are rational, therefore, the irrational number. If it were a rational number, then the rationality of the number would follow from this, but it is not rational.
If the expression, which is given a number, contains several irrational numbers, root signs, logarithms, trigonometric functions, numbers π, e, etc., then it is required to prove the irrationality or rationality of a given number in each specific case. However, there are a number of already obtained results that can be used. Let's list the main ones.
It is proved that a k-th root of an integer is a rational number only if the number under the root is the k-th power of another integer, in other cases such a root defines an irrational number. For example, the numbers and are irrational, since there is no integer whose square is 7, and there is no integer whose raising to the fifth power gives the number 15. And the numbers and are not irrational, since and .
As for logarithms, sometimes it is possible to prove their irrationality by contradiction. For example, let's prove that log 2 3 is an irrational number.
Let's say that log 2 3 is a rational number, not an irrational one, that is, it can be represented as an ordinary fraction m/n . and allow us to write the following chain of equalities: . The last equality is impossible, since on its left side odd number, and even on the right side. So we came to a contradiction, which means that our assumption turned out to be wrong, and this proves that log 2 3 is an irrational number.
Note that lna for any positive and non-unit rational a is an irrational number. For example, and are irrational numbers.
It is also proved that the number e a for any non-zero rational a is irrational, and that the number π z for any non-zero integer z is irrational. For example, numbers are irrational.
Irrational numbers are also the trigonometric functions sin , cos , tg and ctg for any rational and non-zero value of the argument. For example, sin1 , tg(−4) , cos5,7 , are irrational numbers.
There are other proven results, but we will restrict ourselves to those already listed. It should also be said that in proving the above results, the theory associated with algebraic numbers and transcendent numbers.
In conclusion, we note that one should not make hasty conclusions about the irrationality of the given numbers. For example, it seems clear that an irrational number in irrational degree is an irrational number. However, this is not always the case. As a confirmation of the voiced fact, we present the degree. It is known that - an irrational number, and also proved that - an irrational number, but - a rational number. You can also give examples of irrational numbers, the sum, difference, product and quotient of which are rational numbers. Moreover, the rationality or irrationality of the numbers π+e , π−e , π e , π π , π e and many others has not yet been proven.
Bibliography.
- Mathematics. Grade 6: textbook. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
- Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
irrational number- This real number, which is not rational, that is, cannot be represented as a fraction, where are integers, . An irrational number can be represented as an infinite non-repeating decimal.
The set of irrational numbers is usually denoted by a capital Latin letter in bold without shading. Thus: , i.e. set of irrational numbers is difference of sets of real and rational numbers.
On the existence of irrational numbers, more precisely segments that are incommensurable with a segment of unit length, the ancient mathematicians already knew: they knew, for example, the incommensurability of the diagonal and the side of the square, which is equivalent to the irrationality of the number.
Properties
- Any real number can be written as an infinite decimal fraction, while irrational numbers and only they are written as non-periodic infinite decimal fractions.
- Irrational numbers define Dedekind cuts in the set of rational numbers that have no largest number in the lower class and no smallest number in the upper class.
- Every real transcendental number is irrational.
- Every irrational number is either algebraic or transcendental.
- The set of irrational numbers is everywhere dense on the real line: between any two numbers there is an irrational number.
- The order on the set of irrational numbers is isomorphic to the order on the set of real transcendental numbers.
- The set of irrational numbers is uncountable, is a set of the second category.
Examples
| Irrational numbers - ζ(3) - √2 - √3 - √5 - - - - - |
Irrational are:
Irrationality Proof Examples
Root of 2
Assume the contrary: it is rational, that is, it is represented as an irreducible fraction, where is an integer, and is a natural number. Let's square the supposed equality:
.From this it follows that even, therefore, even and . Let where the whole. Then
Therefore, even, therefore, even and . We have obtained that and are even, which contradicts the irreducibility of the fraction . Hence, the original assumption was wrong, and is an irrational number.
Binary logarithm of the number 3
Assume the contrary: it is rational, that is, it is represented as a fraction, where and are integers. Since , and can be taken positive. Then
But it's clear, it's odd. We get a contradiction.
e
Story
The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manawa (c. 750 BC - c. 690 BC) found that the square roots of some natural numbers, such as 2 and 61 cannot be explicitly expressed.
The first proof of the existence of irrational numbers is usually attributed to Hippasus of Metapontus (c. 500 BC), a Pythagorean who found this proof by studying the lengths of the sides of a pentagram. In the time of the Pythagoreans, it was believed that there is a single unit of length, sufficiently small and indivisible, which is an integer number of times included in any segment. However, Hippasus argued that there is no single unit of length, since the assumption of its existence leads to a contradiction. He showed that if the hypotenuse of an isosceles right triangle contains an integer number of unit segments, then this number must be both even and odd at the same time. The proof looked like this:
- The ratio of the length of the hypotenuse to the length of the leg of an isosceles right triangle can be expressed as a:b, where a and b selected as the smallest possible.
- According to the Pythagorean theorem: a² = 2 b².
- As a² even, a must be even (since the square of an odd number would be odd).
- Insofar as a:b irreducible b must be odd.
- As a even, denote a = 2y.
- Then a² = 4 y² = 2 b².
- b² = 2 y², therefore b is even, then b even.
- However, it has been proven that b odd. Contradiction.
Greek mathematicians called this ratio of incommensurable quantities alogos(inexpressible), but according to the legends, Hippasus was not paid due respect. There is a legend that Hippasus made the discovery while on a sea voyage and was thrown overboard by other Pythagoreans "for creating an element of the universe, which denies the doctrine that all entities in the universe can be reduced to whole numbers and their ratios." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the underlying assumption that numbers and geometric objects are one and inseparable.
Definition of an irrational number
Irrational numbers are those numbers that, in decimal notation, are infinite non-periodic decimal fractions.
So, for example, numbers obtained by taking the square root of natural numbers are irrational and are not squares of natural numbers. But not all irrational numbers are obtained by extracting square roots, because the number "pi" obtained by dividing is also irrational, and you are unlikely to get it when trying to extract the square root from a natural number.
Properties of irrational numbers
Unlike numbers written in infinite decimal fractions, only irrational numbers are written in non-periodic infinite decimal fractions.
The sum of two non-negative irrational numbers can eventually be a rational number.
Irrational numbers define Dedekind sections in the set of rational numbers, in the lower class of which there is no largest number, and in the upper class there is no smaller one.
Any real transcendental number is irrational.
All irrational numbers are either algebraic or transcendental.
The set of irrational numbers on the line are densely packed, and between any two of its numbers there is bound to be an irrational number.
The set of irrational numbers is infinite, uncountable and is a set of the 2nd category.
When performing any arithmetic operation on rational numbers, except division by 0, its result will be a rational number.
When adding a rational number to an irrational number, the result is always an irrational number.
When adding irrational numbers, we can get a rational number as a result.
The set of irrational numbers is not even.
Numbers are not irrational
Sometimes it is quite difficult to answer the question of whether a number is irrational, especially in cases where the number is in the form of a decimal fraction or in the form numeric expression, root or logarithm.
Therefore, it will not be superfluous to know which numbers are not irrational. If we follow the definition of irrational numbers, then we already know that rational numbers cannot be irrational.
Irrational numbers are not:
First, all natural numbers;
Second, integers;
Thirdly, ordinary fractions;
Fourth, different mixed numbers;
Fifth, these are infinite periodic decimal fractions.
In addition to all of the above, any combination of rational numbers that is performed by the signs of arithmetic operations, such as +, -, , :, cannot be an irrational number, since in this case the result of two rational numbers will also be a rational number.
Now let's see which of the numbers are irrational:
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