- π
Thus, many ir rational numbers there is a difference I = R ∖ Q (\displaystyle \mathbb (I) =\mathbb (R) \backslash \mathbb (Q) ) sets of real and rational numbers.
The existence of irrational numbers, more precisely, segments incommensurable with a segment of unit length, was already known to ancient mathematicians: they knew, for example, the incommensurability of the diagonal and the side of a square, which is equivalent to the irrationality of the number 2 (\displaystyle (\sqrt (2))).
Properties
- The sum of two positive irrational numbers can be a rational number.
- Irrational numbers define Dedekind sections in the set of rational numbers that do not have a largest number in the lower class and do not have a smallest number in the upper class.
- The set of irrational numbers is dense everywhere on the number line: between any two different 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. [ ]
Algebraic and transcendental numbers
Every irrational number is either algebraic or transcendental. The set of algebraic numbers is a countable set. Since the set of real numbers is uncountable, the set of irrational numbers is uncountable.
The set of irrational numbers is a set of the second category.
Let's square the supposed equality:
2 = m n ⇒ 2 = m 2 n 2 ⇒ m 2 = 2 n 2 (\displaystyle (\sqrt (2))=(\frac (m)(n))\Rightarrow 2=(\frac (m^(2 ))(n^(2)))\Rightarrow m^(2)=2n^(2)).Story
Antiquity
The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manava (ca. 750-690 BC) figured out that square roots some natural numbers, such as 2 and 61, cannot be expressed explicitly [ ] .
The first proof of the existence of irrational numbers, or more precisely the existence of incommensurable segments, is usually attributed to the Pythagorean Hippasus of Metapontum (approximately 470 BC). At the time of the Pythagoreans, it was believed that there was a single unit of length, sufficiently small and indivisible, which included an integer number of times in any segment [ ] .
There is no exact data on which number was proven irrational by Hippasus. According to legend, he found it by studying the lengths of the sides of the pentagram. Therefore, it is reasonable to assume that this was the golden ratio since this is the ratio of the diagonal to the side in a regular pentagon.
Greek mathematicians called this ratio of incommensurable quantities alogos(unspeakable), but according to the legends they did not pay due respect to Hippasus. 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 that denies the doctrine that all entities in the universe can be reduced to integers and their ratios.” The discovery of Hippasus challenged Pythagorean mathematics serious problem, destroying the underlying assumption of the entire theory that numbers and geometric objects are one and inseparable.
Later, Eudoxus of Cnidus (410 or 408 BC - 355 or 347 BC) developed a theory of proportions that took into account both rational and irrational relationships. This served as the basis for understanding the fundamental essence of irrational numbers. Quantity began to be considered not as a number, but as a designation of entities, such as line segments, angles, areas, volumes, time intervals - entities that can change continuously (in the modern sense of the word). Magnitudes were contrasted with numbers, which can only change by “jumps” from one number to the next, for example, from 4 to 5. Numbers are made up of the smallest indivisible quantity, while quantities can be reduced indefinitely.
Since no quantitative value was correlated with magnitude, Eudoxus was able to cover both commensurate and incommensurable quantities when defining a fraction as the ratio of two quantities, and proportion as the equality of two fractions. By removing quantitative values (numbers) from the equations, he avoided the trap of having to call an irrational quantity a number. Eudoxus's theory allowed Greek mathematicians to make incredible progress in geometry, providing them with the necessary logical basis for working with incommensurable quantities. The tenth book of Euclid's Elements is devoted to the classification of irrational quantities.
Middle Ages
The Middle Ages were marked by the adoption of concepts such as zero, negative numbers, integers and fractions, first by Indian and then by Chinese mathematicians. Later, Arab mathematicians joined in and were the first to consider negative numbers as algebraic objects (along with positive numbers), which made it possible to develop the discipline now called algebra.
Arab mathematicians combined the ancient Greek concepts of “number” and “magnitude” into a single, more general idea of real numbers. They were critical of Euclid's ideas about relations; in contrast, they developed a theory of relations of arbitrary quantities and expanded the concept of number to relations of continuous quantities. In his commentary on Euclid's Book 10 Elements, the Persian mathematician Al Makhani (c. 800 CE) explored and classified quadratic irrational numbers (numbers of the form) and the more general cubic irrational numbers. He defined rational and irrational quantities, which he called irrational numbers. He easily operated with these objects, but talked about them as separate objects, for example:
In contrast to Euclid's concept that quantities are primarily line segments, Al Makhani considered integers and fractions to be rational quantities, and square and cube roots to be irrational. He also introduced the arithmetic approach to the set of irrational numbers, since it was he who showed the irrationality of the following quantities:
The Egyptian mathematician Abu Kamil (c. 850 CE - c. 930 CE) was the first to consider it acceptable to recognize irrational numbers as solutions quadratic equations or coefficients in equations - mainly in the form of square or cubic roots, as well as roots of the fourth degree. In the 10th century, the Iraqi mathematician Al Hashimi derived general evidence(and not visual geometric demonstrations) of the irrationality of the product, quotient and results of other mathematical transformations over irrational and rational numbers. Al Khazin (900 AD - 971 AD) gives the following definition of rational and irrational quantity:
| Let a unit quantity be contained in a given quantity one or more times, then this [given] quantity corresponds to a whole number... Every quantity that is half, or a third, or a quarter of a unit quantity, or, when compared with a unit quantity, is three-fifths of it, is rational quantity. And in general, any quantity that is related to a unit as one number is to another is rational. If a quantity cannot be represented as several or a part (l/n), or several parts (m/n) of a unit length, it is irrational, that is, inexpressible except with the help of roots. |
Many of these ideas were later adopted by European mathematicians after the translation of Arabic texts into Latin in the 12th century. Al Hassar, an Arab mathematician from the Maghreb who specialized in Islamic inheritance laws, introduced modern symbolic mathematical notation for fractions in the 12th century, dividing the numerator and denominator by a horizontal bar. The same notation then appeared in the works of Fibonacci in the 13th century. During the XIV-XVI centuries. Madhava from Sangamagrama and representatives of the Kerala School of Astronomy and Mathematics investigated endless rows, converging to some irrational numbers, for example, to π, and also showed the irrationality of some trigonometric functions. Jestadeva presented these results in the book Yuktibhaza. (proving at the same time the existence of transcendental numbers), thereby rethinking the work of Euclid on the classification of irrational numbers. Works on this topic were published in 1872
Continued fractions, closely related to irrational numbers (a continued fraction representing a given number is infinite if and only if the number is irrational), were first explored by Cataldi in 1613, then came to attention again in the work of Euler, and in early XIX century - in the works of Lagrange. Dirichlet also made significant contributions to the development of the theory of continued fractions. In 1761, Lambert used continued fractions to show that π (\displaystyle \pi ) is not a rational number, and also that e x (\displaystyle e^(x)) And tg x (\displaystyle \operatorname (tg) x) are irrational for any non-zero rational x (\displaystyle x). Although Lambert's proof can be called incomplete, it is generally considered to be quite rigorous, especially considering the time it was written. Legendre in 1794, after introducing the Bessel-Clifford function, showed that π 2 (\displaystyle \pi ^(2)) irrational, where does irrationality come from? π (\displaystyle \pi ) follows trivially (a rational number squared would give a rational).
The existence of transcendental numbers was proven by Liouville in 1844-1851. Later, Georg Cantor (1873) showed their existence using a different method, and argued that any interval of the real series contains an infinite number of transcendental numbers. Charles Hermite proved in 1873 that e transcendental, and Ferdinand Lindemann in 1882, based on this result, showed transcendence π (\displaystyle \pi ) Literature
The abstractness of mathematical concepts sometimes emanates so much detachment that the thought involuntarily arises: “Why is all this for?” But, despite the first impression, all theorems, arithmetic operations, functions, etc. - nothing more than a desire to satisfy basic needs. This can be seen especially clearly in the example of the appearance of various sets.
It all started with the appearance of natural numbers. And, although it is unlikely that now anyone will be able to answer how exactly it was, most likely, the legs of the queen of sciences grow from somewhere in the cave. Here, analyzing the number of skins, stones and tribesmen, a person has many “numbers to count.” And that was enough for him. Until some point, of course.
Then the skins and stones had to be divided and taken away. This is how the need arose for arithmetic operations, and with them rational ones, which can be defined as a fraction like m/n, where, for example, m is the number of skins, n is the number of fellow tribesmen.
It would seem that the already discovered mathematical apparatus is quite enough to enjoy life. But it soon turned out that there are cases when the result is not only not an integer, but not even a fraction! And, indeed, the square root of two cannot be expressed in any other way using a numerator and a denominator. Or, for example, the well-known number Pi, discovered by the ancient Greek scientist Archimedes, is also not rational. And over time, such discoveries became so numerous that all numbers that could not be “rationalized” were combined and called irrational.
Properties
The sets considered earlier belong to a set of fundamental concepts of mathematics. This means that they cannot be defined through simpler mathematical objects. But this can be done with the help of categories (from the Greek “statements”) or postulates. In this case, it was best to indicate the properties of these sets.
o Irrational numbers define Dedekind cuts in the set of rational numbers that do not have a largest number in the lower number and do not have a smallest number in the upper one.
o Every transcendental number is irrational.
o Every irrational number is either algebraic or transcendental.
o The set of numbers is dense everywhere on the number line: between any there is an irrational number.
o The set is uncountable and is a set of the second Baire category.
o This set is ordered, that is, for every two different rational numbers a and b, you can indicate which one is less than the other.
o Between every two different rational numbers there is at least one more, and therefore an infinite number of rational numbers.
o Arithmetic operations (addition, multiplication and division) on any two rational numbers are always possible and result in a certain rational number. The exception is division by zero, which is impossible.
o Every rational number can be represented as decimal(finite or infinite periodic).
The ancient mathematicians already knew about a segment of unit length: 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:
Examples of proof of irrationality
Root of 2
Let us assume the opposite: it is rational, that is, it is represented in the form of an irreducible fraction, where and are integers. Let's square the supposed equality:
.It follows that even is even and . Let it be where the whole is. Then
Therefore, even means even and . We found that and are even, which contradicts the irreducibility of the fraction . This means that the original assumption was incorrect, and it is an irrational number.
Binary logarithm of the number 3
Let us assume the opposite: it is rational, that is, it is represented as a fraction, where and are integers. Since , and can be chosen to be positive. Then
But even and odd. We get a contradiction.
e
Story
The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manava (c. 750 BC - c. 690 BC) figured out that the square roots of some natural numbers, such as 2 and 61 cannot be expressed explicitly.
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 the pentagram. At the time of the Pythagoreans, it was believed that there was a single unit of length, sufficiently small and indivisible, which entered any segment an integer number of times. 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. 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 chosen as the smallest possible.
- According to the Pythagorean theorem: a² = 2 b².
- Because a- even, a must be even (since the square of an odd number would be odd).
- Because the a:b irreducible b must be odd.
- Because a even, we denote a = 2y.
- Then a² = 4 y² = 2 b².
- b² = 2 y², therefore b- even, then b even.
- However, it has been proven that b odd. Contradiction.
Greek mathematicians called this ratio of incommensurable quantities alogos(unspeakable), but according to the legends they did not pay due respect to Hippasus. 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 that denies the doctrine that all entities in the universe can be reduced to integers and their ratios.” The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the underlying assumption that numbers and geometric objects were one and inseparable.
see also
Notes
| Numerical systems | |
|---|---|
| Counting sets |
Natural numbers () Integers () |
Understanding numbers, especially natural numbers, is one of the oldest math "skills." Many civilizations, even modern ones, have attributed certain mystical properties due to their enormous importance in the description of nature. Although modern science and mathematics do not confirm these “magical” properties, the importance of number theory is undeniable.
Historically, a variety of natural numbers appeared first, then fairly quickly fractions and positive irrational numbers were added to them. Zero and negative numbers were introduced after these subsets of the set of real numbers. The last set, the set of complex numbers, appeared only with the development of modern science.
In modern mathematics, numbers are not introduced in historical order, although quite close to it.
Natural numbers $\mathbb(N)$
The set of natural numbers is often denoted as $\mathbb(N)=\lbrace 1,2,3,4... \rbrace $, and is often padded with zero to denote $\mathbb(N)_0$.
$\mathbb(N)$ defines the operations of addition (+) and multiplication ($\cdot$) with the following properties for any $a,b,c\in \mathbb(N)$:
1. $a+b\in \mathbb(N)$, $a\cdot b \in \mathbb(N)$ the set $\mathbb(N)$ is closed under the operations of addition and multiplication
2. $a+b=b+a$, $a\cdot b=b\cdot a$ commutativity
3. $(a+b)+c=a+(b+c)$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ associativity
4. $a\cdot (b+c)=a\cdot b+a\cdot c$ distributivity
5. $a\cdot 1=a$ is a neutral element for multiplication
Since the set $\mathbb(N)$ contains a neutral element for multiplication but not for addition, adding a zero to this set ensures that it includes a neutral element for addition.
In addition to these two operations, the “less than” relations ($
1. $a b$ trichotomy
2. if $a\leq b$ and $b\leq a$, then $a=b$ antisymmetry
3. if $a\leq b$ and $b\leq c$, then $a\leq c$ is transitive
4. if $a\leq b$ then $a+c\leq b+c$
5. if $a\leq b$ then $a\cdot c\leq b\cdot c$
Integers $\mathbb(Z)$
Examples of integers:
$1, -20, -100, 30, -40, 120...$
Solving the equation $a+x=b$, where $a$ and $b$ are known natural numbers, and $x$ is an unknown natural number, requires the introduction of a new operation - subtraction(-). If there is a natural number $x$ satisfying this equation, then $x=b-a$. However, this particular equation does not necessarily have a solution on the set $\mathbb(N)$, so practical considerations require expanding the set of natural numbers to include solutions to such an equation. This leads to the introduction of a set of integers: $\mathbb(Z)=\lbrace 0,1,-1,2,-2,3,-3...\rbrace$.
Since $\mathbb(N)\subset \mathbb(Z)$, it is logical to assume that the previously introduced operations $+$ and $\cdot$ and the relations $ 1. $0+a=a+0=a$ there is a neutral element for addition
2. $a+(-a)=(-a)+a=0$ there is an opposite number $-a$ for $a$
Property 5.:
5. if $0\leq a$ and $0\leq b$, then $0\leq a\cdot b$
The set $\mathbb(Z)$ is also closed under the subtraction operation, that is, $(\forall a,b\in \mathbb(Z))(a-b\in \mathbb(Z))$.
Rational numbers $\mathbb(Q)$
Examples of rational numbers:
$\frac(1)(2), \frac(4)(7), -\frac(5)(8), \frac(10)(20)...$
Now consider equations of the form $a\cdot x=b$, where $a$ and $b$ are known integers, and $x$ is an unknown. For the solution to be possible, it is necessary to introduce the division operation ($:$), and the solution takes the form $x=b:a$, that is, $x=\frac(b)(a)$. Again the problem arises that $x$ does not always belong to $\mathbb(Z)$, so the set of integers needs to be expanded. This introduces the set of rational numbers $\mathbb(Q)$ with elements $\frac(p)(q)$, where $p\in \mathbb(Z)$ and $q\in \mathbb(N)$. The set $\mathbb(Z)$ is a subset in which each element $q=1$, therefore $\mathbb(Z)\subset \mathbb(Q)$ and the operations of addition and multiplication extend to this set according to the following rules, which preserve all the above properties on the set $\mathbb(Q)$:
$\frac(p_1)(q_1)+\frac(p_2)(q_2)=\frac(p_1\cdot q_2+p_2\cdot q_1)(q_1\cdot q_2)$
$\frac(p-1)(q_1)\cdot \frac(p_2)(q_2)=\frac(p_1\cdot p_2)(q_1\cdot q_2)$
The division is introduced as follows:
$\frac(p_1)(q_1):\frac(p_2)(q_2)=\frac(p_1)(q_1)\cdot \frac(q_2)(p_2)$
On the set $\mathbb(Q)$, the equation $a\cdot x=b$ has a unique solution for each $a\neq 0$ (division by zero is undefined). This means that there is an inverse element $\frac(1)(a)$ or $a^(-1)$:
$(\forall a\in \mathbb(Q)\setminus\lbrace 0\rbrace)(\exists \frac(1)(a))(a\cdot \frac(1)(a)=\frac(1) (a)\cdot a=a)$
The order of the set $\mathbb(Q)$ can be expanded as follows:
$\frac(p_1)(q_1)
The set $\mathbb(Q)$ has one important property: Between any two rational numbers there are infinitely many other rational numbers, therefore there are no two adjacent rational numbers, unlike the sets of natural numbers and integers.
Irrational numbers $\mathbb(I)$
Examples of irrational numbers:
$0.333333...$
$\sqrt(2) \approx 1.41422135...$
$\pi\approx 3.1415926535...$
Since between any two rational numbers there are infinitely many other rational numbers, it is easy to erroneously conclude that the set of rational numbers is so dense that there is no need to expand it further. Even Pythagoras made such a mistake in his time. However, his contemporaries already refuted this conclusion when studying solutions to the equation $x\cdot x=2$ ($x^2=2$) on the set of rational numbers. To solve such an equation, it is necessary to introduce the concept of a square root, and then the solution to this equation has the form $x=\sqrt(2)$. An equation like $x^2=a$, where $a$ is a known rational number and $x$ is an unknown one, does not always have a solution on the set of rational numbers, and again the need arises to expand the set. A set of irrational numbers arises, and numbers such as $\sqrt(2)$, $\sqrt(3)$, $\pi$... belong to this set.
Real numbers $\mathbb(R)$
The union of the sets of rational and irrational numbers is the set of real numbers. Since $\mathbb(Q)\subset \mathbb(R)$, it is again logical to assume that the introduced arithmetic operations and relations retain their properties on the new set. The formal proof of this is very difficult, so the above-mentioned properties of arithmetic operations and relations on the set of real numbers are introduced as axioms. In algebra, such an object is called a field, so the set of real numbers is said to be an ordered field.
In order for the definition of the set of real numbers to be complete, it is necessary to introduce an additional axiom that distinguishes the sets $\mathbb(Q)$ and $\mathbb(R)$. Suppose that $S$ is a non-empty subset of the set of real numbers. An element $b\in \mathbb(R)$ is called the upper bound of a set $S$ if $\forall x\in S$ holds $x\leq b$. Then we say that the set $S$ is bounded above. The smallest upper bound of the set $S$ is called the supremum and is denoted $\sup S$. The concepts of lower bound, set bounded below, and infinum $\inf S$ are introduced similarly. Now the missing axiom is formulated as follows:
Any non-empty and upper-bounded subset of the set of real numbers has a supremum.
It can also be proven that the field of real numbers defined in the above way is unique.
Complex numbers$\mathbb(C)$
Examples of complex numbers:
$(1, 2), (4, 5), (-9, 7), (-3, -20), (5, 19),...$
$1 + 5i, 2 - 4i, -7 + 6i...$ where $i = \sqrt(-1)$ or $i^2 = -1$
The set of complex numbers represents all ordered pairs of real numbers, that is, $\mathbb(C)=\mathbb(R)^2=\mathbb(R)\times \mathbb(R)$, on which the operations of addition and multiplication are defined as follows way:
$(a,b)+(c,d)=(a+b,c+d)$
$(a,b)\cdot (c,d)=(ac-bd,ad+bc)$
There are several forms of writing complex numbers, of which the most common is $z=a+ib$, where $(a,b)$ is a pair of real numbers, and the number $i=(0,1)$ is called the imaginary unit.
It is easy to show that $i^2=-1$. Extending the set $\mathbb(R)$ to the set $\mathbb(C)$ allows us to determine the square root of negative numbers, which was the reason for introducing the set of complex numbers. It is also easy to show that a subset of the set $\mathbb(C)$, given by $\mathbb(C)_0=\lbrace (a,0)|a\in \mathbb(R)\rbrace$, satisfies all the axioms for real numbers, therefore $\mathbb(C)_0=\mathbb(R)$, or $R\subset\mathbb(C)$.
The algebraic structure of the set $\mathbb(C)$ with respect to the operations of addition and multiplication has the following properties:
1. commutativity of addition and multiplication
2. associativity of addition and multiplication
3. $0+i0$ - neutral element for addition
4. $1+i0$ - neutral element for multiplication
5. Multiplication is distributive with respect to addition
6. There is a single inverse for both addition and multiplication.
The set of irrational numbers is usually denoted by a capital letter I (\displaystyle \mathbb (I) ) in bold style without filling. Thus: I = R ∖ Q (\displaystyle \mathbb (I) =\mathbb (R) \backslash \mathbb (Q) ), that is, the set of irrational numbers is the difference between the sets of real and rational numbers.
The existence of irrational numbers, more precisely, segments incommensurable with a segment of unit length, was already known to ancient mathematicians: they knew, for example, the incommensurability of the diagonal and the side of a square, which is equivalent to the irrationality of the number.
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Irrational are:
Examples of proof of irrationality
Root of 2
Let's assume the opposite: 2 (\displaystyle (\sqrt (2))) rational, that is, represented as a fraction m n (\displaystyle (\frac (m)(n))), Where m (\displaystyle m) is an integer, and n (\displaystyle n)- natural number .
Let's square the supposed equality:
2 = m n ⇒ 2 = m 2 n 2 ⇒ m 2 = 2 n 2 (\displaystyle (\sqrt (2))=(\frac (m)(n))\Rightarrow 2=(\frac (m^(2 ))(n^(2)))\Rightarrow m^(2)=2n^(2)).Story
Antiquity
The concept of irrational numbers was implicitly adopted by Indian mathematicians in the 7th century BC, when Manava (c. 750 BC - c. 690 BC) figured out that the square roots of some natural numbers, such as 2 and 61 cannot be expressed explicitly [ ] .
The first proof of the existence of irrational numbers is usually attributed to Hippasus of Metapontus (c. 500 BC), a Pythagorean. At the time of the Pythagoreans, it was believed that there was a single unit of length, sufficiently small and indivisible, which included an integer number of times in any segment [ ] .
There is no exact data on which number was proven irrational by Hippasus. According to legend, he found it by studying the lengths of the sides of the pentagram. Therefore, it is reasonable to assume that this was the golden ratio [ ] .
Greek mathematicians called this ratio of incommensurable quantities alogos(unspeakable), but according to the legends they did not pay due respect to Hippasus. 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 that denies the doctrine that all entities in the universe can be reduced to integers and their ratios.” The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the underlying assumption that numbers and geometric objects were one and inseparable.