We have previously shown that $1\frac25$ is close to $\sqrt2$. If it were exactly equal to $\sqrt2$, . Then the ratio is $\frac(1\frac25)(1)$, which can be turned into an integer ratio $\frac75$ by multiplying the top and bottom of the fraction by 5, and would be the desired value.
But, unfortunately, $1\frac25$ is not the exact value of $\sqrt2$. A more accurate answer, $1\frac(41)(100)$, gives us 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. Greek mathematicians spent a lot of time and effort to calculate the exact value of $\sqrt2$, but they never succeeded. They were unable to represent the ratio $\frac(\sqrt2)(1)$ as a ratio of integers.
Finally, the great Greek mathematician Euclid proved that no matter how much the accuracy of calculations increases, it is impossible to obtain the exact value of $\sqrt2$. There is no fraction that, when squared, will give the result 2. They say that Pythagoras was the first to come to this conclusion, but this inexplicable fact amazed the scientist so much that he swore himself and took an oath from his students to keep this discovery 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 remember 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 whole and fractional numbers, both positive and negative, are rational.
As it turns out, most square roots are irrational numbers. Only numbers in the series of square numbers have rational square roots. These numbers are also called perfect squares. Rational numbers are also fractions made from these perfect squares. For example, $\sqrt(1\frac79)$ is a rational number since $\sqrt(1\frac79)=\frac(\sqrt16)(\sqrt9)=\frac43$ or $1\frac13$ (4 is the root the square root of 16, and 3 is the square root of 9).
The set of irrational numbers is usually denoted by a capital letter I (\displaystyle \mathbb (I) ) in bold style without shading. 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.
What are irrational numbers? Why are they called that? Where are they used and what are they? Few people can answer these questions without thinking. But in fact, the answers to them are quite simple, although not everyone needs them and in very rare situations
Essence and designation
Irrational numbers are infinite non-periodic numbers. The need to introduce this concept is due to the fact that to solve new problems that arise, the previously existing concepts of real or real, integer, natural and rational numbers were no longer sufficient. For example, in order to calculate which quantity is the square of 2, you need to use non-periodic infinite decimals. In addition, many simple equations also have no solution without introducing the concept of an irrational number.
This set is denoted as I. And, as is already clear, these values cannot be represented as a simple fraction, the numerator of which will be an integer, and the denominator will be
For the first time, one way or another, Indian mathematicians encountered this phenomenon in the 7th century when it was discovered that the square roots of some quantities cannot be indicated explicitly. And the first proof of the existence of such numbers is attributed to the Pythagorean Hippasus, who did this while studying an isosceles right triangle. Some other scientists who lived before our era made a serious contribution to the study of this set. The introduction of the concept of irrational numbers entailed a revision of the existing mathematical system, which is why they are so important.
origin of name
If ratio translated from Latin is “fraction”, “ratio”, then the prefix “ir”
gives this word the opposite meaning. Thus, the name of the set of these numbers indicates that they cannot be correlated with an integer or fraction and have a separate place. This follows from their essence.Place in the general classification
Irrational numbers, along with rational numbers, belong to the group of real or real numbers, which in turn belong to complex numbers. There are no subsets, but there are algebraic and transcendental varieties, which will be discussed below.
Properties
Since irrational numbers are part of the set of real numbers, all their properties that are studied in arithmetic (they are also called basic algebraic laws) apply to them.
a + b = b + a (commutativity);
(a + b) + c = a + (b + c) (associativity);
a + (-a) = 0 (existence of the opposite number);
ab = ba (commutative law);
(ab)c = a(bc) (distributivity);
a(b+c) = ab + ac (distribution law);
a x 1/a = 1 (existence of a reciprocal number);
The comparison is also carried out in accordance with general laws and principles:
If a > b and b > c, then a > c (transitivity of the relation) and. etc.
Of course, all irrational numbers can be converted using basic arithmetic. There are no special rules for this.
In addition, the Archimedes axiom applies to irrational numbers. It states that for any two quantities a and b, it is true that if you take a as a term enough times, you can beat b.
Usage
Despite the fact that you don’t encounter them very often in everyday life, irrational numbers cannot be counted. There are a huge number of them, but they are almost invisible. Irrational numbers are all around us. Examples that are familiar to everyone are the number pi, equal to 3.1415926..., or e, which is essentially the base of the natural logarithm, 2.718281828... In algebra, trigonometry and geometry, they have to be used constantly. By the way, the famous meaning of the “golden ratio”, that is, the ratio of both the larger part to the smaller part, and vice versa, also
belongs to this set. The lesser known “silver” one too.
On the number line they are located very densely, so that between any two quantities classified as rational, an irrational one is sure to occur.
There are still a lot of unsolved problems associated with this set. There are criteria such as the measure of irrationality and the normality of a number. Mathematicians continue to study the most significant examples to determine whether they belong to one group or another. For example, it is believed that e is a normal number, i.e., the probability of different digits appearing in its notation is the same. As for pi, research is still underway regarding it. The measure of irrationality is a value that shows how well a given number can be approximated by rational numbers.
Algebraic and transcendental
As already mentioned, irrational numbers are conventionally divided into algebraic and transcendental. Conditionally, since, strictly speaking, this classification is used to divide the set C.
This designation hides complex numbers, which include real or real numbers.
So, algebraic is a value that is the root of a polynomial that is not identically equal to zero. For example, the square root of 2 would be in this category because it is a solution to the equation x 2 - 2 = 0.
All other real numbers that do not satisfy this condition are called transcendental. This variety includes the most famous and already mentioned examples - the number pi and the base of the natural logarithm e.
Interestingly, neither one nor the other were originally developed by mathematicians in this capacity; their irrationality and transcendence were proven many years after their discovery. For pi, the proof was given in 1882 and simplified in 1894, ending a 2,500-year debate about the problem of squaring the circle. It has still not been fully studied, so modern mathematicians have something to work on. By the way, the first fairly accurate calculation of this value was carried out by Archimedes. Before him, all calculations were too approximate.
For e (Euler's or Napier's number), proof of its transcendence was found in 1873. It is used in solving logarithmic equations.
Other examples include the values of sine, cosine, and tangent for any algebraic nonzero value.
The material in this article provides initial information about irrational numbers. First we will give the definition of irrational numbers and explain it. Below we give examples of irrational numbers. Finally, let's look at some approaches to figuring out whether a given number is irrational or not.
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Definition and examples of irrational numbers
When studying decimals, we separately considered infinite non-periodic decimals. Such fractions arise when measuring decimal lengths of segments that are incommensurable with a unit segment. We also noted that infinite non-periodic decimal fractions cannot be converted to ordinary fractions (see converting ordinary fractions to decimals and vice versa), therefore, these numbers are not rational numbers, they represent the so-called irrational numbers.
So we come to definition of irrational numbers.
Definition.
Numbers that represent infinite non-periodic decimal fractions in decimal notation are called irrational numbers.
The voiced definition allows us to give 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 threes separating eights increases by two each time).
It should be noted that irrational numbers are quite rarely found in the form of endless non-periodic decimal fractions. They are usually found in the form , etc., as well as in the form of specially entered letters. The most famous examples of irrational numbers in this notation are the arithmetic square root of two, the number “pi” π=3.141592..., the number e=2.718281... and the golden number.
Irrational numbers can also be defined in terms of real numbers, which combine rational and irrational numbers.
Definition.
Irrational numbers are real numbers that are not rational numbers.
Is this number irrational?
When a number is given not as a decimal fraction, but as some root, logarithm, etc., then answering the question of whether it is irrational is quite difficult in many cases.
Undoubtedly, when answering the question posed, it is very useful to know which numbers are not irrational. From the definition of irrational numbers it follows that irrational numbers are not rational numbers. Thus, irrational numbers are NOT:
- finite and infinite periodic decimal fractions.
Also, any composition of rational numbers connected by the signs of arithmetic operations (+, −, ·, :) is not an irrational number. This is because the sum, difference, product and quotient of two rational numbers is a rational number. For example, the values of expressions and are rational numbers. Here we note that if such expressions contain one single irrational number among the rational numbers, then the value of the entire expression will be an irrational number. For example, in the expression the number is irrational, and the remaining numbers are rational, therefore it is an irrational number. If it were a rational number, then the rationality of the number would follow, but it is not rational.
If the expression that specifies the number contains several irrational numbers, root signs, logarithms, trigonometric functions, numbers π, e, etc., then it is necessary to prove the irrationality or rationality of the given number in each specific case. However, there are a number of results already obtained that can be used. Let's list the main ones.
It has been proven that a kth root of an integer is a rational number only if the number under the root is the kth power of another integer; in other cases, such a root specifies 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 are not irrational, since and .
As for logarithms, it is sometimes possible to prove their irrationality using the method of contradiction. As an example, let's prove that log 2 3 is an irrational number.
Let's assume 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 on the right side – even. So we came to a contradiction, which means that our assumption turned out to be incorrect, and this proved that log 2 3 is an irrational number.
Note that lna for any positive and non-one rational a is an irrational number. For example, and are irrational numbers.
It is also proven 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 , tan(−4) , cos5,7 are irrational numbers.
There are other proven results, but we will limit ourselves to those already listed. It should also be said that when proving the above results, the theory associated with algebraic numbers And transcendental numbers.
In conclusion, we note that we should not make hasty conclusions regarding the irrationality of the given numbers. For example, it seems obvious that an irrational number to an irrational degree is an irrational number. However, this is not always the case. To confirm the stated fact, we present the degree. It is known that - is an irrational number, and it has also been proven that - is an irrational number, but is 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 have not yet been proven.
Bibliography.
- Mathematics. 6th grade: educational. 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 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by 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 those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Understanding numbers, especially natural numbers, is one of the oldest math "skills." Many civilizations, even modern ones, have attributed certain mystical properties to numbers due to their enormous importance in describing 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 multiplicationSince 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:
$\sqrt(2) \approx 1.41422135...$
$\pi\approx 3.1415926535...$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.
