Understanding numbers, especially natural numbers, is one of the oldest mathematical "skills". Many civilizations, even modern ones, have attributed some mystical properties to numbers because of their great importance in describing nature. Although modern science and mathematics do not support these "magic" properties, the significance of number theory is undeniable.
Historically, a lot of natural numbers first appeared, then pretty soon 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 entered in historical order, although in a rather 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 zero-padded to denote $ \ mathbb (N) _0 $.
In $ \ mathbb (N) $, the operations of addition (+) and multiplication ($ \ cdot $) are defined 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 $ distributiveness
5. $ a \ cdot 1 = a $ is the neutral element for multiplication
Since the set $ \ mathbb (N) $ contains a neutral element for multiplication, but not for addition, adding zero to this set ensures that it includes a neutral element for addition.
In addition to these two operations, on the set $ \ mathbb (N) $, the relations "less than" ($
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 transitivity
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:
$1, -20, -100, 30, -40, 120...$
The solution of 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 $ that satisfies 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 extending 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 additions
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 unknown. For the solution to be possible, it is necessary to enter 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 must be expanded. Thus, we introduce 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 is $ q = 1 $, therefore $ \ mathbb (Z) \ subset \ mathbb (Q) $ and the operations of addition and multiplication are extended to this set according to the following rules, which preserve all of 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) $
Division is introduced in this way:
$ \ 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 not defined). This means that there is an inverse $ \ 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 extended 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, in contrast to the sets of naturals and integers.
Irrational numbers $ \ mathbb (I) $
Examples of irrational numbers:
$ \ sqrt (2) \ approx 1.41422135 ... $
$ \ pi \ approx 3.1415926535 ... $
In view of the fact that there are infinitely many other rational numbers between any two rational numbers, it is easy to make an erroneous conclusion that the set of rational numbers is so dense that there is no need for its further expansion. Even Pythagoras made such a mistake in his time. However, already his contemporaries refuted this conclusion when studying solutions of 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 of the type $ x ^ 2 = a $, where $ a $ is a known rational number, and $ x $ is an unknown, does not always have a solution on the set of rational numbers, and again there is a need to expand the set. A set of irrational numbers arises, and such numbers 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, therefore 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 they say that the set of real numbers is 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 distinguishing the sets $ \ mathbb (Q) $ and $ \ mathbb (R) $. Suppose that $ S $ is a non-empty subset of the set of real numbers. The element $ b \ in \ mathbb (R) $ is called the upper bound of the set $ S $ if $ \ forall x \ in S $ is true $ x \ leq b $. Then the set $ S $ is said to be bounded above. The smallest upper bound of the set $ S $ is called the supremum and is denoted by $ \ sup S $. The concepts of a lower bound, a set bounded from below, and an infinum $ \ inf S $ are introduced similarly. The missing axiom is now formulated as follows:
Any non-empty and upper-bounded subset of the set of real numbers has a supremum.
You can also prove 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 notation for complex numbers, the most common of which is $ z = a + ib $, where $ (a, b) $ is a pair of real numbers, and the number $ i = (0,1) $ is called an 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 a set of complex numbers. It is also easy to show that a subset of the set $ \ mathbb (C) $, defined as $ \ mathbb (C) _0 = \ lbrace (a, 0) | a \ in \ mathbb (R) \ rbrace $, satisfies all the axioms for real numbers, hence $ \ 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.commutability 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 element for both addition and multiplication.
What are irrational numbers? Why are they called that? Where are they used and what are they? Few can answer these questions without hesitation. 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 The need to introduce this concept is due to the fact that the previously existing concepts of real or real, integer, natural and rational numbers were not enough to solve new emerging problems. For example, in order to figure out how square 2 is, you need to use non-periodic infinite decimal fractions. In addition, many of the simplest equations also do not have a solution without introducing the concept of an irrational number.
This set is denoted as I. And, as it is already clear, these values cannot be represented as a simple fraction, in the numerator of which there will be an integer, and in the denominator -
For the first time, one way or another, Indian mathematicians faced this phenomenon in the 7th century when it was discovered that the square roots of some quantities could not be indicated explicitly. And the first proof of the existence of such numbers is attributed to the Pythagorean Hippasus, who did this in the process of studying an isosceles right triangle. Some 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 in Latin is "fraction", "ratio", then the prefix "ir"
gives this word the opposite meaning. Thus, the name of the set of these numbers suggests that they cannot be correlated with whole or fractional numbers, they 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 are complex. 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, then all their properties that are studied in arithmetic (they are also called basic algebraic laws) are applicable to them.
a + b = b + a (commutativity);
(a + b) + c = a + (b + c) (associativity);
a + (-a) = 0 (existence of the opposite number);
ab = ba (displacement law);
(ab) c = a (bc) (distributivity);
a (b + c) = ab + ac (distribution law);
a x 1 / a = 1 (existence of a reciprocal);
The comparison is also carried out in accordance with general laws and principles:
If a> b and b> c, then a> c (the 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 action of the Archimedes axiom extends to irrational numbers. It says that for any two quantities a and b, the statement is true that by taking a as a term a sufficient number of times, you can exceed b.
Usage
Despite the fact that in ordinary life you do not often have to deal with them, irrational numbers cannot be counted. There are a lot of them, but they are almost invisible. We are surrounded by irrational numbers everywhere. Examples familiar to everyone are 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 greater part to the lesser, and vice versa, also
refers to this set. The less well-known "silver" is also.
On the number line, they are very densely located, so that between any two quantities referred to the set of rational ones, an irrational one is necessarily encountered.
There are still a lot of unresolved problems associated with this set. There are criteria such as the measure of irrationality and the normality of a number. Mathematicians continue to examine the most significant examples for belonging to one group or another. For example, it is considered that e is a normal number, that is, the probability of different digits appearing in its record is the same. As for pi, research is underway on it. The measure of irrationality is a quantity that shows how well a particular 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 that include real or real.
So, algebraic is a value that is a root of a polynomial that is not identically zero. For example, the square root of 2 would be in this category because it is the 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 second was originally deduced by mathematicians in this capacity, their irrationality and transcendence were proved many years after their discovery. For pi, the proof was presented in 1882 and simplified in 1894, ending the 2,500 year controversy over the problem of squaring the circle. It is still not fully understood, so modern mathematicians have something to work on. By the way, the first sufficiently accurate calculation of this value was carried out by Archimedes. Before him, all calculations were too rough.
For e (Euler's or Napier's number), evidence of its transcendence was found in 1873. It is used to solve logarithmic equations.
Other examples include sine, cosine, and tangent values for any algebraic nonzero values.
The ancient mathematicians already knew with a segment of unit length: they knew, for example, the incommensurability of the diagonal and the side of a square, which is tantamount to the irrationality of a number.
Irrational are:
Examples of proof of irrationality
Root of 2
Suppose the opposite: rational, that is, represented as an irreducible fraction, where and are integers. Let's square the assumed equality:
.Hence it follows that even means even and. Let where is the whole. Then
Therefore, even means even and. We got that and are even, which contradicts the irreducibility of the fraction. This means that the initial assumption was wrong, and - an irrational number.
Binary logarithm of 3
Suppose the opposite: rational, that is, represented as a fraction, where and are integers. Since, and can be chosen positive. Then
But even and odd. We get a contradiction.
e
History
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 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 side lengths of the pentagram. At the time of the Pythagoreans, it was believed that there is a single unit of length, sufficiently small and indivisible, which enters any segment an integer number of times. However, Hippasus proved 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-angled 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.
- By the Pythagorean theorem: a² = 2 b².
- As a² even, a must be even (since the square of an odd number would be odd).
- Because the 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 aalogos(unspeakable), however, according to the legends, they did not give Hippas the respect he deserved. Legend has it that Hippasus made a 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 whole numbers and their relationships." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the assumption underlying the entire theory that numbers and geometric objects are one and inseparable.
see also
Notes (edit)
| Number systems | |
|---|---|
| Counting multitudes |
Integers () |
A lot of irrational numbers are usually indicated by a capital Latin letter I (\ displaystyle \ mathbb (I)) in bold, no fill. 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 ancient mathematicians already knew about the existence of irrational numbers, more precisely, segments incommensurable with a segment of unit length: they knew, for example, the incommensurability of the diagonal and the side of a square, which is equivalent to the irrationality of a number.
Collegiate YouTube
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Irrational are:
Examples of proof of irrationality
Root of 2
Suppose 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 assumed equality:
2 = mn ⇒ 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)).History
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 explicitly expressed [ ] .
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 is a single unit of length, sufficiently small and indivisible, which is an integer number of times included in any segment [ ] .
There is no exact data about the irrationality of which number was proved 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 it was the golden ratio [ ] .
Greek mathematicians called this ratio of incommensurable quantities aalogos(ineffable), however, according to the legends, they did not give Hippas the respect he deserved. Legend has it that Hippasus made a 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 whole numbers and their relationships." The discovery of Hippasus posed a serious problem for Pythagorean mathematics, destroying the assumption underlying the entire theory that numbers and geometric objects are one and indivisible.
This property plays an important role in solving differential equations. So, for example, the only solution to the differential equation
is the function
where c is an arbitrary constant.
- 1. Number e irrational and even transcendental. Its transcendence was only proven in 1873 by Charles Hermite. It is assumed that e- a normal number, that is, the probability of the appearance of different digits in its record is the same.
- 2. Number e is a computable (and hence arithmetic) number.
Euler's formula, in particular
5. t. N. "Poisson integral" or "Gauss integral"
8. Introducing Catalan:
9. Representation through the work:
10. Through Bell's numbers:
11. The measure of irrationality of a number e is equal to 2 (which is the smallest possible value for irrational numbers).
Proof of irrationality
Let's pretend that
where a and b are natural numbers. Considering this equality and considering the expansion in a series:
we get the following equality:
We represent this sum as the sum of two terms, one of which is the sum of the members of the series by n from 0 to a, and the second is the sum of all other members of the series:
Now let's move the first sum to the left side of the equality:
We multiply both sides of the resulting equality by. We get
Now let's simplify the resulting expression:
Consider the left-hand side of the resulting equality. Obviously, the number is integer. A number is also an integer, since (it follows that all numbers of the form are integers). Thus, the left side of the resulting equality is an integer.
Let's move on to the right side now. This sum has the form
According to Leibniz's feature, this series converges, and its sum S is a real number between the first term and the sum of the first two terms (with signs), i.e.
Both of these numbers lie between 0 and 1. Therefore, that is, - the right side of equality - cannot be an integer. We got a contradiction: an integer cannot be equal to a number that is not an integer. This contradiction proves that the number e is not rational and therefore irrational.