The sum of irrational numbers is an irrational number. Essence and designation

Integers

Natural numbers definition is integers positive numbers. Natural numbers are used to count objects and for many other purposes. These are the numbers:

This is a natural series of numbers.
Is zero a natural number? No, zero is not a natural number.
How many natural numbers exists? Exists infinite set natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It is impossible to specify it, because there is an infinite number of natural numbers.

The sum of natural numbers is a natural number. So, adding natural numbers a and b:

The product of natural numbers is a natural number. So, the product of natural numbers a and b:

c is always a natural number.

Difference of natural numbers There is not always a natural number. If the minuend is greater than the subtrahend, then the difference of the natural numbers is a natural number, otherwise it is not.

The quotient of natural numbers is not always a natural number. If for natural numbers a and b

where c is a natural number, this means that a is divisible by b. In this example, a is the dividend, b is the divisor, c is the quotient.

The divisor of a natural number is a natural number by which the first number is divisible by a whole.

Every natural number is divisible by one and itself.

Prime natural numbers are divisible only by one and themselves. Here we mean divided entirely. Example, numbers 2; 3; 5; 7 is only divisible by one and itself. These are simple natural numbers.

One is not considered a prime number.

Numbers that are greater than one and that are not prime are called composite numbers. Examples of composite numbers:

One is not considered a composite number.

The set of natural numbers is one, prime numbers and composite numbers.

The set of natural numbers is denoted Latin letter N.

Properties of addition and multiplication of natural numbers:

commutative property of addition

associative property of addition

(a + b) + c = a + (b + c);

commutative property of multiplication

associative property of multiplication

(ab) c = a (bc);

distributive property of multiplication

A (b + c) = ab + ac;

Whole numbers

Integers are the natural numbers, zero, and the opposites of the natural numbers.

The opposite of natural numbers are negative integers, for example:

1; -2; -3; -4;...

The set of integers is denoted by the Latin letter Z.

Rational numbers

Rational numbers are whole numbers and fractions.

Any rational number can be represented as a periodic fraction. Examples:

1,(0); 3,(6); 0,(0);...

From the examples it is clear that any integer is periodic fraction with period zero.

Any rational number can be represented as a fraction m/n, where m is an integer number,n natural number. Let's imagine the number 3,(6) from the previous example as such a fraction.

The set of all natural numbers is denoted by the letter N. Natural numbers are the numbers that we use to count objects: 1,2,3,4, ... In some sources, the number 0 is also considered a natural number.

The set of all integers is denoted by the letter Z. Integers are all natural numbers, zero and negative numbers:

1,-2,-3, -4, …

Now we add to the set of all integers the set of all ordinary fractions: 2/3, 18/17, -4/5 and so on. Then we get a set of all rational numbers.

Set of rational numbers

The set of all rational numbers is denoted by the letter Q. The set of all rational numbers (Q) is the set consisting of numbers of the form m/n, -m/n and the number 0. In as n,m can be any natural number. It should be noted that all rational numbers can be represented as a finite or infinite PERIODIC decimal fraction. The converse is also true that any finite or infinite periodic decimal fraction can be written as a rational number.

But what about, for example, the number 2.0100100010...? It is infinitely NON-PERIODIC decimal. And it does not apply to rational numbers.

IN school course In algebra, only real (or real) numbers are studied. Plenty of everyone real numbers denoted by the letter R. The set R consists of all rational and all irrational numbers.

The concept of irrational numbers

Irrational numbers- it's all endless decimals non-periodic fractions. Irrational numbers do not have a special designation.

For example, all numbers obtained by extracting the square root of natural numbers that are not squares of natural numbers will be irrational. (√2, √3, √5, √6, etc.).

But do not think that irrational numbers are obtained only by extracting square roots. For example, the number “pi” is also irrational, and it is obtained by division. And no matter how hard you try, you won’t be able to get it by extracting Square root from any natural number.

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 a discovery while in sea ​​voyage, and was thrown overboard by the 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 challenged Pythagorean mathematics serious problem, destroying the underlying assumption of the entire theory that numbers and geometric objects are 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 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. Last set, set complex numbers, appeared only with the development of modern science.

In modern mathematics, numbers are not entered into 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. $a+(-a)=(-a)+a=0$ exists 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 the introduction of a 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.