This article contains basic information about real numbers. First, the definition of real numbers is given and examples are given. The position of the real numbers on the coordinate line is shown next. And in conclusion, it is analyzed how real numbers are given in the form of numerical expressions.
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Definition and examples of real numbers
Real numbers as expressions
From the definition of real numbers, it is clear that real numbers are:
- any natural number;
- any integer ;
- any ordinary fraction (both positive and negative);
- any mixed number;
- any decimal fraction (positive, negative, finite, infinite periodic, infinite non-periodic).
But very often real numbers can be seen in the form , etc. Moreover, the sum, difference, product, and quotient of real numbers are also real numbers (see operations with real numbers). For example, these are real numbers.
And if you go further, then from real numbers using arithmetic signs, root signs, degrees, logarithmic, trigonometric functions, etc. you can compose all kinds of numerical expressions, the values of which will also be real numbers. For example, expression values 
and 
are real numbers.
In conclusion of this article, we note that the next step in expanding the concept of number is the transition from real numbers to complex numbers.
Bibliography.
- Vilenkin N.Ya. etc. Mathematics. Grade 6: textbook for educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).
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Repetition of junior high school
Integral
Derivative
Volumes of bodies
Solids of revolution
Method of coordinates in space
Rectangular coordinate system. Relationship between vector coordinates and point coordinates. The simplest problems in coordinates. Scalar product of vectors.
The concept of a cylinder. The surface area of a cylinder. The concept of a cone.
The surface area of a cone. Sphere and ball. The area of the sphere. Mutual arrangement of sphere and plane.
The concept of volume. The volume of a rectangular parallelepiped. Volume of a straight prism, cylinder. The volume of the pyramid and cone. The volume of the ball.
Section III. Beginnings of mathematical analysis
Derivative. Derivative of a power function. Differentiation rules. Derivatives of some elementary functions. The geometric meaning of the derivative.
Application of the derivative to the study of functions Increasing and decreasing function. Extrema of the function. Application of the derivative to plotting graphs. The largest, smallest values of the function.
Primitive. Rules for finding primitives. The area of a curvilinear trapezoid and the integral. Calculation of integrals. Calculation of areas using integrals.
Training tasks for exams
Section I. Algebra
Number is an abstraction used to quantify objects. Numbers arose in primitive society in connection with the need for people to count objects. Over time, with the development of science, the number has become the most important mathematical concept.
To solve problems and prove various theorems, you need to understand what types of numbers are. The main types of numbers include: natural numbers, integers, rational numbers, real numbers.
Natural numbers are numbers obtained by the natural counting of objects, or rather, by their numbering ("first", "second", "third" ...). The set of natural numbers is denoted by the Latin letter N (you can remember, based on the English word natural). We can say that N =(1,2,3,....)
By complementing natural numbers with zero and negative numbers (i.e., numbers opposite to natural numbers), the set of natural numbers is expanded to the set of integers.
Integers are numbers from the set (0, 1, -1, 2, -2, ....). This set consists of three parts - natural numbers, negative integers (the opposite of natural numbers) and the number 0 (zero). Integers are denoted by the Latin letter Z. We can say that Z=(1,2,3,....). Rational numbers are numbers that can be expressed as a fraction, where m is an integer and n is a natural number.
There are rational numbers that cannot be written as a finite decimal fraction, for example. If, for example, you try to write a number as a decimal fraction using the well-known division corner algorithm, you get an infinite decimal fraction. An infinite decimal is called periodical, repeating number 3 - her period. A periodic fraction is briefly written as follows: 0, (3); reads: "Zero integers and three in the period."
In general, a periodic fraction is an infinite decimal fraction, in which, starting from a certain decimal place, the same digit or several digits are repeated - the period of the fraction.
For example, a decimal is periodic with a period of 56; reads "23 integers, 14 hundredths and 56 in the period."
So, every rational number can be represented as an infinite periodic decimal fraction.
The converse statement is also true: each infinite periodic decimal fraction is a rational number, since it can be represented as a fraction, where is an integer, is a natural number.
Real (real) numbers are numbers that are used to measure continuous quantities. The set of real numbers is denoted by the Latin letter R. Real numbers include rational numbers and irrational numbers. Irrational numbers are numbers that are obtained by performing various operations on rational numbers (for example, extracting a root, calculating logarithms), but are not rational at the same time. Examples of irrational numbers are .
Any real number can be displayed on the number line:
For the sets of numbers listed above, the following statement is true: the set of natural numbers is included in the set of integers, the set of integers is included in the set of rational numbers, and the set of rational numbers is included in the set of real numbers. This statement can be illustrated using Euler circles.
Exercises for self-solving
If the number α cannot be represented as an irreducible fraction $$\frac(p)(q)$$, then it is called irrational.
An irrational number is written as an infinite non-periodic decimal fraction.
The fact of the existence of irrational numbers will be demonstrated by an example.
Example 1.4.1. Prove that there is no rational number whose square is 2.
Decision. Suppose there exists an irreducible fraction $$\frac(p)(q)$$ such that $$(\frac(p)(q))^(2)=2$$
or $$p^(2)=2q^(2)$$. It follows that $$p^(2)$$ is a multiple of 2, and hence p is a multiple of 2. Otherwise, if p is not divisible by 2, i.e., $$p=2k-1$$, then $$p^(2)=(2k-1)^(2)=4k^(2)-4k+1$$ is not divisible by 2 either. Therefore, $$ p=2k$$ $$\Rightarrow$$ $$p^(2)=4k^(2)$$ $$\Rightarrow$$ $$4k^(2)=2q^(2)$$ $$\ Rightarrow$$ $$q^(2)=2k^(2)$$.
Since $$q^(2)$$ is a multiple of 2, then q is also a multiple of 2, i.e. $$q=2m$$.
So, the numbers p and q have a common factor - the number 2, which means that the fraction $$\frac(p)(q)$$ is reduced.
This contradiction means that the assumption made is false, thus the statement is proved.
The set of rational and irrational numbers is called the set of real numbers.
In the set of real numbers, the operations of addition and multiplication are axiomatically introduced: any two real numbers a and b are assigned the number $$a+b$$ and the product $$a\cdot b$$.
In addition, relations "greater than", "less than" and equality are introduced in this set:
$$a>b$$ if and only if a - b is a positive number;
$$a
Let us list the main properties of numerical inequalities.
1. If $$a>b$$ and $$b>c$$ $$\Rightarrow$$ $$a>c$$.
2. If $$a>b$$ and $$c>0$$ $$\Rightarrow$$ $$ac>bc$$.
3. If $$a>b$$ and $$c<0$$ $$\Rightarrow$$ $$ac
5. If a, b, c, d are positive numbers such that $$a>b$$ and $$c>d$$ $$\Rightarrow$$ $$ac>bd$$.
Consequence. If a and b are positive numbers and $$a>b$$ $$\Rightarrow$$ $$a^(2)>b^(2)$$.
6. If $$a>b$$ and $$c>d$$ $$\Rightarrow$$ $$a+c>b+d$$.
7. If $$a>0$$, $$b>0$$ and $$a>b$$ $$\Rightarrow$$ $$\frac(1)(a)<\frac{1}{b}$$.
Geometric interpretation of real numbers.
Let's take a straight line l, see fig. 1.4.1, and fix a point O on it - the origin.
Point O divides the line into two parts - rays. The ray directed to the right is called the positive ray, and the ray directed to the left is called the negative ray. On the straight line, we mark the segment taken as a unit of length, i.e. enter scale.
Rice. 1.4.1. Geometric interpretation of real numbers.
A straight line with a selected origin, positive direction and scale is called a number line.
Each point of the number line can be associated with a real number according to the following rule:
- point O will be assigned zero;
– each point N on the positive ray is assigned a positive number a, where a is the length of the segment ON ;
– each point M on the negative ray is assigned a negative number b, where $$b=-\left | OM \right |$$ (the length of the segment OM, taken with a minus sign).
Thus, a one-to-one correspondence is established between the set of all points of the real number line and the set of real numbers, i.e. :
1) each point on the number line is assigned one and only one real number;
2) different points are assigned different numbers;
3) there is not a single real number that does not correspond to any point on the number line.
Example 1.4.2. On the number line, mark the points corresponding to the numbers:
1) $$1\frac(5)(7)$$ 2) $$\sqrt(2)$$ 3) $$\sqrt(3)$$
Decision. 1) In order to mark the fractional number $$\frac(12)(7)$$, you need to construct a point corresponding to $$\frac(12)(7)$$.
To do this, you need to divide a segment of length 1 into 7 equal parts. We solve this problem in this way.
We draw an arbitrary ray from t.O and set aside 7 equal segments on this ray. Get
segment OA, and from point A we draw a straight line to the intersection with 1.
Rice. 1.4.2. Division of a single segment into 7 equal parts.
The straight lines drawn parallel to the straight line A1 through the ends of the laid-off segments divide the segment of unit length into 7 equal parts (Fig. 1.4.2). This makes it possible to construct a point representing the number $$1\frac(5)(7)$$ (Fig.1.4.3).
Rice. 1.4.3. A point on the number axis corresponding to the number $$1\frac(5)(7)$$.
2) The number $$\sqrt(2)$$ can be obtained like this. We construct a right triangle with unit legs. Then the length of the hypotenuse is $$\sqrt(2)$$; this segment is set aside from O on the number line (Fig. 1.4.4).
3) To construct a point remote from PO at a distance of $$\sqrt(3)$$ (to the right), it is necessary to construct a right-angled triangle with legs of length 1 and $$\sqrt(2)$$. Then its hypotenuse has the length $$\sqrt(2)$$, which allows you to specify the desired point on the real axis.
For real numbers, the concept of a module (or absolute value) is defined.
Rice. 1.4.4. The point on the number axis corresponding to the number $$\sqrt(2)$$.
The modulus of a real number a is called:
is the number itself, if a is a positive number;
- zero if a- zero;
– -a, if a- a negative number.
The absolute value of a number a denoted by $$\left | a \right |$$.
The definition of the module (or absolute value) can be written as
| $$\left | a \right |=\left\(\begin(matrix)a, a\geq0\\-a, a<0\end{matrix}\right.$$ | (1.4.1) |
Geometrically, the module of the number a means the distance on the number line from the origin O to the point corresponding to the number a.
We note some properties of the module.
1. For any number a the equality $$\left | a \right |=\left | -a \right |$$.
2. For any numbers a and b equalities are true
$$\left | ab \right |=\left | a \right |\cdot \left | b \right |$$; $$\left | \frac(a)(b) \right |=\frac(\left | a \right |)(\left | b \right |)$$ $$(b\neq 0)$$; $$\left | a \right |^(2)=a^(2)$$.
3. For any number a the inequality $$\left | a \right |\geq 0$$.
4. For any number a the inequality $$-\left | a\right |\leq a\leq \left | a \right |$$.
5. For any numbers a and b the inequality
$$\left | a+b \right |\leq \left | a \right |+\left | b \right |$$
Consider the following numerical sets.
If $$a
2) the interval (a; b) is the set of all real numbers α
, for each of which is true: $$a<\alpha
Similarly, you can enter a half-interval.
In some cases, one speaks of "gaps", meaning by this either a ray, or a segment, or an interval, or a half-interval.
A bunch of R all real numbers are denoted as follows: $$(-\infty; \infty)$$.
For any real number a, we introduce the concept of a degree with a natural exponent n, namely
$$a^(n)=\underbrace (a\cdot a\cdot a\cdot a...a)$$, $$n\geq 2$$ and $$a^(1)=a$$.
Let be a is any non-zero number, then by definition $$a^(0)=1$$.
The zero power of zero is not defined.
Let be a- any non-zero number, m is any integer. Then the number $$a^(m)$$ is determined by the rule:
$$a^(m)=\left\(\begin(matrix)a, m=1;\\\underbrace(a\cdot a\cdot a\cdot a...a), m\in N, m \geq2;\\1, m=0;\\\frac(1)(a^(n)), m=-n, n\in N\end(matrix)\right.$$
wherein a m is called a degree with an integer exponent.
Before defining the concept of a degree with a rational exponent, we introduce the concept of an arithmetic root.
Arithmetic root degree n (n ∈ N, n > 2) non-negative number a called a non-negative number b such that b n = a. Number b denoted as $$b\sqrt[n](a)$$.
Properties of arithmetic roots ( a > 0, b > 0, n, m, k- integers.)
| 1. $$\sqrt[n](ab)=\sqrt[n](a)\cdot \sqrt[n](b)$$ | 5. $$\sqrt[n](\sqrt[k](a))=\sqrt(a)$$ |
| 2. $$(a)^(\frac(k)(n))=\sqrt[n](a^(k))$$ | 6. $$\sqrt[n](a^(m))=\sqrt(a^(mk))$$ |
| 3. $$(\sqrt[n](a))^(k)=\sqrt[n](a^(k))$$ | 7. $$\sqrt(a^(2))=\left | a \right |$$ |
| 4. $$\sqrt[n](\frac(a)(b))=\frac(\sqrt[n](a))(\sqrt[n](b)) (b\neq 0)$$ | 8. $$\sqrt(a^(2n))=\left | a \right |$$ |
Let be a< 0
, a n is a natural number greater than 1. If n is an even number, then the equality b n = a does not hold for any real value b. This means that in the field of real numbers it is impossible to determine the root of an even degree from a negative number. If n is an odd number, then there is only one real number b such that b n = a. This number is denoted √n a and is called the odd root of a negative number.
Using the definition of raising to an integer power and the definition of an arithmetic root, we give a definition of a degree with a rational exponent.
Let be a is a positive number and $$r=\frac(p)(q)$$ is a rational number, and q- natural number.
positive number
$$b=\sqrt[q](a^(p))$$
is called the power of a with exponent r and is denoted as
$$b=a^(r)$$, or $$a^(\frac(p)(q))=\sqrt[q](a^(r))$$, here $$q\in N $$, $$q\geq2$$.
Consider the basic properties of a degree with a rational exponent.
Let be a and b are any positive numbers, r 1 and r 2 are any rational numbers. Then the following properties are true:
| 1. $$(ab)^(r_(1))=a^(r_(1))\cdot b^(r_(1))$$ | |
| 2. $$(\frac(a)(b))^(r_(1))=\frac(a^(r_(1)))(b^(r_(1)))$$ | |
| 3. $$a^(r_(1))\cdot a^(r_(2))=a^(r_(1)+r_(2))$$ | |
| 4. $$\frac(a^(r_(1)))(a^(r_(2)))=a^(r_(1)-r_(2))$$ | |
| 5. $$(a^(r_(1)))^(r_(2))=a^(r_(1)r_(2))$$ | (1.4.2) |
| 6. $$a^(0)=1$$ | |
| 7. If $$a>1$$ and $$r_(1)>0\Rightarrow a^(r_(1))> 1$$ | |
| 8. If $$0< a< 1$$ и $$r_{1}>0\Rightarrow 0< a^{r_{1}}< 1$$ | |
| 9. If $$a>1$$ and $$r_(1)>r_(2)\Rightarrow a^(r_(1))> a^(r_(2))$$ | |
| 10. If $$0< a< 1$$ и $$r_{1}>r_(2)\Rightarrow a^(r_(1))> a^(r_(2))$$ |
The concept of the degree of a positive number is generalized for any real exponent α
.
Determining the degree of a positive number a with real exponents α
.
1. If $$\alpha > 0$$ and
1) $$\alpha=m$$, $$m\in N \Rightarrow a^(\alpha)=\left\(\begin(matrix)a, m=1\\\underbrace(a\cdot a\ cdot a\cdot a....a), m\geq 2\end(matrix)\right.$$
2) $$\alpha=\frac(p)(q)$$, where p and q- natural numbers $$\Rightarrow a^(\alpha)=\sqrt[q](a^(p))$$
3) α is an irrational number, then
a) if a > 1, then a α- number greater than a r i and less than a r k, where r i α
with a disadvantage rk- any rational approximation of a number α
in excess;
b) if 0< a< 1, то a α- a number greater than a r k and less than a r i;
c) if a= 1, then a α = 1.
2. If $$\alpha=0$$, then a α = 1.
3. If $$\alpha<0$$, то $$a^{\alpha}=\frac{1}{a^{\left | \alpha \right |}}$$.
Number a α is called a degree, the number a is the base of the degree, the number α
- exponent.
A power of a positive number with a real exponent has the same properties as a power with a rational exponent.
Example 1.4.3. Calculate $$\sqrt(81)\cdot\sqrt(\frac(16)(6))$$.
Decision. Let's use the root property:
$$\sqrt(81)\cdot\sqrt(\frac(16)(6))=\sqrt(\frac(81\cdot16)(6))=\sqrt(\frac(3^(4)\cdot2 ^(4))(3\cdot2))=\sqrt(3^(3)\cdot2^(3))=6$$
Answer. 6.
Example 1.4.4. Calculate $$6.25^(1.5)-2.25^(1.5)$$
| 1) 4 | 2) 8 | 3) 8,25 | 4) 12,25 |
But are these fractions always periodic? The answer to this question is negative: there are segments whose lengths cannot be expressed by an infinite periodic fraction (that is, a positive rational number) with a chosen unit of length. This was the most important discovery in mathematics, from which it followed that rational numbers are not enough to measure the lengths of segments.
If the unit of length is the length of a side of a square, then the length of the diagonal of this square cannot be expressed by a positive rational number.
From this statement it follows that there are segments whose lengths cannot be expressed as a positive number (with the chosen unit of length), or, in other words, written as an infinite periodic fraction. This means that the infinite decimal fractions obtained by measuring the lengths of segments can be non-periodic.
It is believed that infinite non-periodic decimal fractions are a record of new numbers - positive irrational numbers. Since the concepts of a number and its notation are often identified, they say that infinite periodic decimal fractions are positive irrational numbers.
The set of positive irrational numbers is denoted by the symbol J+.
The union of two sets of numbers: positive rational and positive irrational is called the set of positive real numbers and is denoted by the symbol R+.
Any positive real number can be represented by an infinite decimal fraction - periodic (if it is rational) or non-periodic (if it is irrational).
Actions on positive real numbers are reduced to actions on positive rational numbers. In this regard, for each positive real number, its approximate values \u200b\u200bare introduced in terms of deficiency and excess.
Let two positive real numbers be given a and b, an and bn- according to their approximations in terms of deficiency, a¢n and b¢n are their approximations in excess.
The sum of real numbers a and b a+ b n satisfies the inequality an+ bn ≤ a + b< a¢n + b¢n.
The product of real numbers a and b such a real number is called a× b, which for any natural n satisfies the inequality an× bn ≤
a b
Difference of positive real numbers a and b such a real number is called with, what a= b + c.
Quotient of positive real numbers a and b such a real number is called with, what a= b × s.
The union of the set of positive real numbers with the set of negative real numbers and zero is the set R of all real numbers.
Comparison of real numbers and operations on them are performed according to the rules known from the school mathematics course.
Problem 60. Find the first three decimal places of the sum 0.333… + 1.57079…
Decision. Let's take decimal approximations of terms with four decimal places:
0,3333 < 0,3333… < 0,3334
1,5707 < 1,57079… < 1,5708.
Add up: 1.9040 ≤ 0.333… + 1.57079…< 1,9042.
Therefore, 0.333… + 1.57079…= 1.904…
Task 61. Find the first two decimal places of the product a x b, if a= 1.703604… and b = 2,04537…
Decision. We take decimal approximations of these numbers with three decimal places:
1,703 < a <1,704 и 2,045 < b < 2,046. По определению произведения действительных чисел имеем:
1.703 × 2.045 ≤ a x b < 1,704 × 2,046 или 3,483 ≤ ab < 3,486.
Thus, a x b= 3,48…
Exercises for independent work
1. Write down the decimal approximations of the irrational number π = 3.1415 ... in terms of deficiency and excess with an accuracy of:
a) 0.1; b) 0.01; c) 0.001.
2. Find the first three decimal places of the sum a+ b, if:
a) a = 2,34871…, b= 5.63724…; b) a = , b= π; in) a = ; b= ; G) a = ; b = .
