Laws of arithmetic operations on real numbers. Arithmetic operations on real numbers
Let some number XÎ R + first changed to a, and then on in, and the number X so large that both of these changes do not deduce from the set R + . Let's call sum numbers a and in a real number expressing the resulting change. For example, if you first make a change to 4, and then to 7, the number 12 will go first to 16, and then 16 will go to 23. But in order for 12 to go to 23, you need to change it to 11, which means 4 + 7 = 11, as and should be. If you first make a change to -4, and then to -7, then 12 will go first to 8; and then to 1. But in order to get 1 out of 12, you need to change 12 to -11. It follows that (–4) + (–7) = –11.
In general, if a and in - positive real numbers and
X>a+in, then when changing to - in number X–a goes to ( x – a) – in, those. in X–(a + in).
But to get X – (a + in) needs to be changed. X on the
–(a + b). This shows that (- a) + (–in) = – (a + b).
Consider now the addition of numbers of opposite signs. Let's start with the case when the terms are opposite numbers. Obviously, if we change the number X first on a, and then on - a, then we get again X. In other words, x +(a +(–a)) = X. Since, on the other hand, and X+ 0 = X, then you have to put a +(–a) = 0. So, the sum of opposite numbers is equal to zero.
Now let's find the sum a+ (–in) in the general case (we assume that a and in are positive numbers, so in negative). If a a> in, then
a = (a–in)+ in, and that's why a+ (–in) = (a–in)+in+ (–in).
But successive changes in the number X on the a– in, in and - in can be replaced by changing to a–in(changes to in and - in cancel each other out). Therefore, we put a +(–in) = a–in, if a> in. It is obvious that at a> in and (- in) +a = a–in.
Let now a<in. In this case we have - in = (–a)+ (–(in– a)), and that's why a + (–in) = a + (–a) + (–(in– a)) = – (in – a). So, at a < in must be put a + (–in) = – (in – a). The same result will be obtained when adding - in and a: (–in) + a = –(in– a).
The resulting addition rules for real numbers can be formulated as the following definition.
Definition.When added two real numbers of the same sign, you get a number of the same sign, the modulus of which is equal to the sum of the moduli of the terms. When adding numbers of different signs, a number is obtained whose sign coincides with the sign of the term with a larger module, and the module is equal to the difference between the greater and lesser modules of the terms. The sum of opposite numbers is zero, and adding to zero does not change the number.
It is easy to check that the addition in R has the properties of commutativity, associativity and contractibility. It can be seen from the above definition that zero is a neutral element with respect to addition , those.
a + 0= a.
Subtraction in multitude R is defined as the inverse operation of addition. Because every number in in R has the opposite number in, such that in+ (–in) = 0, then the subtraction of the number in is equivalent to adding to a number c: a–in=a+ (–in).
Indeed, for any a and in we have:
(a + (–in)) + in = a+ ((–in) + in) = a, and this means that a– in = a + (–in).
For positive numbers a and in, such that a>in, their difference
a–in was a change that in goes into a. By analogy with this we will call for any real numbers a and in number a–in change that translates in in a. It takes point 0 to point a – in. As for positive real numbers, this change is geometrically represented by a directed segment coming from the point in exactly a. Its length is equal to the distance from the origin to the point
a–in, those. modulo number a–in. We have proved the following important assertion:
The length of the segment from a point in exactly a, is equal to | a–in|.
We introduce into the set R
order relation. We will assume that
a> in if and only if the difference a– in positive. It is easy to prove that this relation is antisymmetric and transitive, i.e. is a strict order relation. However, for any a and in from R
one and only one of the following is true: a= in, a< in, in< a, those. order relation in R
linearly. Because the a– 0 = a, then a> 0 if aÎ R
+
, and a< 0, еслиaÎ R-
.
It is easy to prove that if a> in, then for any WithÎ R
we have
a+ With> in+ With.
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
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.
It follows from this statement that there are segments whose lengths cannot be expressed by a positive number (with the chosen unit of length), or, in other words, cannot be 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 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…
Solution. 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…
Solution. 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.
In this way, 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 = .
REAL NUMBERS II
§ 46 Addition of real numbers
So far, we can only add rational numbers to each other. As we know,
But what is the meaning of the sum of two numbers, of which at least one is irrational, we still do not know this. We now have to define what is meant by the sum α + β two arbitrary real numbers α and β .
For example, consider the numbers 1 / 3 and √2. Let's represent them in the form of infinite decimal fractions
1 / 3 = 0,33333...;
√2 =1,41421... .
First, we add the corresponding decimal approximations of these numbers with a disadvantage. These approximations, as noted at the end of the previous section, are rational numbers. And we already know how to add such numbers:
0+1 = 1
0,3+1,4= 1,7
0,33+1,41 = 1,74
0,333 + 1,414 = 1,747
0,3333 + 1,4142= 1,7475
0,33333 + 1,41421 = 1,74754
.................................................................
Then we add the corresponding decimal approximations of these numbers with an excess:
1 +2 = 3
0,4+ 1,5 = 1,9
0,34+ 1,42= 1,76
0,334 + 1,415 = 1,749
0,3334 + 1,4143=1,7477
0,33334+ 1,41422= 1,74756
..........................................................
It can be proved* that there exists, moreover, a unique real number γ , which is greater than all sums of decimal approximations of the numbers 1 / 3 and √2 with a disadvantage, but less than all sums of decimal approximations of these numbers with an excess:
* A rigorous proof of this fact is beyond the scope of our program and therefore is not given here.
1 < γ < 3
1,7 < γ < 1,9
1,74 < γ < 1,76
1,747 < γ < 1,749
1,7475 < γ < 1,7477
1,74754 < γ < 1,74756
By definition, this number γ and is taken as the sum of the numbers 1 / 3 and √2:
γ = 1 / 3 + √2
It's obvious that γ = 1,7475....
The sum of any other positive real numbers, at least one of which is irrational, can be defined similarly. The essence of the matter will not change even if one of the terms, and perhaps both, are negative.
So, if numbers α and β are rational, then their sum is found by the rule of addition of rational numbers(see § 36).
If at least one of them is irrational, then the sum α + β a real number is called which is greater than all the sums of the corresponding decimal approximations of these numbers with a disadvantage, but less than all the sums of the corresponding decimal approximations of these numbers with an excess.
The action of addition thus defined obeys the following two laws:
1) commutative law:
α + β = β + α
2) association law:
(α + β ) + γ = α + (β + γ ).
We will not prove this. Students can do this on their own. We only note that in the proof we will have to use the fact already known to us: the addition of rational numbers is subject to commutative and associative laws (see § 36).
Exercises
327. Present these amounts as decimal fractions, indicating at least three correct digits after the busy:
a) √2 + √3 ; d) √2 + (- √3 ) g) 3/4 + (-√5 );
b) √2 + 5/8; e) (- 1/3) + √5 h) 1/3 + √2 + √3.
c) (-√2) + √3; f) 11/9 + (- √5);
328. Find the first few decimal approximations (with and without excess) for real numbers:
a) 1 / 2 + √7 b) √3 + √7 c) √3 + (-√7)
329. Proceeding from the definition of the sum of real numbers, prove that for any number α
α + (- α ) = 0.
330. Is the sum of two infinite non-periodic fractions always a non-periodic fraction? Explain the answer with examples.
1. The concept of an irrational number. Infinite decimal non-periodic fractions. The set of real numbers.
2. Arithmetic operations on real numbers. Laws of addition and multiplication.
3. Extension of real positive numbers to the set of real numbers. Properties of the set of real numbers.
4. Approximate numbers. Rules for rounding real numbers and actions with approximate numbers. Calculations with the help of a microcalculator.
5. Key Findings
Real numbers
One of the sources of the appearance of decimal fractions is the division of natural numbers, the other is the measurement of quantities. Let us find out, for example, how decimal fractions can be obtained when measuring the length of a segment.
Let X- the segment whose length is to be measured, e- single cut. Cut length X denote by the letter X, and the length of the segment e- letter E. Let the segment X comprises n segments equal to e₁ and cut X₁, which is shorter than the segment e(Fig. 130), i.e. n ∙E < X < (n + 1) ∙E. Numbers n and n+ 1 are approximate values of the length of the segment X at unit length E with a deficiency and with an excess up to 1.
To get an answer with greater accuracy, take the segment e₁ is a tenth of the segment e and we will put it in the segment X₁. In this case, two cases are possible.
1) The segment e₁ fit into the segment X₁ precisely n once. Then the length n segment X expressed as a final decimal: X = (n+n₁\10) ∙E= n, n₁∙E. For example, X= 3.4∙E.
2) Cut X₁ turns out to consist of n segments equal to e₁, and a segment X₂, which is shorter than the segment e₁. Then n,n₁∙E < X < n,n₁n₁′∙ E, where n,n₁ and n,n₁n₁′ - approximate values of the segment length X with a deficiency and with an excess with an accuracy of 0.1.
It is clear that in the second case the process of measuring the length of a segment X you can continue by taking a new unit segment e₂ - hundredth of the segment e.
In practice, this process of measuring the length of a segment will end at some stage. And then the result of measuring the length of the segment will be either a natural number or a final decimal fraction. If we imagine this process of measuring the length of a segment ideally (as they do in mathematics), then two outcomes are possible:
1) At the k-th step, the measurement process will end. Then the length of the segments will be expressed as a final decimal fraction of the form n,n₁… n k.
2) The described process for measuring the length of a segment X continues indefinitely. Then the report about it can be represented by the symbol n,n₁… n k..., which is called an infinite decimal.
How to be sure of the possibility of a second outcome? To do this, it is enough to measure the length of such a segment, for which it is known that its length is expressed, for example, by a rational number 5. If it turned out that as a result of measuring the length of such a segment, a final decimal fraction is obtained, then this would mean that the number 5 can be represented as a final decimal fraction, which is impossible: 5 \u003d 5.666 ....
So, when measuring the lengths of segments, infinite decimal fractions can be obtained. 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.
Theorem. 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.
Proof. Let the length of the side of the square be expressed by the number 1. Suppose the opposite of what needs to be proved, i.e., that the length of the diagonal AC of the square ABCB is expressed as an irreducible fraction. Then, according to the Pythagorean theorem, the equality would hold
1²+ 1² = . It follows from it that m² = 2n². So, m² is an even number, then the number m is even (the square of an odd number cannot be even). So, m = 2p. Replacing the number m by 2p in the equation m² = 2n², we get that 4p² = 2n², i.e. 2p² = n². It follows that n² is even, hence n is an even number. Thus, the numbers m and n are even, which means that the fraction can be reduced by 2, which contradicts the assumption that it is irreducible. The established contradiction proves that 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 rational number.
It follows from the proved theorem that there are segments whose lengths cannot be expressed by 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 non-periodic decimal fractions are positive irrational numbers.
We arrived at the concept of a positive irrational number through the process of measuring the lengths of segments. But irrational numbers can also be obtained by extracting roots from some rational numbers. So √2, √7, √24 are irrational numbers. Irrational are also lg 5, sin 31, the numbers π = 3.14..., e= 2.7828... and others.
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+. Thus, Q+ ∪ J + = R+. With the help of Euler circles, these sets are depicted in Figure 131.
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.
Addition and multiplication of positive real numbers have the properties of commutativity and associativity, and multiplication is distributive with respect to addition and subtraction.
Using positive real numbers, you can express the result of measuring any scalar quantity: length, area, mass, etc. But in practice, it is often necessary to express by a number not the result of measuring a quantity, but its change. Moreover, its change can occur in different ways - it can increase, decrease or remain unchanged. Therefore, in order to express the change in magnitude, other numbers are needed besides positive real numbers, and for this it is necessary to expand the set R + by adding the number 0 (zero) and negative numbers to it.
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 to us from the school mathematics course.
Exercises
1. Describe the process of measuring the length of a segment, if the report on it is presented as a fraction:
a) 3.46; b) 3,(7); c) 3.2(6).
2. The seventh part of a single segment fits into segment a 13 times. Will the length of this segment be represented by a finite or infinite fraction? Periodic or non-periodic?
3. A set is given: (7; 8; √8; 35.91; -12.5; -√37; 0; 0.123; 4136).
Can it be divided into two classes: rational and irrational?
4. It is known that any number can be represented by a point on the coordinate line. Do points with rational coordinates exhaust the entire coordinate line? What about points with real coordinates?
99. Main conclusions § 19
When studying the material of this paragraph, we have clarified many concepts known from the school course of mathematics, linking them with the measurement of the length of a segment. These are concepts such as:
fraction (correct and incorrect);
equal fractions;
irreducible fraction;
positive rational number;
equality of positive rational numbers;
mixed fraction;
infinite periodic decimal;
infinite non-periodic decimal;
irrational number;
real number.
We found out that the relation of equality of fractions is an equivalence relation and took advantage of this, defining the concept of a positive rational number. We also found out how addition and multiplication of positive rational numbers are connected with measuring the lengths of segments and obtained formulas for finding their sum and product.
The definition of the relation "less than" on the set Q+ made it possible to name its main properties: it is ordered, dense, it does not contain the smallest and largest number.
We have proved that the set Q+ of positive rational numbers satisfies all the conditions that allow it to be considered an extension of the set N of natural numbers.
By introducing decimal fractions, we proved that any positive rational number can be represented by an infinite periodic decimal fraction.
Infinite non-periodic fractions are considered records of irrational numbers.
If we unite the sets of positive rational and irrational numbers, then we get the set of positive real numbers: Q+ ∪ J + = R+.
If we add negative real numbers and zero to positive real numbers, then we get the set R of all real numbers.
