Integer icon. Whole numbers. Definition. Operations on integers

What does integer mean

So, consider what numbers are called integers.

Thus, integers will denote such numbers: $0$, $±1$, $±2$, $±3$, $±4$, etc.

The set of natural numbers is a subset of the set of integers, i.e. any natural will be an integer, but not any integer is a natural number.

Integer positive and integer negative numbers

Definition 2

a plus.

The numbers $3, 78, 569, 10450$ are positive integers.

Definition 3

are signed integers minus.

Numbers $−3, −78, −569, -10450$ are negative integers.

Remark 1

The number zero does not refer to either positive integers or negative integers.

Whole positive numbers are integers greater than zero.

Whole negative numbers are integers less than zero.

The set of natural integers is the set of all positive integers, and the set of all opposites of natural numbers is the set of all negative integers.

Integer non-positive and integer non-negative numbers

All positive integers and the number zero are called integer non-negative numbers.

Integer non-positive numbers are all negative integers and the number $0$.

Remark 2

In this way, whole non-negative number are integers greater than zero or equal to zero, and non-positive integer are integers less than zero or equal to zero.

For example, non-positive integers: $−32, −123, 0, −5$, and non-negative integers: $54, 123, 0.856 342.$

Description of changing values ​​using integers

Integers are used to describe changes in the number of any items.

Consider examples.

Example 1

Suppose a store sells a certain number of items. When the store receives $520$ of items, the number of items in the store will increase, and the number of $520$ shows a positive change in the number. When the store sells $50$ items, the number of items in the store will decrease, and the number $50$ will express a negative change in the number. If the store will neither bring nor sell the goods, then the number of goods will remain unchanged (i.e., we can talk about a zero change in the number).

In the above example, the change in the number of goods is described using the integers $520$, $−50$, and $0$, respectively. A positive value of the integer $520$ indicates a positive change in the number. A negative value of the integer $−50$ indicates a negative change in the number. The integer $0$ indicates the immutability of the number.

Integers are convenient to use, because no explicit indication of an increase in number or decrease is needed - the sign of the integer indicates the direction of the change, and the value indicates a quantitative change.

Using integers, you can express not only a change in quantity, but also a change in any value.

Consider an example of a change in the cost of a product.

Example 2

An increase in cost, for example, by $20$ rubles is expressed using a positive integer $20$. Decreasing the cost, for example, by $5$ rubles is described using a negative integer $−5$. If there are no cost changes, then such a change is determined using the integer $0$.

Separately, consider the value of negative integers as the size of the debt.

Example 3

For example, a person has $5,000 rubles. Then, using a positive integer $5,000$, you can show the number of rubles that he has. A person has to pay a rent in the amount of $7,000 rubles, but he does not have that kind of money; in this case, such a situation is described by a negative integer $−7,000$. In this case, the person has $−7,000$ rubles, where "-" indicates debt, and the number $7,000$ shows the amount of debt.

These are the numbers that are used in counting: 1, 2, 3...etc.

Zero is not natural.

Natural numbers are usually denoted by the symbol N.

Whole numbers. Positive and negative numbers

Two numbers that differ only in sign are called opposite, for example, +1 and -1, +5 and -5. The "+" sign is usually not written, but it is assumed that a "+" is in front of the number. Such numbers are called positive. Numbers preceded by a "-" sign are called negative.

The natural numbers, their opposites and zero are called whole numbers. The set of integers is denoted by the symbol Z.

Rational numbers

These are finite fractions and infinite periodic fractions. For example,

The set of rational numbers is denoted Q. All integers are rational.

Irrational numbers

An infinite non-periodic fraction is called an irrational number. For example:

The set of irrational numbers is denoted J.

Real numbers

The set of all rational and all irrational numbers is called set of real (real) numbers.

Real numbers are denoted by the symbol R.

Rounding numbers

Consider the number 8,759123... . To round to the nearest integer means to write down only that part of the number that is before the decimal point. Rounding to tenths means writing down the whole part and after the decimal point one digit; round to hundredths - two digits after the decimal point; up to thousandths - three digits, etc.

Number- the most important mathematical concept that has changed over the centuries.

The first ideas about the number arose from counting people, animals, fruits, various products, etc. The result is natural numbers: 1, 2, 3, 4, ...

Historically, the first extension of the concept of number is the addition of fractional numbers to a natural number.

Shot called a part (share) of a unit or several equal parts of it.

Designated: , where m,n- whole numbers;

Fractions with denominator 10 n, where n is an integer, they are called decimal: .

Among decimal fractions, a special place is occupied by periodic fractions: - pure periodic fraction, - mixed periodic fraction.

Further expansion of the concept of number is already caused by the development of mathematics itself (algebra). Descartes in the 17th century introduces the concept negative number.

Numbers whole (positive and negative), fractional (positive and negative) and zero are called rational numbers. Any rational number can be written as a finite and periodic fraction.

To study continuously changing variables, it turned out to be necessary to expand the concept of number - the introduction of real (real) numbers - by adding irrational numbers to rational numbers: irrational numbers are infinite decimal non-periodic fractions.

Irrational numbers appeared when measuring incommensurable segments (side and diagonal of a square), in algebra - when extracting roots, an example of a transcendental, irrational number is π, e .

Numbers natural(1, 2, 3,...), whole(..., –3, –2, –1, 0, 1, 2, 3,...), rational(represented as a fraction) and irrational(not representable as a fraction ) form a set real (real) numbers.

Separately in mathematics, complex numbers are distinguished.

Complex numbers arise in connection with the problem of solving squares for the case D< 0 (здесь D is the discriminant of the quadratic equation). For a long time, these numbers did not find physical use, which is why they were called "imaginary" numbers. However, now they are very widely used in various fields of physics and technology: electrical engineering, hydro- and aerodynamics, elasticity theory, etc.

Complex numbers are written as: z= a+ bi. Here a and b – real numbers, a i – imaginary unit.e. i 2 = -1. Number a called abscissa, a b-ordinate complex number a+ bi. Two complex numbers a+ bi and a-bi called conjugate complex numbers.

Properties:

1. Real number a can also be written as a complex number: a+ 0i or a - 0i. For example 5 + 0 i and 5 - 0 i mean the same number 5 .

2. Complex number 0 + bi called purely imaginary number. Recording bi means the same as 0 + bi.

3. Two complex numbers a+ bi and c+ di are considered equal if a= c and b= d. Otherwise, the complex numbers are not equal.

Actions:

Addition. The sum of complex numbers a+ bi and c+ di is called a complex number ( a+ c) + (b+ d)i. In this way, when adding complex numbers, their abscissas and ordinates are added separately.

Subtraction. The difference between two complex numbers a+ bi(reduced) and c+ di(subtracted) is called a complex number ( a-c) + (b-d)i. In this way, when subtracting two complex numbers, their abscissas and ordinates are subtracted separately.

Multiplication. The product of complex numbers a+ bi and c+ di is called a complex number.

(ac-bd) + (ad+ bc)i. This definition stems from two requirements:

1) numbers a+ bi and c+ di must multiply like algebraic binomials,

2) number i has the main property: i 2 = –1.

EXAMPLE ( a + bi)(a-bi)= a 2 +b 2 . Consequently, workof two conjugate complex numbers is equal to a positive real number.

Division. Divide a complex number a+ bi(divisible) to another c+ di (divider) - means to find the third number e+ fi(chat), which, when multiplied by a divisor c+ di, which results in the dividend a+ bi. If the divisor is not zero, division is always possible.

EXAMPLE Find (8+ i) : (2 – 3i) .

Solution. Let's rewrite this ratio as a fraction:

Multiplying its numerator and denominator by 2 + 3 i and doing all the transformations, we get:

Task 1: Add, subtract, multiply and divide z 1 to z 2

Extracting the square root: Solve the equation x 2 = -a. To solve this equation we are forced to use a new type of numbers - imaginary numbers . In this way, imaginary the number is called whose second power is a negative number. According to this definition of imaginary numbers, we can define and imaginary unit:

Then for the equation x 2 = - 25 we get two imaginary root:

Task 2: Solve the equation:

1) x 2 = – 36; 2) x 2 = – 49; 3) x 2 = – 121

Geometric representation of complex numbers. Real numbers are represented by points on the number line:

Here is the point A means number -3, dot B is the number 2, and O-zero. In contrast, complex numbers are represented by points on the coordinate plane. For this, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex number a+ bi will be represented by a dot P with abscissaa and ordinateb. This coordinate system is called complex plane .

module complex number is called the length of the vector OP, depicting a complex number on the coordinate ( comprehensive) plane. Complex number modulus a+ bi denoted by | a+ bi| or) letter r and is equal to:

Conjugate complex numbers have the same modulus.

The rules for drawing up a drawing are almost the same as for a drawing in a Cartesian coordinate system. Along the axes, you need to set the dimension, note:

e
unit along the real axis; Rez

imaginary unit along the imaginary axis. im z

Task 3. Construct the following complex numbers on the complex plane: , , , , , , ,

1. Numbers are exact and approximate. The numbers we encounter in practice are of two kinds. Some give the true value of the quantity, others only approximate. The first is called exact, the second - approximate. Most often it is convenient to use an approximate number instead of an exact number, especially since in many cases the exact number cannot be found at all.

So, if they say that there are 29 students in the class, then the number 29 is exact. If they say that the distance from Moscow to Kyiv is 960 km, then here the number 960 is approximate, since, on the one hand, our measuring instruments are not absolutely accurate, on the other hand, the cities themselves have some extent.

The result of operations with approximate numbers is also an approximate number. By performing some operations on exact numbers (dividing, extracting the root), you can also get approximate numbers.

The theory of approximate calculations allows:

1) knowing the degree of accuracy of the data, assess the degree of accuracy of the results;

2) take data with an appropriate degree of accuracy, sufficient to ensure the required accuracy of the result;

3) rationalize the calculation process, freeing it from those calculations that will not affect the accuracy of the result.

2. Rounding. One source of approximate numbers is rounding. Round off both approximate and exact numbers.

Rounding a given number to some of its digits is the replacement of it with a new number, which is obtained from the given one by discarding all of its digits written to the right of the digit of this digit, or by replacing them with zeros. These zeros are usually underlined or written smaller. To ensure the greatest proximity of the rounded number to the rounded one, the following rules should be used: in order to round the number to one of a certain digit, you must discard all the digits after the digit of this digit, and replace them with zeros in the whole number. This takes into account the following:

1) if the first (left) of the discarded digits is less than 5, then the last remaining digit is not changed (rounding down);

2) if the first discarded digit is greater than 5 or equal to 5, then the last remaining digit is increased by one (rounding up).

Let's show this with examples. Round up:

a) up to tenths of 12.34;

b) up to hundredths of 3.2465; 1038.785;

c) up to thousandths of 3.4335.

d) up to 12375 thousand; 320729.

a) 12.34 ≈ 12.3;

b) 3.2465 ≈ 3.25; 1038.785 ≈ 1038.79;

c) 3.4335 ≈ 3.434.

d) 12375 ≈ 12,000; 320729 ≈ 321000.

3. Absolute and relative errors. The difference between the exact number and its approximate value is called the absolute error of the approximate number. For example, if the exact number 1.214 is rounded to tenths, we get an approximate number of 1.2. In this case, the absolute error of the approximate number 1.2 is 1.214 - 1.2, i.e. 0.014.

But in most cases, the exact value of the quantity under consideration is unknown, but only approximate. Then the absolute error is also unknown. In these cases, indicate the limit that it does not exceed. This number is called the marginal absolute error. They say that the exact value of a number is equal to its approximate value with an error less than the boundary error. For example, the number 23.71 is the approximate value of the number 23.7125 with an accuracy of 0.01, since the absolute approximation error is 0.0025 and less than 0.01. Here the boundary absolute error is equal to 0.01 * .

Boundary absolute error of the approximate number a denoted by the symbol Δ a. Recording

x≈a(±Δ a)

should be understood as follows: the exact value of the quantity x is in between a– Δ a and a+ Δ a, which are called the lower and upper bounds, respectively. X and denote NG x VG X.

For example, if x≈ 2.3 (±0.1), then 2.2<x< 2,4.

Conversely, if 7.3< X< 7,4, тоX≈ 7.35 (±0.05). Absolute or marginal absolute error does not characterize the quality of the measurement. The same absolute error can be considered significant and insignificant, depending on the number that expresses the measured value. For example, if we measure the distance between two cities with an accuracy of one kilometer, then such accuracy is quite sufficient for this change, while at the same time, when measuring the distance between two houses on the same street, such accuracy will be unacceptable. Therefore, the accuracy of the approximate value of a quantity depends not only on the magnitude of the absolute error, but also on the value of the measured quantity. Therefore, the measure of accuracy is the relative error.

Relative error is the ratio of the absolute error to the value of the approximate number. The ratio of the boundary absolute error to the approximate number is called the boundary relative error; denote it like this: Relative and boundary relative errors are usually expressed as a percentage. For example, if measurements show that the distance X between two points is more than 12.3 km, but less than 12.7 km, then the arithmetic mean of these two numbers is taken as an approximate value, i.e. their half-sum, then the boundary absolute error is equal to the half-difference of these numbers. In this case X≈ 12.5 (±0.2). Here, the boundary absolute error is 0.2 km, and the boundary relative

In order to do any job effectively, you need tools to dig, you need a shovel or an excavator; to think you need words. Numbers are tools that allow you to work with quantities.

It seems that we all know what a number is: 1, 2, 3… But let's talk about numbers as tools.

Let's take three objects: an apple, a balloon, the Earth (Fig. 1). What do they have in common? Shape is all balls.

Rice. 1. Illustration for example

Take three other items (Fig. 2). What do they have in common? Color - they are all blue.

Rice. 2. Illustration for example

Let us now take three sets: three cars, three apples, three pencils (Fig. 3). What do they have in common? The number is three.

Rice. 3. Illustration for example

We can put an apple on each car, and stick a pencil in each apple (Fig. 4). A common property of these sets is the number of elements.

Rice. 4. Comparison of sets

However, there are few natural numbers for solving problems, so negative, rational, irrational, etc. were also introduced. Mathematics (especially that part of it that is studied at school) is a kind of mechanism for processing signs.

Take, for example, two piles of sticks, one with seventeen pieces, and the other with twenty-five (Fig. 5). How to find out how many sticks are in both piles?

Rice. 5. Illustration for example

If there is no mechanism, then it is not clear: you can only put the sticks in one pile and count them.

But if the number of sticks is written in the decimal system familiar to us (and), then we can use the mechanisms for addition. For example, we can add numbers in a column (Fig. 6): .

Rice. 6. Stacking

Also, we will not be able to add the numbers written like this: three hundred and seventy-four plus four hundred and eighty-five. But if you write down the numbers in the decimal system, then for addition there is an algorithm - addition in a column (Fig. 7):.

Rice. 7. Stacking

If there is a car, then it is worth building a smooth road, together they are effective. Similarly: if there is a plane, then an airfield is needed. That is, the mechanism itself and the surrounding infrastructure are connected - individually they are much less effective.

In this case, there is a tool - numbers written in a positional system, and an infrastructure has been invented for them: algorithms for performing various actions, for example, addition in a column.

The numbers written in the decimal positional system replaced others (Roman, etc.) precisely because efficient and simple algorithms were invented to work with them.

Let's take a closer look at the decimal positional system. There are two main ideas that underlie it (thanks to which it got its name).

1. Decimal: we count in groups, namely tens.

2. positionality: The contribution of a digit to a number depends on its position. For example, , : the numbers are different, although they consist of the same digits.

These two ideas helped create a system that is easy to perform and write down numbers, since we have a limited set of characters (in this case, numbers) to write an infinite number of numbers.

Emphasize the importance technologies on such an example. Suppose you need to move a heavy load. If you use manual labor, then everything will depend on how strong a person is carrying the load: one will cope, the other will not.

The invention of technology (for example, a car that can carry this load) equalizes the possibilities of people: a fragile girl or a weightlifter can sit behind the wheel, but both of them will be able to cope with the task of moving the load equally effectively. That is, technology can be taught to anyone, not just a specialist.

Addition and multiplication in a column is also a technology. Working with numbers written in the Roman numeral system is a difficult task, only specially trained people could do it. Any fourth grader can add and multiply numbers in the decimal system.

As we have said, people have invented different numbers, and all of them are needed. The next (after natural) important invention are negative numbers. With the help of negative numbers, counting has become easier. How did it happen?

If we subtract the smaller from the larger, then there is no need for negative numbers: it is clear that the larger number contains the smaller. But it turned out that it is worth introducing negative numbers as a separate object. It cannot be seen, touched, but it is useful.

Consider this example: You can do the calculations in a different order: then there is no problem, we have enough natural numbers.

But sometimes there is a need to perform actions sequentially. If we run out of money in our account, we are given a loan. Let us have rubles, and we spent on conversations. There are not enough rubles on the account, it is convenient to write it down with a minus sign, since if we return them, then the account will have:. This idea underlies the invention of such a tool as negative numbers.

In life, we often work with concepts that cannot be touched: joy, friendship, etc. But this does not prevent us from understanding and analyzing them. We can say that these are just invented things. Indeed, they are, but they help people do something. Also, the car was invented by man, but it helps us move. Numbers are also invented by man, but they help to solve problems.

Let's take such an object as a clock (Fig. 8). If you pull out a part from there, it is not clear what it is and why it is needed. Without a watch, this part does not exist. So the negative number exists within mathematics.

Rice. 8. Clock

Often teachers try to indicate what a negative number is. They give an example of a negative temperature (Fig. 9).

Rice. 9. Negative temperature

But this is only a name, a designation, and not the number itself. It was possible to introduce another scale, where the same temperature would be, for example, positive. In particular, negative temperatures on the Celsius scale in the Kelvin scale are expressed as positive numbers: .

That is, there is no negative quantity in nature. However, numbers are not only used to express quantities. Recall the basic functions of the number.

So, we talked about natural and integer numbers. The number is a handy tool that can be used to solve various problems. Of course, for those working inside mathematics, numbers are objects. As for those who make pliers, they are also objects, not tools. We will consider numbers as a tool that allows us to think and work with quantities.