What does an integer mean

So, consider what numbers are called integers.

Thus, such numbers: $ 0 $, $ ± 1 $, $ ± 2 $, $ ± $ 3, $ ± $ 4 $, etc. will be designated.

Lots of natural numbers There are a subset of many integers, i.e. Any natural will be an integer, but not any integer is a natural number.

Whole positive and whole negative numbers

Definition 2.

a plus.

Numbers $ 3,78, 569, $ 10450 - whole positive numbers.

Definition 3.

are integers with a sign minus.

Numbers $ -3, -78, -569, -10450 $ - whole negative numbers.

Note 1.

The number of zero does not apply to any positive, nor to the whole negative numbers.

Whole positive numbers are integers, large zero.

Whole negative numbers are integers smaller than zero.

The set of natural integers is the set of all integers of positive numbers, and the set of all opposite natural numbers is a set of all whole negative numbers.

Interestable and whole non-negative numbers

All whole positive numbers and number zero are called whole non-negative numbers.

Intequently indisputable numbers All are all negative numbers and the number $ 0 $.

Note 2.

In this way, whole nonnegative number are integers, large zero or equal zero, and in an integrity number - integers smaller than zero or equal zero.

For example, integer non-persons: $ -32, -123, 0, -5 $, and whole non-negative numbers: $ 54, 123, 0, 856 342. $

Description of changes in values \u200b\u200busing integers

The integers are used to describe the change in the number of any items.

Consider examples.

Example 1.

Suppose in the store for sale some number of product names. When $ 520 $ items go to the store, the number of product names in the store will increase, and the number $ 520 $ shows a change in the number in positive side. When the store sells $ 50 $ product items, the number of product names in the store will decrease, and the number $ 50 will express the change in the number in negative side. If the store does not bring neither to bring or sell the goods, then the number of goods will remain unchanged (i.e., we can talk about the zero change of the number).

In the above example, the change in the number of goods is described with the use of integers $ 520 $, $ -50 $ and $ 0 $, respectively. The positive value of a whole number of $ 520 $ indicates a change in the number in a positive side. The negative value of an integer number $ -50 $ indicates a change in the number in the negative side. An integer than $ 0 $ indicates the invariance of the number.

Integers conveniently use, because It is not necessary to indicate an increase in the number or decrease - the sign of an integer indicates the direction of change, and the value is on the quantitative change.

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

Consider an example of changing the cost of goods.

Example 2.

Increased cost, for example, $ 20 rubles is expressed with a positive integer $ 20 $. A decrease in cost, for example, $ 5 $ rubles is described using a negative integer $ -5 $. If there is no changes in the cost, then such a change is determined using an integer $ 0 $.

Separately consider the value of negative integers as debt size.

Example 3.

For example, a person has $ 5,000 rubles. Then with the help of a positive number of $ 5,000 $ you can show the number of rubles that he has. A person must pay a rent in the amount of $ 7,000 rubles, but he has no such money, in this case, such a situation is described by a negative integer $ -7,000 $. In this case, a person has $ -7,000 $ rubles, where "-" indicates a debt, and the number of $ 7,000 $ shows the amount of debt.

Integers

Natural numbers definition are whole positive numbers. Natural numbers are used to account with objects and many other purposes. These are these numbers:

This is a natural number of numbers.
Zero natural number? No, zero is not a natural number.
How many natural numbers exist? There is an infinite set of natural numbers.
What is the smallest natural number? Unit is the smallest natural number.
What is the greatest natural number? It is impossible to indicate, because there is an infinite set of natural numbers.

The sum of natural numbers is a natural number. So, the addition of 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.

The difference in natural numbers is not always a natural number. If a reduced more subtracted, then the difference in natural numbers is a natural number, otherwise there is no.

Private natural numbers do not always have a natural number. If for natural numbers a and b

where C is a natural number, then this means that A is divided into B ath. In this example, A is divisible, B is a divider, C - private.

A natural number divider is a natural number that the first number is divided by a focus.

Each natural number is divided into one and on itself.

Simple natural numbers are divided only by one and on themselves. Here I mean, they are divided by a focus. Example, numbers 2; 3; five; 7 are divided only by one and on themselves. These are simple natural numbers.

Unit are not considered a simple number.

Numbers that are more units and which are not simple, called composite. Examples of compound numbers:

Unit are not considered a component.

Many natural numbers make up a unit, simple numbers and constituent numbers.

The set of natural numbers is denoted by the Latin letter N.

Properties of addition and multiplication of natural numbers:

move property of addition

the combination property of addition

(A + B) + C \u003d A + (B + C);

moving property multiplication

full character multiplication

(AB) C \u003d A (BC);

distribution property multiplication

A (B + C) \u003d AB + AC;

Whole numbers

The integers are natural numbers, zero and numbers opposite to natural.

Numbers opposite to natural - these are whole negative numbers, for example:

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

Many integers are denoted by the Latin letter Z.

Rational numbers

Rational numbers are integers and fractions.

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

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

Examples show that any integer is a periodic fraction with a period of zero.

Any rational number can be represented in the form of fractions M / N, where M integer, n natural number. Imagine in the form of such a fraction number 3, (6) from the previous example.

In this article, we define many integers, consider which integers are called positive, and which are negative. We also show how integers are used to describe the change in some values. Let's start with the definition and examples of integers.

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Whole numbers. Definition, examples

First we remember about natural numbers ℕ. The name itself suggests that these are numbers that are naturally used for an account from time immemorial. In order to embrace the concept of integers, we need to expand the definition of natural numbers.

Definition 1. Whole numbers

The integers are natural numbers, the numbers opposite to them, and the number of zero.

Many integers are denoted by the letter ℤ.

The set of natural numbers ℕ is a subset of integers ℤ. Any natural number is integer, but not any integer is natural.

It follows from the definition that any of the numbers 1, 2, 3 is integer. . , Number 0, as well as numbers - 1, - 2, - 3 ,. .

In accordance with this, we give examples. Numbers 39, - 589, 10000000, - 1596, 0 are integers.

Let the coordinate straight line be horizontally and directed to the right. Look at her to clearly imagine the location of the integers on a straight line.

The beginning of the reference on the coordinate direct corresponds to the number 0, and the points lying on both sides of the zero correspond to positive and negative integers. Each point corresponds to a single integer.

Any point is direct, the coordinate of which is an integer, you can get, postponing some of the coordinates a number of single segments.

Positive and negative integers

Of all the integers, it is logical to allocate positive and negative integers. Let's give them definition.

Definition 2. Positive integers

Positive integers are integer numbers with a plus sign.

For example, the number 7 is an integer with a plus sign, that is, a positive integer. On the coordinate direct, this number lies to the right of the point of reference, for which the number 0 is accepted. Other examples of positive integers: 12, 502, 42, 33, 100,500.

Definition 3. Negative integers

Negative integers are integers with a minus sign.

Examples of whole negative numbers: - 528, - 2568, - 1.

The number 0 divides positive and negative integers and itself is neither positive or negative.

Any number opposite to a positive integer, due to the definition, is a negative integer. Fair and reverse. The number inverse to any negative integer is a positive integer.

You can give other formulations of definitions of negative and positive integers, using their comparison with zero.

Definition 4. Positive integers

Positive integers are integers that are more zero.

Definition 5. Negative integers

Negative integers are integers that are less than zero.

Accordingly, the positive numbers are the right to start the reference on the coordinate direct, and the negative integers are left from zero.

Earlier, we have already said that natural numbers are a subset of whole. Create this moment. Many natural numbers constitute entire positive numbers. In turn, a plurality of negative integers is a multitude of numbers opposite to natural.

Important!

Any natural number can be called a whole, but any integer cannot be called natural. Answering the question whether negative numbers are natural, you need to safely talk - no, are not.

Non-positive and non-negative integers

Let's give the definition.

Definition 6. Non-negative integers

Non-negative integers are positive integers and the number of zero.

Definition 7. Invasive integers

Invalid integers are negative integers and number zero.

As we see, the number of zero is neither positive nor negative.

Examples of non-negative integers: 52, 128, 0.

Examples of non-positive integers: - 52, - 128, 0.

Non-negative number is a number, more or equal to zero. Accordingly, an inseminious integer is a number smaller or equal to zero.

The terms "non-positive number" and "non-negative number" are used for brevity. For example, instead of saying that the number A is an integer that is greater than or equal to zero, one can say: a is a non-negative number.

Use integers when describing changes in values

What are integers for? First of all, with their help it is convenient to describe and determine the change in the number of any objects. Let us give an example.

Let some crankshafts be stored in the warehouse. If another 500 crankshafts bring to the warehouse, then their number will increase. The number 500 just expresses the change (increase) of the number of parts. If then from the warehouse will drive 200 parts, then this number will also characterize the change in the number of crankshafts. This time, towards the decrease.

If nothing will be taken from the warehouse, and nothing will be brought, then the number 0 will indicate the number of parts.

The obvious ease of use of integers in contrast to the natural is that their sign clearly indicates the direction of changes in the value (increase or decrease).

A decrease in temperature by 30 degrees can be characterized by a negative number - 30, and an increase in 2 degrees is a positive integer number 2.

We give another example using integers. This time, imagine that we must give someone 5 coins. Then, we can say that we possess - 5 coins. The number 5 describes the amount of debt, and the "minus" sign says that we must give coins.

If we need 2 coins to one person, and 3 - another, then the total debt (5 coins) can be calculated according to the rule of addition of negative numbers:

2 + (- 3) = - 5

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Information of this article forms the general idea of whole numbers. First, the definition of integers is given and examples are given. Next, integers are considered on a numeric line, where it becomes clear what numbers are called integer positive numbers, and which are integer negative. After that, it is shown how with the help of integers, changes are described, and all negative numbers are considered in the sense of debt.

Navigating page.

Integers - definition and examples

Definition.

Whole numbers - These are natural numbers, the number of zero, as well as the numbers opposed to natural.

The definition of integers argues that any of the numbers 1, 2, 3, ..., the number 0, as well as any of the numbers -1, -2, -3, ... is whole. Now we can easily bring examples of integers. For example, the number 38 is an integer, the number 70 040 is also an integer, zero is an integer (we recall that zero is not a natural number, zero is an integer), the number -999, -1, -8 934 832 - also are examples of integers numbers.

All integers are conveniently represented as a sequence of integers, which has the following form: 0, ± 1, ± 2, ± 3, ... The sequence of integers can be recorded and so: …, −3, −2, −1, 0, 1, 2, 3, …

From the definition of integers it follows that the set of natural numbers is a subset of many integers. Therefore, any natural number is integer, but not any integer is natural.

Integers on the coordinate direct

Definition.

Whole positive numbers - These are integers that are more zero.

Definition.

Whole negative numbers - These are integers that are less than zero.

Calibly positive and negative numbers can also be determined by their position on the coordinate direct. On the horizontal coordinate direct point, whose coordinates are whole positive numbers, lie to the right of reference. In turn, the points with the whole negative coordinates are located to the left of the point O.

It is clear that the set of all integers positive numbers is a set of natural numbers. In turn, the set of all whole negative numbers are the set of all numbers opposite to natural numbers.

Separately, we will draw your attention to the fact that any natural number we can boldly be called the whole, and any integer we can call natural. Natural we can name only any integer positive number, as the whole negative numbers and zero are not natural.

Interestable and whole non-negative numbers

Let us give the definition of integer inseparable numbers and integer non-negative numbers.

Definition.

All the whole positive numbers along with the number of zero called whole non-negative numbers.

Definition.

Interesting numbers - These are all whole negative numbers along with a number of 0.

In other words, a non-negative number is an integer that is greater than zero, either equal to zero, and an integer indifference number is an integer that is less than zero or equal to zero.

Examples of integer non-quantities are the numbers -511, -10 030, 0, -2, and as examples of integer non-negative numbers, we give numbers 45, 506, 0, 900 321.

Most often, the terms "whole inhabitants" and "whole non-negative numbers" are used for shortness of presentation. For example, instead of the phrase "Number A is a whole, and a more zero or equal to zero," one can say "a - a non-negative number".

Description of changes in values \u200b\u200busing integers

It's time to talk about what the whole numbers are needed.

The main purpose of integers is that with their help it is convenient to describe the change in the number of any items. Tell me on the examples.

Let there be a number of details in the warehouse. If the warehouse is also brought to the warehouse, for example, 400 parts, the number of parts in the warehouse will increase, and the number 400 expresses this change in the amount in a positive side (upwards). If it is taken from the warehouse, for example, 100 parts, then the number of parts in the warehouse will decrease, and the number 100 will express the change in the amount in the negative side (up to the reduction). There will be no details on the warehouse, and they will not take part from the warehouse, then we can talk about the number of parts (that is, it can be about zero change in quantity).

In the examples given, the change in the number of parts can be described using integers 400, -100 and 0, respectively. A positive integer 400 shows a change in the number in a positive side (increase). A negative integer -100 expresses a change in quantity in the negative side (decrease). An integer 0 shows that the amount remains unchanged.

Ease of use of integers compared to the use of natural numbers is that it is not necessary to explicitly indicate the number of or decreases, - an integer determines the change in quantitatively, and the value of an integer indicates the direction of change.

The integers can also express not only the change in the quantity, but also a change in any value. We will deal with this on the example of a change in temperature.

Increased temperature, let's say, 4 degrees are expressed by a positive integer number 4. A decrease in temperature, for example, by 12 degrees can be described by a negative integer -12. And the invariance of temperature is its change, determined by an integer 0.

Separately, you need to say about the interpretation of negative integers as the amount of debt. For example, if we have 3 apples, then a positive number 3 shows the number of apples we own. On the other hand, if we need to give 5 apples to anyone, and we do not have them in stock, then this situation can be described using a negative integer -5. In this case, we "possess" -5 apples, a minus sign indicates a debt, and the number 5 determines the debt quantitatively.

Understanding a negative integer as debt allows, for example, justify the rule of addition of negative integers. Let us give an example. If someone has 2 apples to one person and one apple - another, then the total debt is 2 + 1 \u003d 3 apples, so -2 + (- 1) \u003d - 3.

Bibliography.

• Vilenkin N.Ya. and others. mathematics. Grade 6: Textbook for general educational institutions.

The number is the abstraction used for the quantitative characteristics of objects. The numbers arose still in primitive society due to the need of people to consider objects. Over time, as science develops, the number has become the most important mathematical concept.

To solve problems and evidence various theorems It is necessary to understand what kinds of numbers are. The main types of numbers include: natural numbers, integers, rational numbers, valid numbers.

Integers - These are the numbers received with the natural score of the items, and rather with their numbering ("first", "second", "third" ...). Many natural numbers are denoted by the Latin letter N. (You can remember, relying on the English word Natural). We can say that N. ={1,2,3,....}

Whole numbers - These are the numbers from the set (0, 1, -1, 2, -2, ....). This set consists of three parts - natural numbers, negative integers (opposite natural numbers) and the number 0 (zero). Integers are indicated by the Latin letter Z. . We can say that Z. ={1,2,3,....}.

Rational numbers - These are the numbers representable in the form of a fraction, where M is an integer, and N is a natural number. Latin letter is used to designate rational numbers Q. . All natural and integers are rational. Also as examples of rational numbers can be given: ,,,.

Valid (real) numbers - These are the numbers that are used to measure continuous values. The set of valid numbers is denoted by the Latin letter R. The actual numbers include rational numbers and irrational numbers. Irrational numbers are numbers that are obtained by performing various operations with rational numbers (for example, the root extraction, the calculation of logarithms), but are not rational. Examples of irrational numbers are ,,.

Any valid number can be displayed on a numeric direct:

For listed above sets of numbers, the following statement is fair:

That is, many natural numbers are included in many integers. Many integers are included in many rational numbers. And the set of rational numbers is included in many valid numbers. This statement can be illustrated using Euler's circles.