Rational numbers ℹ️ in mathematics, definition, properties, action on them, examples, how to prove that the number is rational

Rational numbers what is

Rational numbers can be discussed to infinity, finding new chips and tolerant errors in understanding.

In order to avoid problems with such numbers, it is worth considering some of these information about them. This will help assimilate the material and provide the necessary knowledge in mathematics.

What constitutes

To begin with, it should be understood what numbers are called rational. Those are considered fractions in the form of a numerator and denominator. Moreover, the latter should not be zero, since division on such a number is considered invalid.

The categories of numbers may be denoted by rational:

What numbers are called rational
  1. Whole numbers, whether positive or negative.
  2. Mathematical fractional expressions of different types.
  3. Combination of ordinary and fractional.
  4. Decimal fractions.
  5. Infinite periodic fractions.

All groups of indicated expressions are represented as a / b fraction. For example, the number 2 can be represented in the form of fractions 2/1, which makes it possible to attribute it to both the whole and rational.

Similarly, in the form of fractions, mixed and endless periodic fractions can be represented. Therefore, for such expressions, the designation is rational numbers.

On the coordinate direct

Previously, when studying negative numbers in school lessons, the concept of coordinate direct was introduced. There are many points on such a line. Especially difficult to solve the search for fractions and mixed indicators, as they Lying between integers in infinite quantities:

Rational number examples
  • For example, the fraction 0.5 is located between zero and unit. If you increase the interval of such a straight line, it is easy to see fractional from 0.1 to 0.9, it costs ½ in the middle. In the same way, mathematical fractions of the form 3/6, 4/8 and so on can be masked.
  • As for the fraction 3/2, it is located on an arithmetic line between unit and a twos. Between them in large numbers there are decimal fractions, including the desired. An increase in certain segments gives an idea that it still lies on the coordinate direct between the integer. As a result, expressions appeared after a semicolon one sign. And such values ​​a great set, including between fractional.
  • But it is possible to find the real place of the infinite periodic fraction only because it goes to infinity. You can find many illustrations of how close the fraction in real terms can be located.

Therefore, when considering what a rational number means on coordinate direct, it is important to know its appearance and is it possible to convert to another. Often it is necessary to find a separate property or illustrate the task using specific segments.

If worth minus

When schoolchildren passed the theme of multiplication and divisions, they became known: in the role of dividers and divisibles can act as negative and positive expressions.

What is rational numbers in mathematics

So, variations 6: -2 = -3 and -6: 2 = -3 have the same result, although the minus sign has different parts.

Because Each division can be represented as a fraction , minus is set in a numerator or in the denominator. Either make it common.

Between all three variations, you can put a sign of equality, since their result is the same number.

Each of the rational indicators has the opposite.

For example, for the fraction ½ is -1 and its variations. Both are equidistant to the beginning of the coordinates and are located in the middle.

Translation into fractions

Transfer of a mixed expression to the wrong fraction is carried out using multiplication by the denominator, the fractional part and add to the numerator. The resulting new fraction with the same denominator.

You can consider the algorithm on the next simple example:

Many rational numbers
  • There is 2.5, which should be translated into the wrong fraction.
  • The whole indicator must be multiplied by the channel of the fractional part and add the numerator of the same part.
  • The resulting value can be subtracted as (2 * 2) + 1 = 4 + 1 = 5.
  • 5 will be a numerator, and the denominator will be the same and will turn out 5/2.
  • Return the initial mixed can be highlighted as a whole part.

However, this method is not suitable for a negative value. If you use the former rule and allocate the whole part, then you can get a contradiction of the form: (-2 * 2) + ½ = -3 / 2, although it was necessary to get -5/2.

Therefore, you should define another method. The whole part is multiplied by the denominator of the fractional part. . From the resulting value, the numerator of the fractional part is subtracted. And then it turns out the correct answer.

Thanks to the coordinate direct, it can be understood why mixed -2,5 is located in the left side. Minus indicates a shift to the left in the number of two steps. The hit occurred at point -2. After that, the shift is still half a step and the middle between -3 and -2.

Comparison of numbers among themselves

From previous lessons it is easy to prove that the right to the right is the value, the more it is. And on the contrary, the more left of the situation suggests that the value under consideration is less than another indicator.

The value of which expression is a rational number

For such cases, when the comparison of the numbers is achieved simply, there is such a rule: out of 2 numbers with positive signs, which has more module. And for negative, it is, whose module is less. For example, there are numbers -4 and -2. When comparing modules, one can say that -4 less -2.

At the same time, newcomers often admit the following error : confused by the module and directly the number. After all, the module -3 and module -1 does not indicate that -3 is more -1, but on the contrary. This can be understood from the coordinate direct, where the first is left to the left of the second. If you want to compare the values, it is important to pay attention to the signs. Minus speaks of the negativity of the expression and vice versa.

Some examples

It is somewhat more complicated to relate to mixed numbers, the extraction of the root, fractional values. It will take to change the rules, since it is not always possible to depict them on the coordinate direct. In this regard, it is required to compare them in other ways than at school:

What does the rational number mean
  1. For example, there are two negative values, namely -3/5 and -7/3.
  2. First there are modules in the form of 3/5 and 7/3, which are positive.
  3. Then each is driven to a common denominator who protrudes 15.
  4. Based on the rule for negative values, rational -3/5 more -7/3, as its module is less.

It is easier to compare modules of integer parts, because you can quickly answer the question. It is known that whole parts are more important compared to the fractions. If you note the numbers 15.4 and 2,1212, then the whole part of the first number is more than the second, and therefore fraction.

The situation is somewhat more complicated with an example where there are values ​​of -3.4 and -3.7. Modules of integer numbers are the same, therefore will have to be compared for rational values. Then it turns out that -3.4 more is -3.7, since its module is less.

When comparing the simple and periodic fraction, the latter should be translated into the standard one. So, 0, (3) becomes 3/9. Comparing, translate the fractions to the total denominator 0, (3) and 4/8, it turns out 24/72 and 36/72. Naturally, 24/72 <36/72. That is, a module 4/8 larger module 0, (3), it means that it is considered large.

Rational numbers are an extensive topic. Their study is considered rather difficult, demanding to take into account many nuances and explanations of the main points, actions with arithmetic numbers and so on. Despite the seeming simplicity, the program for determining what numbers are rational and comparisons are compiled, taking into account the presence of fractional parts, signs after a comma and before expression.

It depends on the search for the correct answer and the solution of the overall task, including the search for interest and volumes.

Rational indicators may relate to assistants in the transition to complex sections in this course of mathematics and give an idea of ​​natural and decimal numerical expressions in general and in particular on unusual cases.

Everyone heard about rational numbers, but not everyone understands that they represent. In fact, everything is simple.

Source: Yandex.
Source: Yandex.

Rational number - This is the result of dividing two integers. For example, the number 2 is the result of dividing 4 and 2, and the number 0.2 is 2 divided by 10. Any rational number we can present for yourself in the form of a fraction M / N. where mis an integer n- Natural number.

What do rational numbers look like? It can be:

  • Fractions (1/2, 5/10)
  • Integers (1, 2, 5)
  • Mixed numbers
  • Decimal fractions (0.14, 4,1)
  • Endless periodic fractions (for example, when dividing 10 to 3, we get 3,33333 ...)

Q - Designation of a set of rational numbers.

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Properties of rational numbers

  • Each natural number is rational.
  • Each whole number is rational.
  • Rational numbers follow the rule Breathtaking and moving Properties. That is, from changes in places of terms of the sum value not to change.

a + b = b + a

(A + B) + C = A + (B + C)

a + 0 = a

a + (- a) = 0

Examples:

2 + 3 = 5 and 3 + 2 = 5, it means 2 + 3 = 3 + 2.

14+ (1 + 4) = 19 and (14 + 1) + 4 = 19, which means 14+ (1 + 4) = (14 + 1) +4

  • Also these laws are stored when multiplying.

a × b = b × a

a × (b × c) = (a × b) × c

a × 1 = a

A × 1 / a = 1

A × 0 = 0

A × B = 0

Examples:

3x4 = 12 and 4x3 = 12, it means 3x4 = 4x3

5x (2x3) = 30 and (5x2) x3 = 30, it means 5x (2x3) = (5x2) x3

  • For rational numbers, the distribution law of multiplication will be equitable.

(A + B) × C = AC + BC

(A - B) × C = AC - BC

Examples:

(4 + 7) x5 = 55 and 4x5 + 7x5 = 55, which means (4 + 7) x5 = 4x5 + 7x5

Irrational numbers and roots

In order to better understand what kind of rational numbers are, you should know what numbers are not. Or rather, what numbers will be irrational. Such numbers can not be written in the form of a simple fraction:

  • The number of pi, which is approximately 3.14. It can be represented as a fraction, but this value will be only approximate.
  • Some roots. For example, the root of 2 or from 99 cannot be written as a fraction
  • Golden section, which is approximately equal to 1.61. Here the situation is the same as with the number of Pi.
  • The number of Euler, which is approximately 2,718, is also not rational.
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Most irrational numbers are found among the roots, but not all irrational roots. For example, the root of number 4 is the number 2, and it can be represented as a fraction. That is, the root of among 4 is a rational number.

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What is rational numbers

January 14, 2021.

Hello, dear blog readers KtonanovenKogo.ru. Today we will talk about mathematical terms.

And this time we will tell all about rational numbers. They are necessarily entering the school program, and children begin to study them in grade 6.

The word "rational" is familiar to many. And under it implies something "logical" and "right". In fact, it is.

Rational numbers are ...

The term has a Latin roots, and translated "Ratio" means "number", "calculation", "reason", "reasoning" and "numbering". But there are other translations - "fraction" and "division".

Rational number - any number that can be shown in the form of fractions A / B . Here a is an integer, and B is natural.

It is worth reminding that:

  1. Whole numbers - These are all possible numbers as negative and positive. And it also applies zero. The main condition - they should not be fractional. That is, -15, 0 and +256 can be called integers, and 2.5 or -3.78 - no.
  2. Integers - These are the numbers that are used with the score, that is, they have "natural origin." This is a series of 1, 2, 3, 4, 5, and so on to infinity. But zero and negative numbers, as well as fractional - do not belong to natural.

And if you apply these definitions, then we can say that:

The rational number is generally all possible numbers except infinite non-periodic decimal fractions. Among them are natural and integers, ordinary and finite decimal fractions, as well as endless periodic fractions.

Scheme

History of study of rational numbers

It is not known when people began to study the fractions. There is an opinion that many thousand years ago. And all began with a banal division. For example, someone had to be divided, but it did not work on equal parts. But it turned out any other, and how much in the appendage.

Most likely, the fraction was studied in ancient Egypt, and in ancient Greece. The then mathematics far advanced in science. And it is difficult to assume that this topic remained them not studied. Although, unfortunately, none of the works were not found specific instructions on rational numbers.

Mathematician

But it is officially believed that the concept of decimal fraction appeared in Europe in 1585. This mathematical term in his writings perpetuated by a Dutch engineer and mathematician Simon Stevein.

Before science, he was an ordinary merchant. And most likely, it was in trading cases that often faced fractional numbers. What then described in his book "Tenth".

In it, Stevech not only explained the usefulness of decimal fractions, but also in every way promoted their use. For example, in a system of measures to accurately determine the value of something.

Varieties of rational numbers

We have already written that the concepts of rational numbers fall almost all possible options. Now consider the existing options in more detail:

  1. Integers . Any number from 1 and to infinity can be represented as a fraction. It is enough to remember the simple mathematical rule. If you divide the number per unit, then the same number will be. For example, 5 = 5/1, 27 = 27/1, 136 = 136/1 and so on.
  2. Whole numbers . Exactly the same logic, as in the case of natural numbers, acts here. Negative numbers can also be represented as a fraction with division per unit. And it will also be in relation to zero. For example, -356 = -356/1, -3 = -3/1, 0 = 0/1 and so on.
  3. Ordinary fractions . This directly refers to the definition of rational numbers. For example, 6/11, 2/5, -3/10 and so on.
  4. Infinite periodic fractions . These are the numbers that, after the comma, the infinite many signs and their sequence repeats. The simplest examples 1/3, 5/6 and so on.
  5. Finite decimal fractions . These are the numbers that can be recorded in two different options, and in which there are a very specific number of semicolons. The easiest example is half. It can be denoted by a shot 0.5 or fraction ½.

All numbers that are included in the concept of rational are called a multitude of rational numbers. In mathematics it is accepted to mark Latin letter Q. .

And graphically it can be portrayed like this:

Numbers

Properties of rational numbers

Rational numbers obey All the main laws of mathematics :

  1. A + B = B + A
  2. A + (B + C) = (a + c) + with
  3. A + 0 = a
  4. A + (-a) = 0
  5. A * B = V * A
  6. A * 1 = A
  7. A * 0 = 0
  8. (A + C) * C = A * C + V * C
  9. (A - c) * C = A * C - V * with

For the sake of interest, you can try to substitute any numbers instead of letters and make sure that these laws are true.

Instead of imprisonment

Once there are rational numbers in mathematics, it means that they should be opposite. So there are - they are called irrational . These are numbers that can not be written in the form of ordinary fraction.

These numbers belong to the mathematical constant "PI". Many know that it is equal to 3.14 and an infinite number of decimal signs, and their sequence is never repeated.

Irrational numbers

Also, the irrational numbers relates many roots. This applies to those who do not obtain an integer. The easiest example is the root of 2. But this is the topic for another article.

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The rational number is a number that can be represented as a fraction. Those. If the number can be obtained by dividing two integers (number without fractional part), then this is rational.

This is a number that can be submitted by an ordinary shot M / N., where the numerator M is an integer, and the denominator N is a natural number.

For instance:

  • 1,15 - a rational number of t. It can be represented as 115/100;
  • 0.5 - a rational number because it is 1/2;
  • 0 is a rational number because it is 0/1;
  • 3 - rational number because it is 3/1;
  • 1 - rational number because it is 1/1;
  • 0.33333 ... - rational number, because it is 1/3;
  • -5.4 - the rational number because it is -54/10 = -27/5.

Lots of rational numbers is indicated by the letter "Q" .

The word "rational" originated from Latin "Ratio", which has several values ​​- the number, calculation, numbering, reasoning, mind, etc.

Properties of rational numbers

Suppose A, B and C - any rational numbers.

Movement and combination laws

a + b = b + a, for example: 2 + 3 = 3 + 2;

a + (B + C) = (a + b) + C, for example: 2 + (3 + 4) = (2 + 3) + 4;

a + 0 = a, for example: 2 + 0 = 2;

a + (- a) = 0, for example: 2 + (- 2) = 0

Movement and combination laws when multiplying

a × b = b × a, for example: 2 × 3 = 3 × 2

A × (b × c) = (a × b) × c, for example: 2 × (3 × 4) = (2 × 3) × 4

A × 1 = a, for example: 2 × 1 = 2

a × 1 / a = 1, if a ≠ 0; For example: 2 × 1/2 = 1

a × 0 = 0, for example: 2 × 0 = 0

a × b = 0, it means: or a = 0, or b = 0, or both are zero

Distribution law multiplication

For addition:

(and +b) × s = a с + bсFor example: (2 + 3) × 4 = 2 × 4 + 3 × 4

For subtraction:

(and b) × с = A. с bсFor example: (3 - 2) × 4 = 3 × 4 - 2 × 4

Irrational numbers

Irrational numbers - the opposite of rational numbers, these are those that cannot be written as a simple fraction.

For instance:

  • the number pi = 3,14159 ... it can be written as 22/7, but it will be only about и far from certain 22/7 = 3,142857 ..);
  • √2 and √99 - irrational, since they are impossible to record a fraction (the roots are often irrational, but not always);
  • e (number) = 2.72 - irrational, since it is impossible to record a fraction;
  • The gold cross section φ = 1.618 ... - irrational, since it is impossible to record a fraction.

Lots of irrational numbers is indicated by the letter "I" .

What is the difference between integer, natural and rational numbers

The integers are natural numbers opposite to them numbers (below zero) and zero.

For instance:

All integers are rational Numbers (natural including), because they can be represented as an ordinary fraction.

Lots of integers in mathematics is indicated by the letter Z.

Integers

Natural numbers are only integers starting from 1.

For instance:

This account appeared in a natural way when people still thought on the fingers and did not know the numbers ("I have so many goats, how many fingers on both hands"), so zero is not included in natural numbers.

Lots of Natural numbers in mathematics is indicated by the letter N.

All decimal fractions are rational numbers?

Decimal fractions look like:

These are the usual fractions that the denominator is equal to 10, 100, 1000, etc. Our examples we can write in this form:

3,4 =. 3,4.;

2,19 =. 2,19 ;

0.561 =. 0,561.

This means that any Finite The decimal fraction is a rational number.

Anyone Periodic fraction You can also submit in the form of an ordinary fraction:

(3 repeats)
(3 repeats)

Consequently, any periodic fraction is a rational number.

But endless and non-periodic decimal fractions are not considered rational numbers, since they cannot be shown in the form of an ordinary fraction.

Can remember how the crib is that the number P. (3,14159 ...) irrational . He has a lot of non-refining marks after the comma and it is impossible to imagine in the form of an ordinary fraction.

Roots - rational numbers or irrational?

The overwhelming part of square and cubic roots is irrational numbers. But there are exceptions: if it can be represented as a fraction (by definition of a rational number). For instance:

  • √2 = 1,414214 ... - irrational;
  • √3 = 1.732050 ... - irrational;
  • ∛7 = 1,912931 ... - irrational;
  • √4 = 2 - rational (2 = 2/1);
  • √9 = 3 - rational (3 = 3/1).

The history of rational numbers and fractions

The earliest known mention of irrational numbers was between 800 and 500 BC. e. In Indian Sulba Sutra.

The first proof of the existence of irrational numbers belongs to the ancient Greek philosopher Pythagorean Hippas from the metapont. He proved (most likely geometrically) the irrationality of the square root of 2.

The legend states that Hippas from Metapont opened irrational numbers when he tried to present a square root of 2 in the form of a fraction. However, Pythagoras believed in the absolute number and could not accept the existence of irrational numbers.

It is believed that because of this, there was a conflict between them, which spawned a lot of legends. Many say that this discovery was killed by Hippas.

In the Babylonian records in mathematics, it is often possible to see a six-month number system in which the fractions have already been used. These records were made more than 4,000 years ago, the system was a bit different, as we, but the point is the same.

Egyptians who lived in a later period also had their own way of writing fractions, something similar to: 3⁻⁻ or 5⁻⁻.

Learn more about natural numbers, the number Pi, the numbers of Fibonacci and the Exhibitor.

Determination of rational numbers

Rational number - This is a number that can be represented as a positive or negative ordinary fraction or number of zero. If the number can be obtained by dividing two integers, then this is a rational number.

Rational numbers are those that can be represented as

Type of rational numbers

where the numerator M is an integer, and the denominator N is a natural number.

Rational numbers - These are all natural, integers, ordinary fractions, endless periodic fractions and final decimal fractions.

Many rational numbers It is customary to mark the Latin letter Q.

Examples of rational numbers:

  • Decimal fraction 1.15 is 115/100;
  • decimal fraction 0.2 is 1/2;
  • An integer 0 is 0/1;
  • An integer 6 is 6/1;
  • An integer 1 is 1/1;
  • Infinite periodic fraction 0,33333 ... is 1/3;
  • Mixed number Mixed number- it's 25/10;
  • Negative decimal fraction -3.16 is -316/100.

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Record the child to a free trial lesson: introduce a platform, solve a couple of tasks in an interactive format and make a program of learning.

Properties of rational numbers

Rational numbers have certain laws and a number of properties - consider each of them. Let a, b and c be any rational numbers.

The main properties of action with rational numbers
  • Moving property of addition: A + B = B + a.
  • The combination property of addition: (A + B) + C = A + (B + C).
  • The addition of a rational number and neutral element (zero) does not change this number: a + 0 = a.
  • Each rational number has an opposite number, and their sum is always zero: a + (-a) = 0.
  • Multiplication Movement: AB = BA.
  • The combination property of multiplication: (A * B) * C = A * (B * C).
  • The product of a rational number and one does not change this number: A * 1 = a.
  • Each different rational number has a reverse number. Their product is equal to one: a * a - 1 = 1.
  • The distribution property of multiplication relative to addition: A * (B + C) = A * B + A * C.

In addition to the main listed, there are still a number of properties:

 
  1. The rule of multiplication of rational numbers with different signs: (-a) * b = -ab. Such a phrase will help remember: "Plus there is a minus for a minus, and there is a minus minus."
  2. The rule of multiplication of negative rational numbers: (-a) * (-b) = AB. Remember the phrase will help: "Minus for minus there is a plus."
  3. The rule of multiplying an arbitrary rational number to zero: a * 0 = 0 or 0 * a = 0. We prove this property. We know that 0 = d + (-d) for any rational D, which means A * 0 = A * (D + (-D)). The distribution law allows you to rewrite the expression: a * d + a * (-d), and since A * (-d) = -ad, then A * D + A * (-D) = A * D + (-AD) . This turned out the sum of two opposite numbers, which as a result gives zero, which proves the equality A * 0 = 0.

We listed only the properties of addition and multiplication. On the set of rational numbers, subtraction and division can be recorded as referring to addition and multiplication. That is, the difference (A - B) can be written as the sum of A + (-B), and the private A / B is equal to the product A * B-1, with B ≠ 0.

Definition of the irrational number

Irrational number - This is a valid number that cannot be expressed in the form of dividing two integers, that is, in a rational fraction

rational fraction

It can be expressed in the form of an infinite non-periodic decimal fraction.

Endless periodic decimal fraction - This is such a fraction, the decimal signs of which are repeated in the form of a group of numbers or one and the same number.

Examples:

  • π = 3,1415926 ...
  • √2 = 1,41421356 ...
  • E = 2,71828182 ...
  • √8 = 2.828427 ...
  • -√11 = -3.31662 ...

Designation of the set of irrational numbers: Latin letter I.

Valid or real numbers - These are all rational and irrational numbers: positive, negative and zero.

Properties of irrational numbers:

  • The result of the sum of the irrational number and rational is equal to the irrational number;
  • The result of the multiplication of the irrational number on any rational number (≠ 0) is equal to the irrational number;
  • The result of subtraction of two irrational numbers is equal to an irrational number or rational;
  • The result of the sum or the product of two irrational numbers is rational or irrational, for example: √2 * √8 = √16 = 4).

The difference between integers, natural and rational numbers

Integers - These are the numbers that we use to calculate something specific, tangible: one banana, two notebooks, ten chairs.

But what is exactly not a natural number:

  • Zero is an integer that when adding or subtracting with any numbers as a result will give the same number. Multiplication on zero gives zero.
  • Negative numbers: -1, -2, -3, -4.
  • Drobi: 1/2, 3/4, 5/6.

Whole numbers - These are natural numbers opposite to them and zero.

If two numbers differ from each other - they are called opposite: +2 and -2, +7 and -7. The plus sign is usually not written, and if there is no sign before the number, it means that it is positive. The numbers facing the "minus" sign are called negative.

What numbers are called rational we already know from the first part of the article. Repeat again.

Rational numbers - These are finite fractions and endless periodic fractions.

For instance: An example of rational numbers

Any rational number can be represented in the form of a fraction, in which the numerator belongs to the integers, and the denominator is natural. Therefore, in many rational numbers include many integers and natural numbers.

Many rational numbers

But not all numbers can be called rational. For example, infinite non-periodic fractions do not belong to a set of rational numbers. So √3 or π (PI number) cannot be called rational numbers.

So figured out! And if not quite - come to exciting mathematics lessons at the Skysmart online school. No boring textbooks: the child is waiting for interactive classes, mathematical comics and teachers who will never leave in trouble.

Rational numbers you are already familiar with them, it remains only to summarize and formulate the rules. So what numbers are called rational numbers? Consider in detail in this topic lesson.

The concept of rational numbers.

Definition: Rational numbers - These are the numbers that can be represented as fraction \ (\ frac {m} {n} \), where M is an integer, and N is a natural number.

In other words, you can say:

Rational numbers - These are all natural numbers, integers, ordinary fractions, endless periodic fractions and finite decimal fractions.

We will analyze every item in detail.

  1. Any natural number can be represented as a fraction, for example, the number 5 = \ (\ FRAC {5} {1} \).
  2. Any integer can be represented as a fraction, for example, numbers 4, 0 and -2. We obtain 4 = \ (\ frac {4} {1} \), 0 = \ (\ FRAC {0} {1} \) and -2 = \ (\ FRAC {-2} {1} \).
  3. Ordinary fractions are already recorded in rational form, for example, \ (\ FRAC {6} {11} \) and \ (\ FRAC {9} {2} \).
  4. Infinite periodic fractions, for example, 0.8 (3) = \ (\ FRAC {5} {6} \).
  5. Finite decimal fractions, for example, 0.5 = \ (\ FRAC {5} {10} = \ FRAC {1} {2} \).

Many rational numbers.

Recall that the set of natural numbers is denoted by the Latin letter of N. Specification of integers is indicated by the Latin letter Z.A. The set of rational numbers is indicated by the Latin letter Q.

In many rational numbers, many integers and natural numbers include the meaning of rational numbers.

In the figure you can show a variety of rational numbers.

Many rational numbers

But not all numbers are rational. There are still many different numbers, which in the future you will study. The reflective unreasonal fractions do not belong to the set of rational numbers. For example, the number e, \ (\ sqrt {3} \) or the number \ (\ pi \) (the number Pi is read) are rational numbers.

Questions on the topic "Rational numbers": What expression is a rational number from numbers \ (\ sqrt {5}, -0. (3), 15, \ FRAC {34} {1569}, \ sqrt {6} \)? Answer: The root of 5 this expression can not be submitted in the form of course a fraction or infinite periodic fraction, therefore this number is not rational In the form of a fraction, therefore it is a rational number. The number 15 can be represented as a fraction \ (\ FRAC {15} {1} \), therefore it is a rational number. These \ (\ FRAC {34} {1569} \) is a rational number . Anti-6 This expression cannot be submitted in the form of course a fraction or infinite periodic fraction, so this number is not rational.

Write a number 1 as a rational number? Answer: To write down as a rational number 1, it is necessary to present it in the form of fraction 1 = \ (\ FRAC {1} {1} \).

Prove that the number \ (\ sqrt {0.0049} \) is rational? Evidence: \ (\ SQRT {0,0049} = 0.07 \)

Is a simple number under the root of a rational number? Answer: No. For example, any simple number under the root 2, 3, 5, 7, 11, 13, ... not taken out of the root and cannot be represented in the form of course the fraction or infinite periodic fraction, therefore is not a rational number.

The topic of rational numbers is quite extensive. You can talk about it infinitely and writing whole works, every time surprised by new chips.

In order to avoid mistakes in the future, in this lesson we will be a little deeper in the theme of rational numbers, I draw the necessary information from it and move on.

What is a rational number

Rational number is a number that can be represented as a fraction A divided by bwhere a - This is a fraction numerator, b- denominator of the fraci. Moreover bIt should not be zero because the division is not allowed.

The following categories of numbers include rational numbers:

  • integers (for example -2, -1, 0 1, 2, etc.)
  • Ordinary fractions (for example a halfone thirdthree-quartersetc.)
  • Mixed numbers (for example two integers one secondone whole two thirdminus two integer one thirdetc.)
  • decimal fractions (for example 0.2, etc.)
  • Infinite periodic fractions (for example 0, (3), etc.)

Each number of this category may be represented as a fraction A divided by b .

Examples:

Example 1. An integer 2 can be represented as a fraction The first two. So the number 2 refers not only to integer numbers, but also to rational.

Example 2. Mixed number two integers one secondcan be represented as a fraction Five second. This fraction is obtained by the transfer of a mixed number to the wrong fraction

Translation of two integer one second to the wrong fraction

So mixed number two integers one secondrefers to rational numbers.

Example 3. Decimal fraction 0,2 can be represented as a fraction Two tenths. This fraction turned out by the transfer of decimal fraction 0.2 to an ordinary fraction. If you are having difficulty at this moment, repeat the topic of decimal fractions.

Since the decimal fraction 0.2 can be represented as a fraction Two tenthsIt means that it also refers to rational numbers.

Example 4. Infinite periodic fraction 0, (3) can be represented as a fraction Three ninths. This fraction is obtained by transferring a clean periodic fraction in an ordinary fraction. If you are having difficulty at this moment, repeat the subject of periodic fractions.

Since the endless periodic fraction 0, (3) can be represented as a fraction Three ninthsIt means that it also refers to rational numbers.

In the future, all the numbers that can be represented in the form of a fraction, we will increasingly be called in one phrase - rational numbers .

Rational numbers on the coordinate direct

The coordinate direct we considered when the negative numbers were studied. Recall that this is a straight line on which there are many numbers. As follows:

Coordinate Direct Figure 1

This figure shows a small fragment of the coordinate direct from -5 to 5.

Mark on the coordinate direct integers of the species 2, 0, -3 is not difficult.

It is much more interesting things with the rest of the numbers: with ordinary fractions, mixed numbers, decimal fractions, etc. These numbers lie between the integers and these numbers are infinitely a lot.

For example, we note on the coordinate direct rational number a half. This number is located exactly between zero and unit

One second on the coordinate direct

Let's try to understand why the fraction a halfSuddenly settled between zero and unit.

As mentioned above, there are other numbers between integers - ordinary fractions, decimal fractions, mixed numbers, etc. For example, if you increase the section in the coordinate line from 0 to 1, then you can see the following picture

Coordinate straight from zero to one

It can be seen that there are already other rational numbers between the integers 0 and 1, which are familiar to decimal fractions for us. Our fraction is visible here a halfwhich is located there, where and the decimal fraction is 0.5. Attentive consideration of this picture gives the answer to the question of why the fraction a halfIt is located there.

Fraction a halfmeans divided 1 to 2. And if divided 1 to 2, then we get 0.5

Unit divided into two fifth

The decimal fraction 0.5 can be masked and under the other fractions. From the main property of the fraction, we know that if the numerator and denomoter of the fraci multiply or split into the same number, then the fraction value will not change.

If the numerator and denominator a halfmultiply by any number, for example, by number 4, then we will get a new fraction Four eighths, and this fraction as well as a halfequal to 0.5

Four divided for eight equals zero as many as five tenths

And therefore on the coordinate shot Four eighthscan be located in the same place where the fraction was located a half

Four eighth on the coordinate direct

Example 2. Let's try to note on the coordinate rational number Three seconds. This number is located exactly between numbers 1 and 2

three second on the coordinate direct

The value of the fraci Three secondsEqual 1.5

Three divided into two will be one whole five tenths

If you increase the area of ​​the coordinate direct from 1 to 2, then we will see the following picture:

coordinate direct from one to two

It can be seen that there are already other rational numbers between integers 1 and 2, which are familiar to the decimal fractions for us. Our fraction is visible here Three secondswhich is located there, where and the decimal fraction 1.5.

We increased certain segments on the coordinate direct to see the other numbers lying on this segment. As a result, we found decimal fractions that had one digit after a comma.

But these were not the only numbers lying on these segments. The numbers lying on the coordinate direct is infinitely a lot.

It is not difficult to guess that there are already other decimal fractions between decimal fractions having a decimal fraction, having two digits after a comma. In other words, hundredth parts of the segment.

For example, let's try to see the numbers that lie between decimal fractions 0.1 and 0.2

Coordinate straight from zero to one tenth to two tenths

Another example. Decimal fractions having two digits after a comma and lying between zero and a rational number of 0.1 look like this:

coordinate straight from zero to zero one tenth

Example 3. Note on the coordinate direct rational number One fiftieth. This rational number will be very close to zero

one fiftieth on the coordinate direct

The value of the fraci One fiftiethEqual 0.02

Unit separated by fifty equals zero as many as two hundredths

If we increase the segment from 0 to 0.1, then we will see where the rational number is accurate. One fiftieth

One fiftieth on a coordinate direct from 0 to 0.1

It can be seen that our rational number One fiftiethIt is located there, where and the decimal fraction is 0.02.

Example 4. Note on the coordinate direct rational number 0, (3)

The rational number 0, (3) is an infinite periodic fraction. His fractional part never ends, she is infinite

0,33333 .... And so on to infinity ..

And since in numbers 0, (3) the fractional part is infinite, this means that we will not be able to find the exact place on the coordinate direct, where this number is located. We can only specify this place approximately.

The rational number is 0.33333 ... will be very close to the usual decimal fraction 0.3

zero whole and three in the period on the coordinate direct

This drawing does not show the exact location of the number 0, (3). This is only an illustration showing how the periodic fraction 0, (3) can be placed closely to a conventional decimal fraction 0.3.

Example 5. Note on the coordinate direct rational number two integers one second. This rational number will be located in the middle between numbers 2 and 3

Two whole and one second on the coordinate direct

two integers one secondit is 2 (two integers) and a half(a half). Fraction a halfdifferently also called "half". Therefore, we noted on the coordinate direct two whole segments and another half of the segment.

If you translate a mixed number two integers one secondIn the wrong fraction, then we get an ordinary fraction Five second. This fraction on the coordinate direct will be located there, where and the fraction two integers one second

Five second on the coordinate direct

The value of the fraci Five secondEqually 2.5

Five divided into two will be one whole five tenths

If you increase the area of ​​the coordinate straight line from 2 to 3, then we will see the following picture:

Five second on the coordinate direct from two to three

It can be seen that our rational number Five secondLocated there, where and the decimal fraction 2.5

Minus before a rational number

In the previous lesson, which was called multiplication and division of integers, we learned to share integers. The role of a divide and divider could stand both positive and negative numbers.

Consider the simplest expression

(-6): 2 = -3

In this expression, divisible (-6) is a negative number.

Now consider the second expression

6: (-2) = -3

Here, a negative number is a divider (-2). But in both cases we get the same answer -3.

Considering that any division can be written in the form of a fraction, we can also review the examples also written in the form of a fraction:

minus six divided into two equals minus three

six divided into minus two equals minus three

And since in both cases the fraction value is the same, minus standing either in a numerator either in the denominator can be made with a general, putting it before the fraction

minus six divided into two or minus six seconds equal to minus three

six divided into minus two or minus six seconds equal to minus three

Therefore, between expressions minus six divided into two    и six divided into minus two    и  Minus six secondYou can put a sign of equality because they carry the same meaning

minus six divided into two equals six divided into minus two equals minus six second

In the future, working with fractions if the minus will meet us in a numerator or in the denominator, we will make this minus common, putting it before the fraud.

Opposite rational numbers

As well as an integer, the rational number has its opposite number.

For example, for a rational number a halfThe opposite number is Minus one second. It is located on the coordinate direct symmetrical location. a halfrelative to the start of coordinates. In other words, both of these numbers are equidistant from the beginning of the coordinates.

minus one second and one second on the coordinate direct

Translation of mixed numbers in incorrect fractions

We know that in order to translate a mixed number in the wrong fraction, you need to multiply the denominator of the fractional part and add to the fractional part. The resulting number will be the numerator of the new fraction, and the denominator remains the same ..

For example, we translate the mixed number two integers one secondIn the wrong shot

Multiply a whole part to the denominator of the fractional part and add a fractional part number:

(2 × 2) + 1

Calculate this expression:

(2 × 2) + 1 = 4 + 1 = 5

The resulting number 5 will be the numerator of a new fraction, and the denominator will remain the same:

Five second

The fully given procedure is written as follows:

Translation of two integer one second to the wrong fraction

To return the original mixed number, it is enough to highlight the whole part in the fraction Five second

Allocation of the whole part in the fraction five second

But this method of translating the mixed number to the wrong fraction is applicable only if the mixed number is positive. For a negative number, this method will not work.

Consider a fraction Minus five second. We highlight in this fraction a whole part. Receive minus two integer one second

allocation of the whole part in the crushed minus five second

To return the initial fraction Minus five secondneed to translate a mixed number minus two integer one secondIn the wrong fraction. But if we use the old rule, namely, we will multiply the integer on the denominator of the fractional part and to add the number of the fractional part to the resulting number, we will obtain the following contradiction:

translation minus two integer one second to the wrong fraction

We received a fraction Minus three seconds, and had to get a fraction Minus five second .

We conclude that mixed number minus two integer one secondIn the wrong fraction translated incorrectly:

minus two integer one second

To properly translate a negative mixed number in the wrong fraction, you need to multiply by the denominator of the fractional part, and from the resulting number subtract Sliver fractional part. In this case, we will all fall into place

The correct translation of the minus of two integer one second to the wrong fraction

Negative mixed number minus two integer one secondis the opposite for a mixed number two integers one second. If a positive mixed number two integers one secondlocated on the right side and looks like

Two whole and one second on the coordinate direct

then negative mixed number minus two integer one secondwill be located on the left side of symmetrically two integers one secondThe relative start of the coordinates

Minus two integer one second and two whole and one second on the coordinate direct

And if two integers one secondread as "two whole and one second", then minus two integer one secondReading as "Minus two whole and minus one second" . Since numbers -2 and Minus one secondLocked on the left side of the coordinate direct - they are both negative.

Any mixed number can be written in deployment. Positive mixed number two integers one secondIn the deployment, written as Two plus one second.

A negative mixed number minus two integer one secondrecorded as minus two whole minus one second

Now we can understand why a mixed number minus two integer one secondIt is located on the left side of the coordinate direct. Minus before two indicates that we moved from zero for two steps left, as a result, turned out to be at the point where the number -2 is

minus two on the coordinate direct

Then, starting from the number -2, they moved to the left Minus one secondStep. And since the value Minus one secondEqually -0.5, then our step will be half from the full step.

minus two and minus one second on the coordinate direct

As a result, we will find me in the middle between numbers -3 and -2

minus two integers and minus one second on the coordinate direct

Example 2. Allocate in incorrect fraction minus twenty seven fifthsWhole part, then the resulting mixed number back to transfer to the wrong fraction

We will execute the first part of the task, namely, we allocate in the wrong fraction minus twenty seven fifthsWhole part

Allocation of the whole part in the crushed minus twenty seven fifth

We will execute the second part of the task, namely I translate the resulting mixed number minus five whole two fifthsIn the wrong fraction. For this, multiply the whole part to the denominator of the fractional part and from the resulting number, the fractional part number will be subtracted:

Transfer minus five integer two fifths in the wrong fraction

If there is no desire to be confused and get used to the new rule, then you can make a mixed number in brackets, and minus leave behind the bracket. Then it will be possible to apply an old good rule: multiply a whole part to the denominator of the fractional part and to add a fractional part number to the resulting number.

Perform the previous task in this way, namely I translate the mixed number minus five whole two fifthsIn the wrong shot

Translation minus five integer two fifths in the wrong fraction solution with brackets

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