Proper fraction. Fraction - what is it? Types of fractions

Proper fraction

Quarters

Orderliness. a And b there is a rule that allows you to uniquely identify one and only one of three relationships between them: “< », « >" or " = ". This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relation as two integers and ; two non-positive numbers a And b are related by the same relationship as two non-negative numbers and ; if suddenly a non-negative, but b- negative, then a > b.
src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">
Adding Fractions Addition operation. a And b For any rational numbers there is a so-called summation rule c summation rule. At the same time, the number itself called amount a And b numbers and is denoted by , and the process of finding such a number is called summation .
. The summation rule has the following form: Addition operation. a And b For any rational numbers Multiplication operation. multiplication rule summation rule c summation rule. At the same time, the number itself , which assigns them some rational number amount a And b work and is denoted by , and the process of finding such a number is also called multiplication .
. The multiplication rule looks like this: Transitivity of the order relation. a , b And summation rule For any triple of rational numbers a If b And b If summation rule less a If summation rule, That a, and if b And b, and if summation rule less a, and if summation rule equals
. 6435">Commutativity of addition. Changing the places of rational terms does not change the sum.
Associativity of addition. The order in which three rational numbers are added does not affect the result.
Presence of zero. There is a rational number 0 that preserves every other rational number when added.
The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
Commutativity of multiplication. Changing the places of rational factors does not change the product.
Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
Availability of unit. There is a rational number 1 that preserves every other rational number when multiplied.
Presence of reciprocal numbers. Any rational number has an inverse rational number, which when multiplied by gives 1.
Distributivity of multiplication relative to addition. To the left and right parts rational inequality you can add the same rational number.
/pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0"> Axiom of Archimedes. a Whatever the rational number a, you can take so many units that their sum exceeds

.

src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0"> Additional properties All other properties inherent in rational numbers are not distinguished as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proven based on the given basic properties or directly by the definition of some mathematical object. Such

additional properties

so many. It makes sense to list only a few of them here.

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Countability of a set

Numbering of rational numbers To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it is enough to give an algorithm that enumerates rational numbers, i.e., establishes a bijection between the sets of rational and natural numbers. The simplest of these algorithms looks like this. An endless table of ordinary fractions is compiled, on each i-th line in each To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it is enough to give an algorithm that enumerates rational numbers, i.e., establishes a bijection between the sets of rational and natural numbers. j i the th column of which the fraction is located. For definiteness, it is assumed that the rows and columns of this table are numbered starting from one. Table cells are denoted by , where

- the number of the table row in which the cell is located, and

- column number.

The resulting table is traversed using a “snake” according to the following formal algorithm. These rules are searched from top to bottom and the next position is selected based on the first match. In the process of such a traversal, each new rational number is associated with another

Following this algorithm, we can enumerate all positive rational numbers. This means that the set of positive rational numbers is countable. It is easy to establish a bijection between the sets of positive and negative rational numbers by simply assigning to each rational number its opposite. That. the set of negative rational numbers is also countable. Their union is also countable by the property of countable sets. The set of rational numbers is also countable as the union of a countable set with a finite one.

The statement about the countability of the set of rational numbers may cause some confusion, since at first glance it seems that it is much more extensive than the set of natural numbers. In fact, this is not so and there are enough natural numbers to enumerate all rational ones.

Lack of rational numbers

The hypotenuse of such a triangle cannot be expressed by any rational number

Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates the misleading impression that rational numbers can be used to measure any geometric distances. It is easy to show that this is not true.

From the Pythagorean theorem we know that the hypotenuse of a right triangle is expressed as the square root of the sum of the squares of its legs. That. the length of the hypotenuse of an isosceles right triangle with a unit leg is equal to , i.e., the number whose square is 2.

If we assume that a number can be represented by some rational number, then there is such an integer m and such a natural number n, that , and the fraction is irreducible, i.e. numbers m And n- mutually simple.

If , then , i.e. m 2 = 2n 2. Therefore, the number m 2 is even, but the product of two odd numbers is odd, which means that the number itself m also even. So there is a natural number k, such that the number m can be represented in the form m = 2k. Number square m In this sense m 2 = 4k 2, but on the other hand m 2 = 2n 2 means 4 k 2 = 2n 2, or n 2 = 2k 2. As shown earlier for the number m, this means that the number n- even as m. But then they are not relatively prime, since both are bisected. The resulting contradiction proves that it is not a rational number.

Simple mathematical rules and techniques, if they are not used constantly, are forgotten most quickly. Terms disappear from memory even faster.

One of these simple actions– transformation is not proper fraction into the correct one or, in other words, mixed.

Improper fraction

An improper fraction is one in which the numerator (the number above the line) is greater than or equal to the denominator (the number below the line). This fraction is obtained by adding fractions or multiplying a fraction by a whole number. According to the rules of mathematics, such a fraction must be converted into a proper one.

Proper fraction

It is logical to assume that all other fractions are called proper. A strict definition is that a fraction whose numerator is less than its denominator is called proper. A fraction that has an integer part is sometimes called a mixed fraction.

Converting an improper fraction to a proper fraction

First case: the numerator and denominator are equal to each other. The result of converting any such fraction is one. It doesn't matter if it's three-thirds or one hundred and twenty-five one hundred and twenty-fifths. Essentially, such a fraction denotes the action of dividing a number by itself.

Second case: the numerator is greater than the denominator. Here you need to remember the method of dividing numbers with a remainder.
To do this, you need to find the number closest to the numerator value, which is divisible by the denominator without a remainder. For example, you have the fraction nineteen thirds. The closest number that can be divided by three is eighteen. That's six. Now subtract the resulting number from the numerator. We get one. This is the remainder. Write down the result of the conversion: six whole and one third.

But before reducing the fraction to the right kind, you need to check whether it can be shortened.
You can reduce a fraction if the numerator and denominator have a common factor. That is, a number by which both are divisible without a remainder. If there are several such divisors, you need to find the largest one.
For example, all even numbers have such a common divisor - two. And the fraction sixteen-twelfths has one more common divisor - four. This is the greatest divisor. Divide the numerator and denominator by four. Result of reduction: four thirds. Now, as a practice, convert this fraction to a proper one.

Fraction in mathematics, a number consisting of one or more parts (fractions) of a unit. Fractions are part of the field of rational numbers. Based on the way they are written, fractions are divided into 2 formats: ordinary type and decimal .

Numerator of fraction- a number showing the number of shares taken (located at the top of the fraction - above the line). Fraction denominator- a number showing how many shares the unit is divided into (located below the line - at the bottom). , in turn, are divided into: correct And incorrect, mixed And composite are closely related to units of measurement. 1 meter contains 100 cm. Which means that 1 m is divided into 100 equal parts. Thus, 1 cm = 1/100 m (one centimeter is equal to one hundredth of a meter).

or 3/5 (three fifths), here 3 is the numerator, 5 is the denominator. If the numerator is less than the denominator, then the fraction is less than one and is called correct:

If the numerator is equal to the denominator, the fraction is equal to one. If the numerator is greater than the denominator, the fraction is greater than one. In both last cases the fraction is called wrong:

To isolate the largest whole number contained in an improper fraction, you divide the numerator by the denominator. If the division is performed without a remainder, then the improper fraction taken is equal to the quotient:

If division is performed with a remainder, then the (incomplete) quotient gives the desired integer, and the remainder becomes the numerator of the fractional part; the denominator of the fractional part remains the same.

A number containing an integer and a fractional part is called mixed. Fraction mixed number maybe improper fraction. Then you can select the largest integer from the fractional part and represent the mixed number in such a way that fraction became a proper fraction (or disappeared altogether).

Common fractions are divided into \textit (proper) and \textit (improper) fractions. This division is based on a comparison of the numerator and denominator.

Proper fractions

Proper fraction An ordinary fraction $\frac(m)(n)$ is called, in which the numerator is less than the denominator, i.e. $m

Example 1

For example, the fractions $\frac(1)(3)$, $\frac(9)(123)$, $\frac(77)(78)$, $\frac(378567)(456298)$ are correct, so how in each of them the numerator is less than the denominator, which meets the definition of a proper fraction.

There is a definition of a proper fraction, which is based on comparing the fraction with one.

correct, if it is less than one:

Example 2

For example, the common fraction $\frac(6)(13)$ is proper because condition $\frac(6)(13) is satisfied

Improper fractions

Improper fraction An ordinary fraction $\frac(m)(n)$ is called, in which the numerator is greater than or equal to the denominator, i.e. $m\ge n$.

Example 3

For example, the fractions $\frac(5)(5)$, $\frac(24)(3)$, $\frac(567)(113)$, $\frac(100001)(100000)$ are irregular, so how in each of them the numerator is greater than or equal to the denominator, which meets the definition of an improper fraction.

Let's give a definition of an improper fraction, which is based on its comparison with one.

The common fraction $\frac(m)(n)$ is wrong, if it is equal to or greater than one:

\[\frac(m)(n)\ge 1\]

Example 4

For example, the common fraction $\frac(21)(4)$ is improper because the condition $\frac(21)(4) >1$ is satisfied;

the common fraction $\frac(8)(8)$ is improper because the condition $\frac(8)(8)=1$ is satisfied.

Let's take a closer look at the concept of an improper fraction.

Let's take the improper fraction $\frac(7)(7)$ as an example. The meaning of this fraction is to take seven shares of an object, which is divided into seven equal shares. Thus, from the seven shares that are available, the entire object can be composed. Those. the improper fraction $\frac(7)(7)$ describes the whole object and $\frac(7)(7)=1$. So, improper fractions, in which the numerator is equal to the denominator, describe one whole object and such a fraction can be replaced by the natural number $1$.

$\frac(5)(2)$ -- it is quite obvious that from these five second parts you can make up $2$ whole objects (one whole object will be made up of $2$ parts, and to compose two whole objects you need $2+2=4$ shares) and one second share remains. That is, the improper fraction $\frac(5)(2)$ describes $2$ of an object and $\frac(1)(2)$ the share of this object.

$\frac(21)(7)$ -- from twenty-one-sevenths parts you can make $3$ whole objects ($3$ objects with $7$ shares in each). Those. the fraction $\frac(21)(7)$ describes $3$ whole objects.

From the examples considered, we can draw the following conclusion: an improper fraction can be replaced by a natural number if the numerator is divisible by the denominator (for example, $\frac(7)(7)=1$ and $\frac(21)(7)=3$) , or the sum of a natural number and a proper fraction, if the numerator is not completely divisible by the denominator (for example, $\ \frac(5)(2)=2+\frac(1)(2)$). That's why such fractions are called wrong.

Definition 1

The process of representing an improper fraction as the sum of a natural number and a proper fraction (for example, $\frac(5)(2)=2+\frac(1)(2)$) is called separating the whole part from an improper fraction.

When working with improper fractions, there is a close connection between them and mixed numbers.

An improper fraction is often written as a mixed number - a number that consists of a whole number and a fraction part.

To write an improper fraction as a mixed number, you must divide the numerator by the denominator with a remainder. The quotient will be the integer part of the mixed number, the remainder will be the numerator of the fractional part, and the divisor will be the denominator of the fractional part.

Example 5

Write the improper fraction $\frac(37)(12)$ as a mixed number.

Solution.

Divide the numerator by the denominator with a remainder:

\[\frac(37)(12)=37:12=3\ (remainder\ 1)\] \[\frac(37)(12)=3\frac(1)(12)\]

Answer.$\frac(37)(12)=3\frac(1)(12)$.

To write a mixed number as an improper fraction, you need to multiply the denominator by the whole part of the number, add the numerator of the fractional part to the resulting product, and write the resulting amount into the numerator of the fraction. The denominator of the improper fraction will be equal to the denominator of the fractional part of the mixed number.

Example 6

Write the mixed number $5\frac(3)(7)$ as an improper fraction.

Solution.

Answer.$5\frac(3)(7)=\frac(38)(7)$.

Adding mixed numbers and proper fractions

Mixed Number Addition$a\frac(b)(c)$ and proper fraction$\frac(d)(e)$ is performed by adding to a given fraction the fractional part of a given mixed number:

Example 7

Add the proper fraction $\frac(4)(15)$ and the mixed number $3\frac(2)(5)$.

Solution.

Let's use the formula for adding a mixed number and a proper fraction:

\[\frac(4)(15)+3\frac(2)(5)=3+\left(\frac(2)(5)+\frac(4)(15)\right)=3+\ left(\frac(2\cdot 3)(5\cdot 3)+\frac(4)(15)\right)=3+\frac(6+4)(15)=3+\frac(10)( 15)\]

By dividing by the number \textit(5) we can determine that the fraction $\frac(10)(15)$ is reducible. Let's perform the reduction and find the result of the addition:

So, the result of adding the proper fraction $\frac(4)(15)$ and the mixed number $3\frac(2)(5)$ is $3\frac(2)(3)$.

Answer:$3\frac(2)(3)$

Adding mixed numbers and improper fractions

Adding improper fractions and mixed numbers reduces to the addition of two mixed numbers, for which it is enough to isolate the whole part from the improper fraction.

Example 8

Calculate the sum of the mixed number $6\frac(2)(15)$ and the improper fraction $\frac(13)(5)$.

Solution.

First, let's extract the integer part from the improper fraction $\frac(13)(5)$:

Answer:$8\frac(11)(15)$.

Improper fraction

Quarters

Orderliness. a And b there is a rule that allows you to uniquely identify one and only one of three relationships between them: “< », « >" or " = ". This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relation as two integers and ; two non-positive numbers a And b are related by the same relationship as two non-negative numbers and ; if suddenly a non-negative, but b- negative, then a > b.
src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">
Adding Fractions Addition operation. a And b For any rational numbers there is a so-called summation rule c summation rule. At the same time, the number itself called amount a And b numbers and is denoted by , and the process of finding such a number is called summation .
. The summation rule has the following form: Addition operation. a And b For any rational numbers Multiplication operation. multiplication rule summation rule c summation rule. At the same time, the number itself , which assigns them some rational number amount a And b work and is denoted by , and the process of finding such a number is also called multiplication .
. The multiplication rule looks like this: Transitivity of the order relation. a , b And summation rule For any triple of rational numbers a If b And b If summation rule less a If summation rule, That a, and if b And b, and if summation rule less a, and if summation rule equals
. 6435">Commutativity of addition. Changing the places of rational terms does not change the sum.
Associativity of addition. The order in which three rational numbers are added does not affect the result.
Presence of zero. There is a rational number 0 that preserves every other rational number when added.
The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
Commutativity of multiplication. Changing the places of rational factors does not change the product.
Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
Availability of unit. There is a rational number 1 that preserves every other rational number when multiplied.
Presence of reciprocal numbers. Any rational number has an inverse rational number, which when multiplied by gives 1.
Distributivity of multiplication relative to addition. The same rational number can be added to the left and right sides of a rational inequality.
/pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0"> Axiom of Archimedes. a Whatever the rational number a, you can take so many units that their sum exceeds

.

/pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">

additional properties

so many. It makes sense to list only a few of them here.

Src="/pictures/wiki/files/48/0caf9ffdbc8d6264bc14397db34e8d72.png" border="0">

Countability of a set

- the number of the table row in which the cell is located, and

- column number.

In the process of such a traversal, each new rational number is associated with another natural number. That is, the fraction 1/1 is assigned to the number 1, the fraction 2/1 to the number 2, etc. It should be noted that only irreducible fractions are numbered. A formal sign of irreducibility is that the greatest common divisor of the numerator and denominator of the fraction is equal to one.

Lack of rational numbers

The hypotenuse of such a triangle cannot be expressed by any rational number