8 8 is a proper fraction. Shares, ordinary fractions, definitions, designations, examples, actions with fractions. How to represent a mixed number as an improper fraction

Proper fraction

quarters

1. Orderliness. a and b there is a rule that allows you to uniquely identify between them one and only one of the three relations: “< », « >' 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 relation as two non-negative numbers and ; if suddenly a non-negative, and b- negative, then a > b. style="max-width: 98%; height: auto; width: auto;" src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">

   

   summation of fractions

2. addition operation. For any rational numbers a and b there is a so-called summation rule c. However, the number itself c called sum numbers a and b and is denoted , and the process of finding such a number is called summation. The summation rule has the following form: .
3. multiplication operation. For any rational numbers a and b there is a so-called multiplication rule, which puts them in correspondence with some rational number c. However, the number itself c called work numbers a and b and is denoted , and the process of finding such a number is also called multiplication. The multiplication rule is as follows: .
4. Transitivity of the order relation. For any triple of rational numbers a , b and c if a less b and b less c, then a less c, and if a equals b and b equals c, then a equals c. 6435">Commutativity of addition. The sum does not change from changing the places of rational terms.
5. Associativity of addition. The order in which three rational numbers are added does not affect the result.
6. The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
7. The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
8. Commutativity of multiplication. By changing the places of rational factors, the product does not change.
9. Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
10. The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
11. The presence of reciprocals. Any rational number has an inverse rational number, which, when multiplied, gives 1.
12. Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
13. Connection of the order relation with the operation of addition. The same rational number can be added to the left and right sides of a rational inequality. max-width: 98% height: auto; width: auto;" src="/pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
14. Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum will exceed a. style="max-width: 98%; height: auto; width: auto;" src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

All other properties inherent in rational numbers are not singled out as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proved on the basis of the given basic properties or directly by the definition of some mathematical object. There are a lot of such additional properties. It makes sense here to cite just a few of them.

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Set countability

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 suffices to give an algorithm that enumerates rational numbers, that is, establishes a bijection between the sets of rational and natural numbers.

The simplest of these algorithms is as follows. An infinite table of ordinary fractions is compiled, on each i-th line in each j th column of which is a fraction. For definiteness, it is assumed that the rows and columns of this table are numbered from one. Table cells are denoted , where i- the row number of the table in which the cell is located, and j- column number.

The resulting table is managed by a "snake" according to the following formal algorithm.

These rules are searched from top to bottom and the next position is selected by the first match.

In the process of such a bypass, each new rational number is assigned to the next natural number. That is, fractions 1 / 1 are assigned the number 1, fractions 2 / 1 - the number 2, etc. It should be noted that only irreducible fractions are numbered. The formal sign of irreducibility is the equality to unity of the greatest common divisor of the numerator and denominator of the fraction.

Following this algorithm, one 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, simply by 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 bewilderment, since at first glance one gets the impression that it is much larger than the set of natural numbers. In fact, this is not the case, and there are enough natural numbers to enumerate all rational ones.

Insufficiency of rational numbers

The hypotenuse of such a triangle is not expressed by any rational number

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

It is known from the Pythagorean theorem 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., a number whose square is 2.

If we assume that the number is represented by some rational number, then there is such an integer m and such a natural number n, which, moreover, the fraction is irreducible, i.e., the numbers m and n are coprime.

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

The numerator of a 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 parts the unit is divided into (located under the line - in the lower part). , in turn, are divided into: correct and wrong, mixed and composite 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 cases the fraction is called wrong:

To isolate the largest integer contained in an improper fraction, you need to 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 the division is performed with a remainder, then the (incomplete) quotient gives the desired integer, the remainder becomes the numerator of the fractional part; the denominator of the fractional part remains the same.

A number that contains an integer and a fractional part is called mixed. Fraction mixed number maybe improper fraction. Then it is possible to extract the largest integer from the fractional part and represent the mixed number in such a way that the fractional part becomes a proper fraction (or disappears altogether).

Please help. I need to write in words: the property consists of 2700 / 137061 shares ... My version: Two thousand seven hundred One hundred thirty-seven thousand sixty-first shares

Is this really necessary? The fact is that it will be completely impossible to understand what is written in words ...

You can write it like this: a fraction, in the numerator of which the number is such and such, and in the denominator - such and such.

Hello! Is there any special rule about the combination of words with the numeral 1.5? It is in digital form, not the word "one and a half"? The text is not mathematical, but there is no possibility to replace a number with a word. For example: Is the time limit for completing the task 1.5 minutes or 1.5 minutes? After 1.5 years or 1.5 years?

The rule is this: with a mixed number, the noun is ruled by a fraction, not an integer. Wed: 35.5 percent(not: ...percent), 12.6 kilometers(not: ...kilometers), 45.0 seconds. (Rosenthal D. E. Spelling and Literary Editing Handbook. M. 1999. § 164, p. 8.)

Question: Infant mortality was 6.8 per thousand births. - here you need to write /person/ (r.p.) or you need to leave /person/. Eight-tenths of a person, of course, sounds terrible, but here are statistics, there is no way to replace a fraction

The answer of the reference service of the Russian language

Grammatically correct: 6.8 people.

Hello! Tell me, please, HOW and WHY is the fraction "1/130" written? Thanks!

The answer of the reference service of the Russian language

How to write it in words? One hundred and thirty.

Please tell me. where can I find a detailed rule for agreeing fractional numbers with adjectives and nouns (for example: 0.68 hundredths of a square meter? square meter?)?

The answer of the reference service of the Russian language

With a mixed number, the noun is ruled by a fraction, not an integer. Right: 0.68 square meters.

Dear "Diploma", explain why of the two options "two hundred nine and a half thousand" and "two hundred and nine and a half thousand" the first option is correct (this is question No. 285264), and of the options "five and a half meters" and "five and a half meters" is correct 5.5 meters (question no. 285260). Can you explain please!

The answer of the reference service of the Russian language

Right: two hundred nine and a half thousand, five and a half meters. But if we use the numerical form for writing, where there is an integer and a fraction, correctly: 209.5 thousand, 5.5 meters. The noun is ruled by a fraction: two hundred and nine point and five tenths of a thousand, five and five tenths of a metre.

How to speak and write correctly: "two hundred nine and a half thousand" or "two hundred nine and a half thousand"? Which word to focus on: the main number or its fraction?

The answer of the reference service of the Russian language

Right: two hundred nine and a half thousand.

"Two hundred percent of the population" or percent? And more difficult:
"Two hundred point three percent of the population" or percent A?
That is, the question is, from what point does the genitive case begin? If not for the word "population", everything would be clear, since it is the fraction that controls the subsequent noun. But here there are two. That's what I don't understand.

The answer of the reference service of the Russian language

In accordance with the rule, the cardinal number agrees in case with the noun: two hundred percent of the population.

Fractional numbers are used with singular nouns: two hundred and three tenths of a percent of the population (three tenths of (what?) percent).

How do you write years with a fraction u? For example: The average age of the unemployed was 35.1 or YEARS?

The answer of the reference service of the Russian language

Both options are unsuccessful: it is customary to measure the year not in tenths, but in months (35 years and so many months).

Good day!
How to write the fraction 5/31010 in words?
Thanks!

The answer of the reference service of the Russian language

Probably like this: five thirty-one thousand tenths. But why? This is a great inconvenience for both the writer and the reader.

Good afternoon. Thanks for answers! Still, I want to clarify your answer to my last question. You sent an answer, which is correct in the dative case:

Http://gramota.ru/spravka/buro/29_458084 Question No. 274637
Hello. Correct in brackets in both cases?
This year we will support 3.5 thousand families.
Apartments were provided by 35 thousand (AM) families.
patterns
The answer of the reference service of the Russian language
Correct in the dative case: three and a half thousand families; three thousand five hundred families; thirty-five thousand families.

BUT WHAT TO DO WITH THIS YOUR ANSWER? How to distinguish in which case the numeral should be read "three and a half five tenth thousand" and when should be read "three and a half thousand"? Or is it of primary importance here, "thousands of whom or what exactly" - people, units, equipment, apples?

http://www.gramota.ru/spravka/buro/29_386324
Question #256506
was reduced by a total of 16.5 units - what is the correct spelling of "units"?
LYOSHA
The answer of the reference service of the Russian language
Correct: 16.5 units. The noun is ruled by a fraction: five tenths of a unit.

The answer of the reference service of the Russian language

Grammar depends on how the sentence is read. In this case it is preferable: three and a half thousand or three thousand five hundred(hard to read and understand: three and five tenths of a thousand).

Hello,
please tell me how to correctly decline compound numbers, as well as agree on a fraction with the noun "share" (or "shares", plural?) in this case:

"The property consists of 21/85 (twenty-one eighty-fifths) shares of the apartment"

Thanks!

The answer of the reference service of the Russian language

Right: ... twenty-one eighty-fifth.

The numerator of a fraction is a cardinal number ( twenty one), and the denominator is ordinal ( 85th). Word share stands in the singular form, as it refers to a numeral that ends in one.

Please resolve doubts urgently!
With a mixed number, the noun is controlled by a fraction, so the noun is put in the singular, for example: 12.6 kilometers, percent, meters, etc. But what about other nouns (not those that measure something), for example: 9,882 visits or visits? Or is the noun always put in the singular with a fractional numeral?

The answer of the reference service of the Russian language

Yes, similar: 9, 882 (thousandths) visits.

Is the word "Whole" a noun or just an adjective? For example: "single whole" - is the whole a noun or an adjective?

The answer of the reference service of the Russian language

In your example, the word is used as a noun.

CE LOE,-Wow; cf.
1. Mat.
A number without a fraction. Subtract a fraction from a whole.
2.
Something one, indivisible. The park and the architectural ensemble make up one c.Slender, single c.The removal of this episode from the play would violate c.Sacrifice particulars for the sake of the whole.

Which is correct: 5 1/2 meter or 5.5 meter? Why?

The answer of the reference service of the Russian language

The second design option (with a decimal fraction) is more familiar (probably due to greater graphical simplicity).

Instruction

The simplest fractions can be printed by inserting special characters representing some ordinary fractions. To do this, select the "Insert-Symbol" menu item. In the plate with a set of characters that appears, select the sign of the desired fraction (if it is there). Unfortunately, the list of available fraction symbols is very limited in standard fonts with the following values: ?, ?, ?, ?, ?, ?, ?, ?, ?. The set of ready-made fractions may vary depending on the font selected in the "Font" field. However, if some special font provides a large selection of fractions, this does not mean at all that these characters will be displayed in the same way on another.

To print any ordinary, type its numerator, then the oblique sign (/), and after it the denominator of the fraction. To give such a fraction a more natural look, select the numerator, press the right mouse button, select the "Font" line in the drop-down context menu and check the box with the word "superscript". Do the same with the denominator of the fraction. Just put a tick in front of the word "subscript".

You can print a fraction by combining vertical offset and decreasing the font size. Type the numerator and denominator of an ordinary fraction, separating them with a slash. Now select the numerator and select the "Font" item in the context (or main) menu. Specify a font size about one third smaller than the set one (for example, 8 pt instead of 12 pt). Then go to the "Interval" tab and in the "Offset" line, select the value "Up". The offset value can be left at the default. After that, do the same procedure with the denominator. Only "Offset" needs to select "Down".

If the fraction sign (horizontal bar) is used in complex mathematical expressions, then it is better to type such a bar (like the whole expression) using the formula editor. To do this, select the following menu items in sequence: "Insert - Object - Microsoft Equation 3.0". After that, the editor of mathematical formulas will start, where you can print any fraction. If the object "Microsoft Equation 3.0" does not appear in the drop-down menu, then this option was not installed when Word was installed. To do this, insert a disc with Word of the same version and run the installation program. Check the Microsoft Equation 3.0 checkbox and after installation this feature will become available. In Microsoft Word 2007, the formula editor is already built into the taskbar.

You can print a complex fraction in Word in another way. Select the following items in sequence: "Insert - Field - Formula - Eq". Now select the fraction icon in the opened editor.

You can print a fraction using a special "symbolic" formula editor. To do this, press the key combination Ctrl + F9. Then, inside the curly braces that appear, type: eq f(1;2) and press F9. The result is one second, recorded in a classic, "vertical" form. To get the desired fraction, print the numerator instead of one, and the denominator of the fraction instead of two. By the way, the resulting fraction can later be edited with a “normal” formula editor.

In extreme cases, the fraction symbol (horizontal line) can be drawn by yourself. To do this, expand the drawing panel, select the line tool and draw a suitable horizontal line. In order to “append” the numerator and denominator to the resulting line, in the settings of the “text wrapping” option, you must select “before the text” or “behind the text”.

note

Entering a fraction can be significantly accelerated if you use a special field: "Sign Code". for example, to get "one half", enter "00BD" (or "00bd") in this field.

Useful advice

All options are focused on Word 2003 (XP). All other versions are slightly different.

Sources:

• how to reduce a fraction by a fraction
• Making shots at home

Probably every person, being a student, at least once in his life, wrote an essay. Students who write abstracts on topics related to calculus have most likely encountered the problem of adding formulas and fractional numbers in a text editor. The Microsoft Office software package has objects called "Microsoft Equation" that allow you to compose a mathematical expression of any complexity.

You will need

• Microsoft Office Word 2007 software.

Instruction

As a result of these actions, a place is now added to the document we are editing to create an additional formula.

In the main menu, the "Designer" tab opens in front of you. In the "Structures" group, click the "Fraction" item, in which you need to select the desired item from the drop-down list with the name "Vertical simple fraction".

After completing the previous step and adding a special place in the document to create a formula, it is possible to insert a template for a vertical fraction. To do this, click on the box that is in the numerator of the fraction and add to it the expression that is in the numerator of your first fraction. After all these actions, click on the box that is in the denominator of the fraction, and add to it the expression that is in the denominator of the first fraction.

After creating the first fraction that has been successfully added to the document, click to the right of it and add a "+" sign.

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The fraction is one of the elements of formulas, for the input of which in the word processor Word there is a Microsoft Equation tool. With it, you can enter any complex mathematical or physical formulas, equations and other elements that include special characters.

Instruction

To launch the Microsoft Equation tool, you need to go to the address: "Insert" -> "Object", in the dialog box that opens, on the first tab from the list, select Microsoft Equation and click "OK" or double-click on the selected item. After launching the editor, a toolbar will open in front of you and an input field will be displayed: a rectangle in a dotted one. The toolbar is divided into sections, each of which contains a set of action signs or expressions. When you click on one of the sections, a list of the tools in it will expand. From the list that opens, select the desired symbol and click on it. Once selected, the specified character will appear in a selected rectangle in the document.

The section that contains elements for writing fractions is located in the second line of the toolbar. When you hover your mouse cursor over it, you will see the tooltip "Fraction and Radical Patterns". Click a section once and expand the list. The drop-down menu has templates for fractions with horizontal and slash. Among the options that appear, you can choose the one that suits your task. Click on the desired option. After clicking, in the input field that opened in the document, a fraction symbol and places for entering the numerator and denominator, framed by a dotted line, will appear. The default cursor is automatically placed in the field for entering the numerator. Enter the numerator. In addition to numbers, you can also enter mathematical symbols, letters, or action signs. They can be entered both from the keyboard and from the corresponding sections of the Microsoft Equation toolbar. After the numerator water, press the TAB key to move to the denominator. You can also go by clicking the mouse in the field for entering the denominator. Once the formula is written, click anywhere in the document with the mouse pointer, the toolbar will close, and fraction input will be completed. To edit a fraction, double-click on it with the left mouse button.

If, when you open the menu "Insert" -> "Object", you did not find the Microsoft Equation tool in the list, you need to install it. Run the installation disc, disc image, or Word distribution file. In the installer window that appears, select "Add or remove components. Adding or removing individual components" and click "Next". In the next window, check the item "Advanced application settings". Click next. In the next window, find the list item "Office Tools" and click on the plus sign on the left. In the expanded list, we are interested in the item "Formula Editor". Click on the icon next to "Formula Editor" and, in the menu that opens, click "Run from my computer". After that, click "Update" and wait until the required component is installed.

Fractional numbers are divided into two groups according to the form of writing, one of which is called "ordinary" fractions, and the other - "decimal". If there are no problems with writing decimal fractions in text documents, then the procedure for placing “two-story” ordinary and mixed fractions (a special case of ordinary ones) in the text is a little more complicated. If a normal slash (/) is not enough to separate the numerator and denominator, then you can resort to the capabilities of the Microsoft Office Word word processor.

Instruction

Go to the "Insert" tab of the word processor menu and click on the "Formula" button placed in the "Symbols" command group. Pay attention to the fact that it is necessary to click on the button, and not on the label of the drop-down list placed close to it (on the right). In this way, the "Formula Builder" is launched and an additional tab with the same name is added to the menu, on which the controls of this constructor are located. If you nevertheless open the drop-down “Formula” button, then you can also launch the constructor from it by selecting the “Insert new formula” line at the bottom of the list.

Click the "Fraction" button - it is placed in the first position in the command called "Structures" on the "Designer" tab. This action brings up a list of nine common fraction spellings. Some of them already have the most commonly used special characters in the numerator and denominator by default. Choose the option that suits you best, and Word will place it in the newly created formula frame.

Edit the numerator and denominator of the created fraction. A vertical rectangle with three points adjoins the upper left corner of the frame of the object containing your fraction - you can move the fraction with the mouse by dragging the object over this rectangle. If you need to change the fraction, just click on it to turn on the "Formula Editor".

In the character encoding tables used by the computer, there are signs that represent the simplest fractions. There are only three of them, and you can insert these symbols in the same way as, for example, a copyright sign. There are several ways to paste, the simplest of them is implemented as follows: enter the code of the desired character and press the key combination alt + x. Using the code 00BC, you can write a fraction ¼, the code 00BD puts a fraction ½ in the text, and 00BE - ¾ (all letters in the codes are Latin).

Related videos

Instruction

Click once on the "Insert" menu item, then select the "Symbol" item. This is one of the easiest ways to insert fractions into text. It consists in the following. There are fractions in the set of ready-made symbols. Their number is usually small, but if you need to write ½, not 1/2 in the text, then this option will be the most optimal for you. In addition, the number of fraction characters may depend on the font. For example, for the Times New Roman font, there are slightly fewer fractions than for the same Arial. Vary fonts to find the best option when it comes to simple expressions.

This article is about common fractions. Here we will get acquainted with the concept of a fraction of a whole, which will lead us to the definition of an ordinary fraction. Next, we will dwell on the accepted notation for ordinary fractions and give examples of fractions, say about the numerator and denominator of a fraction. After that, we will give definitions of correct and incorrect, positive and negative fractions, and also consider the position of fractional numbers on the coordinate ray. In conclusion, we list the main actions with fractions.

Page navigation.

Shares of the whole

First we introduce share concept.

Let's assume that we have some object made up of several absolutely identical (that is, equal) parts. For clarity, you can imagine, for example, an apple cut into several equal parts, or an orange, consisting of several equal slices. Each of these equal parts that make up the whole object is called share of the whole or simply shares.

Note that the shares are different. Let's explain this. Let's say we have two apples. Let's cut the first apple into two equal parts, and the second one into 6 equal parts. It is clear that the share of the first apple will be different from the share of the second apple.

Depending on the number of shares that make up the whole object, these shares have their own names. Let's analyze share names. If the object consists of two parts, any of them is called one second part of the whole object; if the object consists of three parts, then any of them is called one third part, and so on.

One second beat has a special name - half. One third is called third, and one quadruple - quarter.

For the sake of brevity, the following share designations. One second share is designated as or 1/2, one third share - as or 1/3; one fourth share - like or 1/4, and so on. Note that the notation with a horizontal bar is used more often. To consolidate the material, let's give one more example: the entry denotes one hundred and sixty-seventh of the whole.

The concept of a share naturally extends from objects to magnitudes. For example, one of the measures of length is the meter. To measure lengths less than a meter, fractions of a meter can be used. So you can use, for example, half a meter or a tenth or thousandth of a meter. Shares of other quantities are applied similarly.

Common fractions, definition and examples of fractions

To describe the number of shares are used common fractions. Let's give an example that will allow us to approach the definition of ordinary fractions.

Let an orange consist of 12 parts. Each share in this case represents one twelfth of a whole orange, that is, . Let's denote two beats as , three beats as , and so on, 12 beats as . Each of these entries is called an ordinary fraction.

Now let's give a general definition of common fractions.

The voiced definition of ordinary fractions allows us to bring examples of common fractions: 5/10 , , 21/1 , 9/4 , . And here are the records do not fit the voiced definition of ordinary fractions, that is, they are not ordinary fractions.

Numerator and denominator

For convenience, in ordinary fractions we distinguish numerator and denominator.

Definition.

Numerator ordinary fraction (m / n) is a natural number m.

Definition.

Denominator ordinary fraction (m / n) is a natural number n.

So, the numerator is located above the fraction bar (to the left of the slash), and the denominator is below the fraction bar (to the right of the slash). For example, let's take an ordinary fraction 17/29, the numerator of this fraction is the number 17, and the denominator is the number 29.

It remains to discuss the meaning contained in the numerator and denominator of an ordinary fraction. The denominator of the fraction shows how many shares one item consists of, the numerator, in turn, indicates the number of such shares. For example, the denominator 5 of the fraction 12/5 means that one item consists of five parts, and the numerator 12 means that 12 such parts are taken.

Natural number as a fraction with denominator 1

The denominator of an ordinary fraction can be equal to one. In this case, we can assume that the object is indivisible, in other words, it is something whole. The numerator of such a fraction indicates how many whole items are taken. Thus, an ordinary fraction of the form m/1 has the meaning of a natural number m. This is how we substantiated the equality m/1=m .

Let's rewrite the last equality like this: m=m/1 . This equality allows us to represent any natural number m as an ordinary fraction. For example, the number 4 is the fraction 4/1, and the number 103498 is the fraction 103498/1.

So, any natural number m can be represented as an ordinary fraction with denominator 1 as m/1 , and any ordinary fraction of the form m/1 can be replaced by a natural number m.

Fraction bar as division sign

The representation of the original object in the form of n shares is nothing more than a division into n equal parts. After the item is divided into n shares, we can divide it equally among n people - each will receive one share.

If we initially have m identical objects, each of which is divided into n shares, then we can equally divide these m objects among n people, giving each person one share from each of the m objects. In this case, each person will have m shares 1/n, and m shares 1/n gives an ordinary fraction m/n. Thus, the common fraction m/n can be used to represent the division of m items among n people.

So we got an explicit connection between ordinary fractions and division (see the general idea of ​​the division of natural numbers). This relationship is expressed as follows: The bar of a fraction can be understood as a division sign, that is, m/n=m:n.

With the help of an ordinary fraction, you can write the result of dividing two natural numbers for which division is not carried out by an integer. For example, the result of dividing 5 apples by 8 people can be written as 5/8, that is, each will get five eighths of an apple: 5:8=5/8.

Equal and unequal ordinary fractions, comparison of fractions

A fairly natural action is comparison of common fractions, because it is clear that 1/12 of an orange is different from 5/12, and 1/6 of an apple is the same as the other 1/6 of this apple.

As a result of comparing two ordinary fractions, one of the results is obtained: the fractions are either equal or not equal. In the first case we have equal common fractions, and in the second unequal common fractions. Let's give a definition of equal and unequal ordinary fractions.

Definition.

equal, if the equality a d=b c is true.

Definition.

Two common fractions a/b and c/d not equal, if the equality a d=b c is not satisfied.

Here are some examples of equal fractions. For example, the common fraction 1/2 is equal to the fraction 2/4, since 1 4=2 2 (if necessary, see the rules and examples of multiplication of natural numbers). For clarity, you can imagine two identical apples, the first is cut in half, and the second - into 4 shares. It is obvious that two-fourths of an apple is 1/2 a share. Other examples of equal common fractions are the fractions 4/7 and 36/63, and the pair of fractions 81/50 and 1620/1000.

And ordinary fractions 4/13 and 5/14 are not equal, since 4 14=56, and 13 5=65, that is, 4 14≠13 5. Another example of unequal common fractions are the fractions 17/7 and 6/4.

If, when comparing two ordinary fractions, it turns out that they are not equal, then you may need to find out which of these ordinary fractions less another, and which more. To find out, the rule for comparing ordinary fractions is used, the essence of which is to bring the compared fractions to a common denominator and then compare the numerators. Detailed information on this topic is collected in the article comparison of fractions: rules, examples, solutions.

Fractional numbers

Each fraction is a record fractional number. That is, a fraction is just a “shell” of a fractional number, its appearance, and the entire semantic load is contained precisely in a fractional number. However, for brevity and convenience, the concept of a fraction and a fractional number are combined and simply called a fraction. Here it is appropriate to paraphrase a well-known saying: we say a fraction - we mean a fractional number, we say a fractional number - we mean a fraction.

Fractions on the coordinate beam

All fractional numbers corresponding to ordinary fractions have their own unique place on , that is, there is a one-to-one correspondence between fractions and points of the coordinate ray.

In order to get to the point corresponding to the fraction m / n on the coordinate ray, it is necessary to postpone m segments from the origin in the positive direction, the length of which is 1 / n fraction of the unit segment. Such segments can be obtained by dividing a single segment into n equal parts, which can always be done using a compass and ruler.

For example, let's show the point M on the coordinate ray, corresponding to the fraction 14/10. The length of the segment with ends at the point O and the point closest to it, marked with a small dash, is 1/10 of the unit segment. The point with coordinate 14/10 is removed from the origin by 14 such segments.

Equal fractions correspond to the same fractional number, that is, equal fractions are the coordinates of the same point on the coordinate ray. For example, one point corresponds to the coordinates 1/2, 2/4, 16/32, 55/110 on the coordinate ray, since all written fractions are equal (it is located at a distance of half the unit segment, postponed from the origin in the positive direction).

On a horizontal and right-directed coordinate ray, the point whose coordinate is a large fraction is located to the right of the point whose coordinate is a smaller fraction. Similarly, the point with the smaller coordinate lies to the left of the point with the larger coordinate.

Proper and improper fractions, definitions, examples

Among ordinary fractions, there are proper and improper fractions. This division basically has a comparison of the numerator and denominator.

Let's give a definition of proper and improper ordinary fractions.

Definition.

Proper fraction is an ordinary fraction, the numerator of which is less than the denominator, that is, if m

Definition.

Improper fraction is an ordinary fraction in which the numerator is greater than or equal to the denominator, that is, if m≥n, then the ordinary fraction is improper.

Here are some examples of proper fractions: 1/4 , , 32 765/909 003 . Indeed, in each of the written ordinary fractions, the numerator is less than the denominator (if necessary, see the article comparison of natural numbers), so they are correct by definition.

And here are examples of improper fractions: 9/9, 23/4,. Indeed, the numerator of the first of the written ordinary fractions is equal to the denominator, and in the remaining fractions the numerator is greater than the denominator.

There are also definitions of proper and improper fractions based on comparing fractions with one.

Definition.

correct if it is less than one.

Definition.

The common fraction is called wrong, if it is either equal to one or greater than 1 .

So the ordinary fraction 7/11 is correct, since 7/11<1 , а обыкновенные дроби 14/3 и 27/27 – неправильные, так как 14/3>1 , and 27/27=1 .

Let's think about how ordinary fractions with a numerator greater than or equal to the denominator deserve such a name - "wrong".

Let's take the improper fraction 9/9 as an example. This fraction means that nine parts of an object are taken, which consists of nine parts. That is, from the available nine shares, we can make up a whole subject. That is, the improper fraction 9/9 essentially gives a whole object, that is, 9/9=1. In general, improper fractions with a numerator equal to the denominator denote one whole object, and such a fraction can be replaced by a natural number 1.

Now consider the improper fractions 7/3 and 12/4. It is quite obvious that from these seven thirds we can make two whole objects (one whole object is 3 shares, then to compose two whole objects we need 3 + 3 = 6 shares) and there will still be one third share. That is, the improper fraction 7/3 essentially means 2 items and even 1/3 of the share of such an item. And from twelve quarters we can make three whole objects (three objects with four parts each). That is, the fraction 12/4 essentially means 3 whole objects.

The considered examples lead us to the following conclusion: improper fractions can be replaced either by natural numbers, when the numerator is divided entirely by the denominator (for example, 9/9=1 and 12/4=3), or the sum of a natural number and a proper fraction, when the numerator is not evenly divisible by the denominator (for example, 7/3=2+1/3 ). Perhaps this is precisely what improper fractions deserve such a name - “wrong”.

Of particular interest is the representation of an improper fraction as the sum of a natural number and a proper fraction (7/3=2+1/3). This process is called the extraction of an integer part from an improper fraction, and deserves a separate and more careful consideration.

It is also worth noting that there is a very close relationship between improper fractions and mixed numbers.

Positive and negative fractions

Each ordinary fraction corresponds to a positive fractional number (see the article positive and negative numbers). That is, ordinary fractions are positive fractions. For example, ordinary fractions 1/5, 56/18, 35/144 are positive fractions. When it is necessary to emphasize the positiveness of a fraction, then a plus sign is placed in front of it, for example, +3/4, +72/34.

If you put a minus sign in front of an ordinary fraction, then this entry will correspond to a negative fractional number. In this case, one can speak of negative fractions. Here are some examples of negative fractions: −6/10 , −65/13 , −1/18 .

The positive and negative fractions m/n and −m/n are opposite numbers. For example, the fractions 5/7 and −5/7 are opposite fractions.

Positive fractions, like positive numbers in general, denote an increase, income, a change in some value upwards, etc. Negative fractions correspond to expense, debt, a change in any value in the direction of decrease. For example, a negative fraction -3/4 can be interpreted as a debt, the value of which is 3/4.

On the horizontal and right-directed negative fractions are located to the left of the reference point. The points of the coordinate line whose coordinates are the positive fraction m/n and the negative fraction −m/n are located at the same distance from the origin, but on opposite sides of the point O .

Here it is worth mentioning fractions of the form 0/n. These fractions are equal to the number zero, that is, 0/n=0 .

Positive fractions, negative fractions, and 0/n fractions combine to form rational numbers.

Actions with fractions

One action with ordinary fractions - comparing fractions - we have already considered above. Four more arithmetic are defined operations with fractions- addition, subtraction, multiplication and division of fractions. Let's dwell on each of them.

The general essence of actions with fractions is similar to the essence of the corresponding actions with natural numbers. Let's draw an analogy.

Multiplication of fractions can be considered as an action in which a fraction is found from a fraction. To clarify, let's take an example. Suppose we have 1/6 of an apple and we need to take 2/3 of it. The part we need is the result of multiplying the fractions 1/6 and 2/3. The result of multiplying two ordinary fractions is an ordinary fraction (which in a particular case is equal to a natural number). Further we recommend to study the information of the article multiplication of fractions - rules, examples and solutions.

Bibliography.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics: textbook for 5 cells. educational institutions.
• Vilenkin N.Ya. etc. Mathematics. Grade 6: textbook for educational institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).