Proper Fraction 1. What is a proper fraction? Proper and improper fractions: rules

While studying the queen of all sciences - mathematics, at some point everyone comes across fractions. Although this concept (like the types of fractions themselves or mathematical operations with them) is not at all complicated, it must be treated carefully, because in real life It will be very useful outside of school. So, let's refresh our knowledge about fractions: what they are, what they are for, what types they are and how to perform various arithmetic operations with them.

Her Majesty fraction: what is it

In mathematics, fractions are numbers, each of which consists of one or more parts of a unit. Such fractions are also called ordinary or simple. As a rule, they are written in the form of two numbers that are separated by a horizontal or slash line, it is called a “fractional” line. For example: ½, ¾.

The upper, or first, of these numbers is the numerator (shows how many parts are taken from the number), and the lower, or second, is the denominator (demonstrates how many parts the unit is divided into).

The fraction bar actually functions as a division sign. For example, 7:9=7/9

Traditionally, common fractions are less than one. While decimals can be larger than it.

What are fractions for? Yes, for everything, because in the real world, not all numbers are integers. For example, two schoolgirls in the cafeteria bought one delicious chocolate bar together. When they were about to share dessert, they met a friend and decided to treat her too. However, now it is necessary to correctly divide the chocolate bar, considering that it consists of 12 squares.

At first, the girls wanted to divide everything equally, and then each would get four pieces. But, after thinking it over, they decided to treat their friend, not 1/3, but 1/4 of the chocolate. And since the schoolgirls did not study fractions well, they did not take into account that in such a situation they would end up with 9 pieces, which are very difficult to divide into two. This fairly simple example shows how important it is to be able to correctly find a part of a number. But in life similar cases much more.

Types of fractions: ordinary and decimal

All mathematical fractions are divided into two large categories: ordinary and decimal. The features of the first of them were described in the previous paragraph, so now it’s worth paying attention to the second.

Decimal is a positional notation of a fraction of a number, which is written in writing separated by a comma, without a dash or slash. For example: 0.75, 0.5.

In fact, a decimal fraction is identical to an ordinary fraction, however, its denominator is always one followed by zeros - hence its name.

The number preceding the comma is an integer part, and everything after it is a fraction. I love it simple fraction can be converted to decimal. So, indicated in the previous example decimals can be written as usual: ¾ and ½.

It is worth noting that both decimal and ordinary fractions can be either positive or negative. If they are preceded by a “-” sign, this fraction is negative, if “+” is a positive fraction.

Subtypes of ordinary fractions

There are these types of simple fractions.

Subtypes of decimal fraction

Unlike a simple fraction, a decimal fraction is divided into only 2 types.

Final - received this name due to the fact that after the decimal point it has a limited (finite) number of digits: 19.25.
An infinite fraction is a number with an infinite number of digits after the decimal point. For example, when dividing 10 by 3, the result will be an infinite fraction 3.333...

Adding Fractions

Carrying out various arithmetic manipulations with fractions is a little more difficult than with ordinary numbers. However, if you understand the basic rules, solving any example with them will not be difficult.

For example: 2/3+3/4. The least common multiple for them will be 12, therefore, it is necessary that this number be in each denominator. To do this, we multiply the numerator and denominator of the first fraction by 4, it turns out 8/12, we do the same with the second term, but only multiply by 3 - 9/12. Now you can easily solve the example: 8/12+9/12= 17/12. The resulting fraction is an incorrect unit because the numerator is greater than the denominator. It can and should be transformed into a correct mixed one by dividing 17:12 = 1 and 5/12.

When mixed fractions are added, operations are performed first with whole numbers, and then with fractions.

If the example contains a decimal fraction and a regular fraction, it is necessary to make both simple, then bring them to the same denominator and add them. For example 3.1+1/2. The number 3.1 can be written as mixed fraction 3 and 1/10 or as incorrect - 31/10. The common denominator for the terms will be 10, so you need to multiply the numerator and denominator of 1/2 by 5 alternately, you get 5/10. Then you can easily calculate everything: 31/10+5/10=35/10. The result obtained is an improper reducible fraction, we bring it into normal form, reducing it by 5: 7/2 = 3 and 1/2, or decimal - 3.5.

When adding 2 decimal fractions, it is important that there are the same number of digits after the decimal point. If this is not the case, you just need to add required amount zeros, because in decimal fractions this can be done painlessly. For example, 3.5+3.005. To solve this problem, you need to add 2 zeros to the first number and then add one by one: 3.500+3.005=3.505.

Subtracting Fractions

When subtracting fractions, you should do the same as when adding: reduce to a common denominator, subtract one numerator from another, and, if necessary, convert the result to a mixed fraction.

For example: 16/20-5/10. The common denominator will be 20. You need to bring the second fraction to this denominator by multiplying both its parts by 2, you get 10/20. Now you can solve the example: 16/20-10/20= 6/20. However, this result applies to reducible fractions, so it is worth dividing both sides by 2 and the result is 3/10.

Multiplying fractions

Dividing and multiplying fractions are much simpler operations than addition and subtraction. The fact is that when performing these tasks, there is no need to look for a common denominator.

To multiply fractions, you simply need to multiply both numerators one by one, and then both denominators. Reduce the resulting result if the fraction is a reducible quantity.

For example: 4/9x5/8. After alternate multiplication, the result is 4x5/9x8=20/72. This fraction can be reduced by 4, so the final answer in the example is 5/18.

How to divide fractions

Dividing fractions is also a simple operation; in fact, it still comes down to multiplying them. To divide one fraction by another, you need to invert the second and multiply by the first.

For example, dividing the fractions 5/19 and 5/7. To solve the example, you need to swap the denominator and numerator of the second fraction and multiply: 5/19x7/5=35/95. The result can be reduced by 5 - it turns out 7/19.

If you need to divide a fraction by a prime number, the technique is slightly different. Initially, you should write this number as an improper fraction, and then divide according to the same scheme. For example, 2/13:5 should be written as 2/13: 5/1. Now you need to turn over 5/1 and multiply the resulting fractions: 2/13x1/5= 2/65.

Sometimes you have to divide mixed fractions. You need to treat them as you would with whole numbers: turn them into improper fractions, reverse the divisor and multiply everything. For example, 8 ½: 3. Convert everything into improper fractions: 17/2: 3/1. This is followed by a 3/1 flip and multiplication: 17/2x1/3= 17/6. Now you should convert the improper fraction to the correct one - 2 whole and 5/6.

So, having figured out what fractions are and how you can perform various arithmetic operations with them, you need to try not to forget about it. After all, people are always more inclined to divide something into parts than to add, so you need to be able to do it correctly.

Proper fraction

Quarters

Orderliness. a And b there is a rule that allows one 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 relation 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. For any rational numbers a And b there is a so-called summation rule c. Moreover, the number itself c called amount numbers a And b and is denoted by , and the process of finding such a number is called summation. The summation rule has next view: .
Multiplication operation. For any rational numbers a And b there is a so-called multiplication rule, which assigns them some rational number c. Moreover, the number itself c called work numbers a And b 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. For any triple of rational numbers a , b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c. 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 multiplication operation is coordinated with the addition operation through the distribution law:
Connection of the order relation with the operation of addition. To the left and right side For a rational inequality, you can add the same rational number. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum exceeds a. 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. There are a lot of such additional properties. 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 is created ordinary fractions, on each i-th line in each j 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 i- the number of the table row in which the cell is located, and j- 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 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.

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. 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 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.

We come across fractions in life much earlier than we begin studying them at school. If we cut a whole apple in half, we get ½ of the fruit. Let's cut it again - it will be ¼. These are fractions. And everything seemed simple. For an adult. For the child (and this topic begin to study at the end of primary school) abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must clearly explain what proper fraction and irregular, ordinary and decimal, what operations can be performed with them and, most importantly, why all this is needed.

What are fractions?

Getting to know new topic at school it starts with ordinary fractions. They are easily recognized by the horizontal line separating the two numbers - above and below. The top one is called the numerator, the bottom one is the denominator. There is also a lowercase option for writing improper and proper ordinary fractions - through a slash, for example: ½, 4/9, 384/183. This option is used when the line height is limited and it is not possible to use a “two-story” entry form. Why? Yes, because it is more convenient. We'll see this a little later.

In addition to ordinary fractions, there are also decimal fractions. It is very simple to distinguish them: if in one case a horizontal or slash is used, in the other a comma is used to separate sequences of numbers. Let's look at an example: 2.9; 163.34; 1.953. We intentionally used a semicolon as a separator to delimit the numbers. The first of them will read like this: “two point nine.”

New concepts

Let's return to ordinary fractions. They come in two types.

The definition of a proper fraction is as follows: it is a fraction whose numerator is less than its denominator. Why is it important? We'll see now!

You have several apples, halved. Total - 5 parts. How would you say: do you have “two and a half” or “five and a half” apples? Of course, the first option sounds more natural, and we will use it when talking with friends. But if we need to calculate how many fruits each person will get, if there are five people in the company, we will write down the number 5/2 and divide it by 5 - from a mathematical point of view, this will be more clear.

So, for naming proper and improper fractions, the rule is this: if a whole part can be distinguished in a fraction (14/5, 2/1, 173/16, 3/3), then it is irregular. If this cannot be done, as in the case of ½, 13/16, 9/10, it will be correct.

The main property of a fraction

If the numerator and denominator of a fraction are simultaneously multiplied or divided by the same number, its value does not change. Imagine: they cut the cake into 4 equal parts and gave you one. They cut the same cake into eight pieces and gave you two. Does it really matter? After all, ¼ and 2/8 are the same thing!

Reduction

Authors of problems and examples in mathematics textbooks often seek to confuse students by offering fractions that are cumbersome to write but can actually be abbreviated. Here is an example of a proper fraction: 167/334, which, it would seem, looks very “scary”. But we can actually write it as ½. The number 334 is divisible by 167 without a remainder - after performing this operation, we get 2.

Mixed numbers

An improper fraction can be represented as a mixed number. This is when the whole part is brought forward and written at the level of the horizontal line. In fact, the expression takes the form of a sum: 11/2 = 5 + ½; 13/6 = 2 + 1/6 and so on.

To take out the whole part, you need to divide the numerator by the denominator. Write the remainder of the division on top, above the line, and the whole part - before the expression. Thus, we get two structural parts: whole units + proper fraction.

You can also carry out the inverse operation - to do this, you need to multiply the integer part by the denominator and add the resulting value to the numerator. Nothing complicated.

Multiplication and division

Oddly enough, multiplying fractions is easier than adding. All that is required is to extend the horizontal line: (2/3) * (3/5) = 2*3 / 3*5 = 2/5.

With division, everything is also simple: you need to multiply the fractions crosswise: (7/8) / (14/15) = 7*15 / 8*14 = 15/16.

Adding Fractions

What to do if you need to perform addition or their denominator is different numbers? It will not work to do the same as with multiplication - here you should understand the definition of a proper fraction and its essence. It is necessary to bring the terms to a common denominator, that is, the lower part of both fractions must have the same numbers.

To do this, you should use the basic property of a fraction: multiply both parts by the same number. For example, 2/5 + 1/10 = (2*2)/(5*2) + 1/10 = 5/10 = ½.

How to choose which denominator to reduce the terms to? This must be the minimum number that is a multiple of both numbers in the denominators of the fractions: for 1/3 and 1/9 it will be 9; for ½ and 1/7 - 14, because there is no smaller value divisible by 2 and 7 without a remainder.

Usage

What are improper fractions used for? After all, it is much more convenient to immediately select the whole part, get a mixed number - and be done with it! It turns out that if you need to multiply or divide two fractions, it is more profitable to use irregular ones.

Let's take the following example: (2 + 3/17) / (37 / 68).

It would seem that there is nothing to cut at all. But what if we write the addition result in the first parentheses as an improper fraction? Look: (37/17) / (37/68)

Now everything falls into place! Let's write the example in such a way that everything becomes obvious: (37*68) / (17*37).

Let's cancel 37 in the numerator and denominator and finally divide the top and bottom by 17. Do you remember the basic rule for proper and improper fractions? We can multiply and divide them by any number as long as we do it for the numerator and denominator at the same time.

So, we get the answer: 4. The example looked complicated, but the answer contains only one number. This happens often in mathematics. The main thing is not to be afraid and follow simple rules.

Common Mistakes

When implementing, a student can easily make one of the common mistakes. Usually they occur due to inattention, and sometimes due to the fact that the material studied has not yet been properly stored in the head.

Often the sum of numbers in the numerator makes you want to reduce its individual components. Let’s say in the example: (13 + 2) / 13, written without parentheses (with a horizontal line), many students, due to inexperience, cross out 13 above and below. But this should not be done under any circumstances, because this is a gross mistake! If instead of addition there was a multiplication sign, we would get the number 2 in the answer. But when performing addition, no operations with one of the terms are allowed, only with the entire sum.

Guys also often make mistakes when dividing fractions. Let's take two proper irreducible fractions and divide by each other: (5/6) / (25/33). The student can mix it up and write the resulting expression as (5*25) / (6*33). But this would happen with multiplication, but in our case everything will be somewhat different: (5*33) / (6*25). We reduce what is possible, and the answer will be 11/10. We write the resulting improper fraction as a decimal - 1.1.

Brackets

Remember that in any mathematical expression the order of operations is determined by the precedence of the operation signs and the presence of parentheses. All other things being equal, the order of actions is counted from left to right. This is also true for fractions - the expression in the numerator or denominator is calculated strictly according to this rule.

After all, this is the result of dividing one number by another. If they are not evenly divided, it becomes a fraction - that's all.

How to write a fraction on a computer

Since standard tools do not always allow creating a fraction consisting of two “tiers,” students sometimes resort to various tricks. For example, they copy the numerators and denominators into the Paint graphic editor and glue them together, drawing a horizontal line between them. Of course, there is a simpler option, which, by the way, provides a lot of additional features, which will be useful to you in the future.

Open Microsoft Word. One of the panels at the top of the screen is called “Insert” - click it. On the right, on the side where the close and minimize window icons are located, there is a “Formula” button. This is exactly what we need!

If you use this function, a rectangular area will appear on the screen in which you can use any mathematical symbols that are not on the keyboard, as well as write fractions in classic form. That is, dividing the numerator and denominator with a horizontal line. You might even be surprised that such a proper fraction is so easy to write.

Learn math

If you are in grades 5-6, then soon knowledge of mathematics (including the ability to work with fractions!) will be required in many school subjects. In almost any problem in physics, when measuring the mass of substances in chemistry, in geometry and trigonometry, you cannot do without fractions. Soon you will learn to calculate everything in your mind, without even writing down expressions on paper, but more and more complex examples. Therefore, learn what a proper fraction is and how to work with it, keep up with curriculum, do your homework on time and you will succeed.

This article is about common fractions. Here we will introduce the concept of a fraction of a whole, which will lead us to the definition of a common fraction. Next we will dwell on the accepted notation for ordinary fractions and give examples of fractions, let’s say about the numerator and denominator of a fraction. After this, we will give definitions of proper and improper, positive and negative fractions, and also consider the position of fractional numbers on coordinate ray. In conclusion, we list the main operations with fractions.

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Shares of the whole

First we introduce concept of share.

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 segments. Each of these equal parts that make up the whole object is called parts of the whole or simply shares.

Note that the shares are different. Let's explain this. Let us have two apples. Cut the first apple into two equal parts, and the second 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 sort it out names of beats. If an object consists of two parts, any of them is called one second part of the whole object; if an object consists of three parts, then any of them is called one third part, and so on.

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

For the sake of brevity, the following were introduced: beat symbols. One second share is designated as or 1/2, one third share is designated 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 reinforce the material, let’s give one more example: the entry denotes one hundred and sixty-seventh part of the whole.

The concept of share naturally extends from objects to quantities. For example, one of the measures of length is the meter. To measure lengths shorter 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. The shares of other quantities are applied similarly.

Common fractions, definition and examples of fractions

To describe the number of shares we use common fractions. Let us give an example that will allow us to approach the definition of ordinary fractions.

Let the orange consist of 12 parts. Each share in this case represents one twelfth of a whole orange, that is, . We denote two beats as , three beats as , and so on, 12 beats we denote as . Each of the given 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 give examples of common fractions: 5/10, , 21/1, 9/4, . And here are the records do not fit the stated definition of ordinary fractions, that is, they are not ordinary fractions.

Numerator and denominator

For convenience, ordinary fractions are distinguished numerator and denominator.

Definition.

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

Definition.

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

So, the numerator is located above the fraction line (to the left of the slash), and the denominator is located below the fraction line (to the right of the slash). For example, let's take the common 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 a fraction shows how many parts one object consists of, and the numerator, in turn, indicates the number of such parts. For example, the denominator 5 of the fraction 12/5 means that one object consists of five shares, and the numerator 12 means that 12 such shares are taken.

Natural number as a fraction with denominator 1

The denominator of a common fraction can be equal to one. In this case, we can consider that the object is indivisible, in other words, it represents something whole. The numerator of such a fraction indicates how many whole objects 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 validity of the equality m/1=m.

Let's rewrite the last equality as follows: 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 103,498 is equal to the fraction 103,498/1.

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

Fraction bar as a division sign

Representing the original object in the form of n shares is nothing more than division into n equal parts. After an 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 between n people, giving each person one share from each of the m objects. In this case, each person will have m shares of 1/n, and m shares of 1/n gives the common fraction m/n. Thus, the common fraction m/n can be used to denote the division of m items between n people.

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

Using an ordinary fraction, you can write the result of dividing two natural numbers for which a whole division cannot be performed. For example, the result of dividing 5 apples by 8 people can be written as 5/8, that is, everyone will get five-eighths of an apple: 5:8 = 5/8.

Equal and unequal fractions, comparison of fractions

A fairly natural action is comparing 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 another 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 unequal. In the first case we have equal common fractions, and in the second – unequal ordinary fractions. Let us 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 multiplying natural numbers). For clarity, you can imagine two identical apples, the first is cut in half, and the second is cut into 4 parts. It is obvious that two quarters of an apple equals 1/2 share. Other examples of equal common fractions are the fractions 4/7 and 36/63, and the pair of fractions 81/50 and 1,620/1,000.

But ordinary fractions 4/13 and 5/14 are not equal, since 4·14=56, and 13·5=65, that is, 4·14≠13·5. Other examples of unequal common fractions are the fractions 17/7 and 6/4.

If, when comparing two common fractions, it turns out that they are not equal, then you may need to find out which of these common fractions less different, and which one - 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 notation fractional number. That is, a fraction is just a “shell” of a fractional number, its appearance, and all the semantic load is contained in the fractional number. However, for brevity and convenience, the concepts of fraction and fractional number are combined and simply called 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 a coordinate ray

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

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

For example, let's show point M on the coordinate ray, corresponding to the fraction 14/10. The length of a segment with ends at point O and the point closest to it, marked with a small dash, is 1/10 of a unit segment. The point with coordinate 14/10 is removed from the origin at a distance of 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, the coordinates 1/2, 2/4, 16/32, 55/110 correspond to one point on the coordinate ray, since all the written fractions are equal (it is located at a distance of half a unit segment laid out from the origin in the positive direction).

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

Proper and improper fractions, definitions, examples

Among ordinary fractions there are proper and improper fractions. This division is based on a comparison of the numerator and denominator.

Let us define proper and improper ordinary fractions.

Definition.

Proper fraction is an ordinary fraction whose numerator 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 comparing natural numbers), so they are correct by definition.

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 comparison of fractions with one.

Definition.

correct, if it is less than one.

Definition.

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

So the common 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 - “improper”.

For example, let's take the improper fraction 9/9. This fraction means that nine parts are taken of an object that consists of nine parts. That is, from the available nine parts we can make up a whole object. That is, improper fraction 9/9 essentially gives the whole item, 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 the natural number 1.

Now consider the improper fractions 7/3 and 12/4. It is quite obvious that from these seven third parts we can compose two whole objects (one whole object consists of 3 parts, then to compose two whole objects we will need 3 + 3 = 6 parts) and there will still be one third part left. That is, the improper fraction 7/3 essentially means 2 objects and also 1/3 of such an object. And from twelve quarter parts 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 evenly by the denominator (for example, 9/9=1 and 12/4=3), or by 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 earned improper fractions the name “irregular.”

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 separating the whole part from an improper fraction, and deserves separate and more careful consideration.

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

Positive and negative fractions

Each common fraction corresponds to a positive fractional number (see the article on 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 you need to highlight the positivity of a fraction, a plus sign is placed in front of it, for example, +3/4, +72/34.

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

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 addition, income, an upward change in any value, etc. Negative fractions correspond to expense, debt, or a decrease in any quantity. For example, the negative fraction −3/4 can be interpreted as a debt whose value is equal to 3/4.

On a horizontal and rightward direction, negative fractions are located to the left of the origin. The points of the coordinate line, the coordinates of which 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.

Operations with fractions

We have already discussed one action with ordinary fractions - comparing fractions - above. Four more arithmetic functions are defined operations with fractions– adding, subtracting, multiplying and dividing fractions. Let's look at each of them.

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

Multiplying fractions can be thought of as the action of finding a fraction from a fraction. To clarify, let's give an example. Let us 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 special case is equal to a natural number). Next, we recommend that you study the information in the article Multiplying Fractions - Rules, Examples and Solutions.

Bibliography.

Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics: textbook for 5th grade. educational institutions.
Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).

Improper fraction

Quarters

Orderliness. a And b there is a rule that allows one 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 relation 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. For any rational numbers a And b there is a so-called summation rule c. Moreover, the number itself c called amount numbers a And b and is denoted by , and the process of finding such a number is called summation. The summation rule has the following form: .
Multiplication operation. For any rational numbers a And b there is a so-called multiplication rule, which assigns them some rational number c. Moreover, the number itself c called work numbers a And b 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. For any triple of rational numbers a , b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c. 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 multiplication operation is coordinated with the addition operation through the distribution law:
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. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum exceeds a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

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Countability of a set

Numbering of rational numbers

The simplest of these algorithms looks like this. An endless table of ordinary fractions is compiled, on each i-th line in each j 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 i- the number of the table row in which the cell is located, and j- 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.

Lack of rational numbers

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

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.