What is called a fraction. Common fractions
The numerator and denominator of a fraction. Types of fractions. Let's continue with fractions. First, a small caveat - we, considering fractions and the corresponding examples with them, for now we will work only with its numerical representation. There are also fractional literal expressions(with and without numbers).However, all the "principles" and rules also apply to them, but we will talk about such expressions separately in the future. I recommend visiting and studying (remembering) the topic of fractions step by step.
The most important thing is to understand, remember and realize that a FRACTION is a NUMBER!!!
Common fraction is a number of the form:
The number located "on top" (in this case m) is called the numerator, the number located below (number n) is called the denominator. Those who have just touched on the topic often get confused - what is the name.
Here's a trick for you, how to remember forever - where is the numerator, and where is the denominator. This technique is associated with verbal-figurative association. Imagine a jar of cloudy water. It is known that as water settles, clean water remains on top, and turbidity (dirt) settles, remember:
CHISSS melt water ABOVE (CHISSS pourer on top)
mud ZZZNNN th water BOTTOM (ZZZNN Amenator below)
So, as soon as it becomes necessary to remember where the numerator is and where the denominator is, then they immediately visually presented a jar of settled water, in which Pure water, and below dirty water. There are other tricks to remember, if they help you, then good.
Examples of ordinary fractions:
What does the horizontal line between numbers mean? This is nothing more than a division sign. It turns out that a fraction can be considered as an example with the action of division. This action is simply recorded in this form. That is, the top number (numerator) is divided by the bottom number (denominator):
In addition, there is another form of recording - a fraction can be written like this (through a slash):
1/9, 5/8, 45/64, 25/9, 15/13, 45/64 and so on...
We can write the above fractions as follows:
The result of the division, as you know, is the number.
Clarified - FRACTION THIS NUMBER !!!
As you have already noticed, in an ordinary fraction, the numerator may be less than the denominator, may be greater than the denominator, and may be equal to it. Here there are many important points, which are understandable intuitively, without any theoretical frills. For example:
1. Fractions 1 and 3 can be written as 0.5 and 0.01. Let's run a little ahead - these are decimal fractions, we'll talk about them a little lower.
2. Fractions 4 and 6 result in an integer 45:9=5, 11:1 = 11.
3. Fraction 5 as a result gives a unit 155:155 = 1.
What conclusions suggest themselves? The following:
1. The numerator, when divided by the denominator, can give a finite number. It may not work, divide by a column 7 by 13 or 17 by 11 - no way! You can divide indefinitely, but we will also talk about this a little lower.
2. A fraction can result in an integer. Therefore, we can represent any integer as a fraction, or rather an infinite series of fractions, look, all these fractions are equal to 2:
More! We can always write any whole number as a fraction - this number itself is in the numerator, one in the denominator:
3. We can always represent a unit as a fraction with any denominator:
*The points indicated are extremely important for working with fractions in calculations and conversions.
Types of fractions.
And now about the theoretical division of ordinary fractions. They are divided into right and wrong.
A fraction whose numerator is less than the denominator is called a proper fraction. Examples:
A fraction whose numerator is greater than or equal to the denominator is called an improper fraction. Examples:
mixed fraction(mixed number).
A mixed fraction is a fraction written as a whole number and a proper fraction and is understood as the sum of this number and its fractional part. Examples:
A mixed fraction can always be represented as an improper fraction and vice versa. Let's go further!
Decimals.
We have already touched on them above, these are examples (1) and (3), now in more detail. Here are examples of decimals: 0.3 0.89 0.001 5.345.
A fraction whose denominator is a power of 10, such as 10, 100, 1000, and so on, is called a decimal. It is not difficult to write the first three indicated fractions as ordinary fractions:
The fourth is a mixed fraction (mixed number):
A decimal has the following notation - withthe integer part begins, then the separator of the integer and fractional parts is a dot or comma and then the fractional part, the number of digits of the fractional part is strictly determined by the dimension of the fractional part: if these are tenths, the fractional part is written as one digit; if thousandths - three; ten-thousandths - four, etc.
These fractions are finite and infinite.
Ending decimal examples: 0.234; 0.87; 34.00005; 5.765.
Examples are endless. For example, the number Pi is an infinite decimal fraction, yet - 0.333333333333…... 0.16666666666…. and others. Also the result of extracting the root from the numbers 3, 5, 7, etc. will be an infinite fraction.
The fractional part can be cyclic (there is a cycle in it), the two examples above are exactly the same, more examples:
0.123123123123…... cycle 123
0.781781781718…... cycle 781
0.0250102501…. cycle 02501
They can be written as 0, (123) 0, (781) 0, (02501).
The number Pi is not a cyclic fraction, like, for example, the root of three.
Below in the examples, words such as “turn over” the fraction will sound - this means that the numerator and denominator are interchanged. In fact, such a fraction has a name - the reciprocal fraction. Examples of reciprocal fractions:
Small summary! Fractions are:
Ordinary (correct and incorrect).
Decimals (finite and infinite).
Mixed (mixed numbers).
That's all!
Sincerely, Alexander.
The numerator, and that by which it is divided is the denominator.
To write a fraction, first write its numerator, then draw a horizontal line under this number, and write the denominator under the line. The horizontal line separating the numerator and denominator is called a fractional bar. Sometimes it is depicted as an oblique "/" or "∕". In this case, the numerator is written to the left of the line, and the denominator to the right. So, for example, the fraction "two-thirds" will be written as 2/3. For clarity, the numerator is usually written at the top of the line, and the denominator at the bottom, that is, instead of 2/3, you can find: ⅔.
To calculate the product of fractions, first multiply the numerator of one fractions to another numerator. Write the result to the numerator of the new fractions. Then multiply the denominators as well. Specify the final value in the new fractions. For example, 1/3? 1/5 = 1/15 (1 × 1 = 1; 3 × 5 = 15).
To divide one fraction by another, first multiply the numerator of the first by the denominator of the second. Do the same with the second fraction (divisor). Or, before performing all the steps, first “flip” the divisor, if it’s more convenient for you: the denominator should be in place of the numerator. Then multiply the denominator of the dividend by the new denominator of the divisor and multiply the numerators. For example, 1/3: 1/5 = 5/3 = 1 2/3 (1 × 5 = 5; 3 × 1 = 3).
Sources:
- Basic tasks for fractions
Fractional numbers allow you to express in different form exact value quantities. You can do the same with fractions. mathematical operations, as with integers: subtraction, addition, multiplication and division. To learn how to decide fractions, it is necessary to remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations after execution, they require reduction of the fractional part of the result.
You will need
- - calculator
Instruction
Look carefully at the numbers. If there are decimals and irregulars among the fractions, it is sometimes more convenient to first perform actions with decimals, and then convert them to the wrong form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which the whole part stands out, lead to the wrong form by multiplying it by the denominator and adding the numerator to the result. Given values will become the new numerator fractions. To extract the whole part from the initially incorrect fractions, divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division becomes the new numerator, the denominator fractions while not changing. For fractions with an integer part, it is possible to perform actions separately, first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summation separately of integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 + (8/12 + 9/12) = 3 + 17/12 = 3 + 1 5/12 = 4 5 /12.
For with fractions. Do the same for the denominators. When dividing one fractions write one fraction on the other, and then multiply its numerator by the denominator of the second. At the same time, the denominator of the first fractions multiplied accordingly by the numerator of the second. At the same time, a kind of reversal of the second fractions(divider). The final fraction will be from the results of multiplying the numerators and denominators of both fractions. Easy to learn fractions, written in the condition in the form of a "four-story" fractions. If it separates two fractions, rewrite them with a ":" delimiter, and continue with normal division.
To get the final result, reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in this case. In this case, there must be integer numbers above and below the line.
note
Don't do arithmetic with fractions that have different denominators. Choose a number such that when the numerator and denominator of each fraction are multiplied by it, as a result, the denominators of both fractions are equal.
When writing fractional numbers, the dividend is written above the line. This quantity is referred to as the numerator of a fraction. Under the line, the divisor, or denominator, of the fraction is written. For example, one and a half kilograms of rice in the form of a fraction will be written as follows: 1 ½ kg of rice. If the denominator of a fraction is 10, it is called a decimal fraction. In this case, the numerator (dividend) is written to the right of the whole part separated by a comma: 1.5 kg of rice. For the convenience of calculations, such a fraction can always be written in the wrong form: 1 2/10 kg of potatoes. To simplify, you can reduce the numerator and denominator values by dividing them by a single whole number. AT this example dividing by 2 is possible. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to do arithmetic with are in the same form.
Shares of a unit and is represented as \frac(a)(b).
Fraction numerator (a)- the number above the line of the fraction and showing the number of shares into which the unit was divided.
Fraction denominator (b)- the number under the line of the fraction and showing how many shares the unit was divided.
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Basic property of a fraction
If ad=bc , then two fractions \frac(a)(b) and \frac(c)(d) are considered equal. For example, fractions will be equal \frac35 and \frac(9)(15), since 3 \cdot 15 = 15 \cdot 9 , \frac(12)(7) and \frac(24)(14), since 12 \cdot 14 = 7 \cdot 24 .
From the definition of the equality of fractions it follows that the fractions will be equal \frac(a)(b) and \frac(am)(bm), since a(bm)=b(am) is a clear example of the use of the associative and commutative properties of multiplication natural numbers In action.
Means \frac(a)(b) = \frac(am)(bm)- looks like this basic property of a fraction.
In other words, we get a fraction equal to the given one by multiplying or dividing the numerator and denominator of the original fraction by the same natural number.
Fraction reduction is the process of replacing a fraction, in which the new fraction is equal to the original, but with a smaller numerator and denominator.
It is customary to reduce fractions based on the main property of a fraction.
For example, \frac(45)(60)=\frac(15)(20)(the numerator and denominator are divisible by the number 3); the resulting fraction can again be reduced by dividing by 5, i.e. \frac(15)(20)=\frac 34.
irreducible fraction is a fraction of the form \frac 34, where the numerator and denominator are relatively prime numbers. The main purpose of fraction reduction is to make the fraction irreducible.
Bringing fractions to a common denominator
Let's take two fractions as an example: \frac(2)(3) and \frac(5)(8) with different denominators 3 and 8 . In order to bring these fractions to a common denominator and first multiply the numerator and denominator of the fraction \frac(2)(3) by 8 . We get the following result: \frac(2 \cdot 8)(3 \cdot 8) = \frac(16)(24). Then multiply the numerator and denominator of the fraction \frac(5)(8) by 3 . We get as a result: \frac(5 \cdot 3)(8 \cdot 3) = \frac(15)(24). So, the original fractions are reduced to a common denominator 24.
Arithmetic operations on ordinary fractions
Addition of ordinary fractions
a) When same denominators The numerator of the first fraction is added to the numerator of the second fraction, leaving the denominator the same. As seen in the example:
\frac(a)(b)+\frac(c)(b)=\frac(a+c)(b);
b) When different denominators fractions are first reduced to a common denominator, and then the numerators are added according to rule a):
\frac(7)(3)+\frac(1)(4)=\frac(7 \cdot 4)(3)+\frac(1 \cdot 3)(4)=\frac(28)(12) +\frac(3)(12)=\frac(31)(12).
Subtraction of ordinary fractions
a) With the same denominators, subtract the numerator of the second fraction from the numerator of the first fraction, leaving the denominator the same:
\frac(a)(b)-\frac(c)(b)=\frac(a-c)(b);
b) If the denominators of the fractions are different, then first the fractions are reduced to a common denominator, and then repeat the steps as in paragraph a).
Multiplication of ordinary fractions
Multiplication of fractions obeys the following rule:
\frac(a)(b) \cdot \frac(c)(d)=\frac(a \cdot c)(b \cdot d),
that is, multiply the numerators and denominators separately.
For example:
\frac(3)(5) \cdot \frac(4)(8) = \frac(3 \cdot 4)(5 \cdot 8)=\frac(12)(40).
Division of ordinary fractions
Fractions are divided in the following way:
\frac(a)(b) : \frac(c)(d)= \frac(ad)(bc),
that is a fraction \frac(a)(b) multiplied by a fraction \frac(d)(c).
Example: \frac(7)(2) : \frac(1)(8)=\frac(7)(2) \cdot \frac(8)(1)=\frac(7 \cdot 8)(2 \cdot 1 )=\frac(56)(2).
Reciprocal numbers
If ab=1 , then the number b is reverse number for number a .
Example: for the number 9, the reverse is \frac(1)(9), as 9 \cdot \frac(1)(9)=1, for the number 5 - \frac(1)(5), as 5 \cdot \frac(1)(5)=1.
Decimals
Decimal is a proper fraction whose denominator is 10, 1000, 10\,000, ..., 10^n .
For example: \frac(6)(10)=0.6;\enspace \frac(44)(1000)=0.044.
In the same way, incorrect numbers with a denominator 10 ^ n or mixed numbers are written.
For example: 5\frac(1)(10)=5.1;\enspace \frac(763)(100)=7\frac(63)(100)=7.63.
In the form of a decimal fraction, any ordinary fraction with a denominator that is a divisor of a certain power of the number 10 is represented.
Example: 5 is a divisor of 100 so the fraction \frac(1)(5)=\frac(1 \cdot 20)(5 \cdot 20)=\frac(20)(100)=0.2.
Arithmetic operations on decimal fractions
Adding decimals
To add two decimal fractions, you need to arrange them so that the same digits and a comma under a comma appear under each other, and then add the fractions as ordinary numbers.
Subtraction of decimals
It works in the same way as addition.
Decimal multiplication
When multiplying decimal numbers just multiply given numbers, not paying attention to commas (as natural numbers), and in the received answer, the comma on the right separates as many digits as there are after the comma in both factors in total.
Let's do the multiplication of 2.7 by 1.3. We have 27 \cdot 13=351 . We separate two digits from the right with a comma (the first and second numbers have one digit after the decimal point; 1+1=2). As a result, we get 2.7 \cdot 1.3=3.51 .
If the result is fewer digits than it is necessary to separate with a comma, then the missing zeros are written in front, for example:
To multiply by 10, 100, 1000, it is necessary to move the decimal point 1, 2, 3 digits to the right in decimal fraction (if necessary, certain number zeros).
For example: 1.47 \cdot 10\,000 = 14,700 .
Decimal division
Dividing a decimal fraction by a natural number is done in the same way as dividing a natural number by a natural number. A comma in the private is placed after the division of the integer part is completed.
If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:
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Consider dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. First of all, we multiply the dividend and the divisor of the fraction by 100, that is, we move the comma to the right in the dividend and divisor by as many characters as there are in the divisor after the decimal point (in this example, two). Then you need to divide the fraction 257.6 by the natural number 112, that is, the problem is reduced to the case already considered:
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It happens that the final decimal fraction is not always obtained when dividing one number by another. The result is an infinite decimal. In such cases, go to ordinary fractions.
2.8: 0.09= \frac(28)(10) : \frac (9)(100)= \frac(28 \cdot 100)(10 \cdot 9)=\frac(280)(9)= 31 \frac(1)(9).
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.
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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 a common 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 smaller 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 the well-known saying: we say a fraction - we mean 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 the 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. 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 an expense, a 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. 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. A part of a unit or several of its parts is called a simple or ordinary fraction. The number of equal parts into which the unit is divided is called the denominator, and the number of parts taken is called the numerator. The fraction is written as: In this case, a is the numerator, b is the denominator. If the numerator is less than the denominator, then the fraction is less than 1 and is called a proper fraction. If the numerator is greater than the denominator, then the fraction is greater than 1, then the fraction is called an improper fraction. If the numerator and denominator of a fraction are equal, then the fraction is equal. 1. If the numerator can be divided by the denominator, then this fraction is equal to the quotient of division: If the division is performed with a remainder, then this improper fraction can be represented by a mixed number, for example: Then 9 is an incomplete quotient (the integer part of the mixed number), To convert a mixed number to a fraction, multiply the integer part of the mixed number by the denominator and add the numerator of the fractional part. The result obtained will be the numerator of an ordinary fraction, and the denominator will remain the same. Fraction expansion. The value of a fraction does not change if its numerator and denominator are multiplied by the same non-zero number. Fraction reduction. The value of a fraction does not change if its numerator and denominator are divided by the same non-zero number. Fraction comparison. Of two fractions with the same numerator, the larger one is the one with the smaller denominator: Of two fractions with the same denominators, the one with the larger numerator is greater: To compare fractions that have different numerators and denominators, it is necessary to expand them, that is, bring them to a common denominator. Consider, for example, the following fractions: Addition and subtraction of fractions. If the denominators of fractions are the same, then in order to add the fractions, it is necessary to add their numerators, and in order to subtract the fractions, it is necessary to subtract their numerators. The resulting sum or difference will be the numerator of the result, while the denominator will remain the same. If the denominators of the fractions are different, you must first reduce the fractions to a common denominator. When adding mixed numbers, their integer and fractional parts are added separately. When subtracting mixed numbers, you must first convert them to the form of improper fractions, then subtract from one another, and then again bring the result, if necessary, to the form of a mixed number. Multiplication of fractions. To multiply fractions, you need to multiply their numerators and denominators separately and divide the first product by the second. Division of fractions. To divide a number by a fraction, you need to multiply that number by its reciprocal. Decimal is the result of dividing one by ten, one hundred, one thousand, etc. parts. First, the integer part of the number is written, then the decimal point is placed on the right. The first digit after the decimal point means the number of tenths, the second - the number of hundredths, the third - the number of thousandths, etc. The numbers after the decimal point are called decimal places. For example: Properties: A periodic decimal contains an infinitely repeating group of digits called a period: 0.321321321321…=0,(321) Adding and subtracting decimals is done in the same way as adding and subtracting integers, you only need to write the corresponding decimal places one under the other. Multiplication of decimal fractions is carried out in several stages: for example: The sum of the numbers of decimal places in the factors is: 2+1=3. Now you need to count 3 digits from the end of the resulting number and put a decimal point: 0.675. Division of decimals. Dividing a decimal by an integer: if the dividend is less than the divisor, then you need to write zero in the integer part of the quotient and put a decimal point after it. Then, without taking into account the decimal point of the dividend, add the next digit of the fractional part to its integer part and again compare the resulting integer part of the dividend with the divisor. If the new number is again less than the divisor, the operation must be repeated. This process is repeated until the resulting dividend is greater than the divisor. After that, division is performed as for integers. If the dividend is greater than or equal to the divisor, first we divide its integer part, write the result of the division in the quotient and put a decimal point. After that, the division continues, as in the case of integers. Dividing one decimal fraction into another: first, the decimal points in the dividend and divisor are transferred by the number of decimal places in the divisor, that is, we make the divisor an integer, and the actions described above are performed. In order to convert a decimal fraction to an ordinary one, it is necessary to take the number after the decimal point as the numerator, and take the k-th power of ten as the denominator (k is the number of decimal places). The non-zero integer part is preserved in the common fraction; the zero integer part is omitted. In order to convert an ordinary fraction to a decimal, it is necessary to divide the numerator by the denominator in accordance with the rules of division. A percentage is a hundredth of a unit, for example: 5% means 0.05. A ratio is the quotient of dividing one number by another. Proportion is the equality of two ratios. For example: The main property of the proportion: the product of the extreme members of the proportion is equal to the product of its middle members, that is, 5x30 = 6x25. Two mutually dependent quantities are called proportional if the ratio of their quantities remains unchanged (proportionality coefficient). Thus, the following arithmetic operations are revealed. The set of rational numbers includes positive and negative numbers (whole and fractional) and zero. A more precise definition of rational numbers, adopted in mathematics, is as follows: a number is called rational if it can be represented as an ordinary irreducible fraction of the form:, where a and b are integers. For a negative number, the absolute value (modulus) is a positive number obtained by changing its sign from "-" to "+"; for a positive number and zero, the number itself. To designate the modulus of a number, two straight lines are used, inside which this number is written, for example: |–5|=5. Let the modulus of a number be given A monomial is the product of two or more factors, each of which is either a number, or a letter, or the power of a letter: 3 x a x b. The coefficient is most often called only a numerical factor. Monomials are said to be similar if they are the same or differ only in coefficients. The degree of a monomial is the sum of the exponents of all its letters. If there are similar ones among the sum of monomials, then the sum can be reduced to a simpler form: 3 x a x b + 6 x a \u003d 3 x a x (b + 2). This operation is called coercion of like terms or parentheses. A polynomial is an algebraic sum of monomials. The degree of a polynomial is the largest of the degrees of the monomials included in the given polynomial. There are the following formulas for abbreviated multiplication: Factoring methods: An algebraic fraction is an expression of the form , where A and B can be a number, a monomial, a polynomial. If two expressions (numeric and alphabetic) are connected by the sign "=", then they are said to form equality. Any true equality, valid for all admissible numerical values of the letters included in it, is called an identity. An equation is a literal equality that is valid for certain values of the letters included in it. These letters are called unknowns (variables), and their values, at which the given equation becomes an identity, are called the roots of the equation. Solving an equation means finding all its roots. Two or more equations are said to be equivalent if they have the same roots. Main types of algebraic equations: The linear equation has ax + b = 0: Equation xn = a, n N: Basic identical transformations: replacement of one expression by another, identically equal to it; transfer of the terms of the equation from one side to the other with opposite signs; multiplication or division of both parts of the equation by the same expression (number) other than zero. A linear equation with one unknown is an equation of the form: ax+b=0, where a and b are known numbers, and x is an unknown value. Systems of two linear equations with two unknowns have the form: Where a, b, c, d, e, f are given numbers; x, y are unknown. Numbers a, b, c, d - coefficients for unknowns; e, f - free members. The solution to this system of equations can be found by two main methods: the substitution method: from one equation we express one of the unknowns through the coefficients and the other unknown, and then we substitute it into the second equation, solving the last equation, we first find one unknown, then we substitute the found value into the first equation and find the second unknown; method of adding or subtracting one equation from another. Operations with roots: The arithmetic root of the nth degree of a non-negative number a is a non-negative number whose n-th power is equal to a. The algebraic root of the nth degree from a given number is the set of all roots from this number. Irrational numbers, unlike rational ones, cannot be represented as an ordinary irreducible fraction of the form m/n, where m and n are integers. These are numbers of a new type that can be calculated with any precision, but cannot be replaced by a rational number. They may appear as a result of geometric measurements, for example: the ratio of the length of the diagonal of a square to the length of its side is equal. A quadratic equation is an algebraic equation of the second degree ax2+bx+c=0, where a, b, c are given numerical or alphabetic coefficients, x is an unknown. If we divide all the terms of this equation by a, as a result we get x2+px+q=0 - the reduced equation p=b/a, q=c/a. Its roots are found by the formula: If b2-4ac>0 then there are two distinct roots, b2-4ac=0 then there are two equal root; b2-4ac Equations containing modules Main types of equations containing modules: Positive and negative fractions
Actions with fractions
1 - remainder (numerator of the fractional part),
5 is the denominator.Actions with fractions
for example:
for example: 
Decimal Properties
Operations with decimals
For example:
For example: 
For example:
Absolute value properties
, for which the properties are valid: 
1) |f(x)| = |g(x)|;
2) |f(x)| = g(x);
3) f1(x)|g1(x)| + f2(x)|g2(x)| + … + fn(x)|gn(x)| =0, n N, where f(x), g(x), fk(x), gk(x) are given functions.
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