Common fractions, regular and improper, mixed and composite. Fractions, ordinary fractions: definitions, notations, examples, actions with fractions What fractions exist

Fractions of a unit and is represented as \frac(a)(b).

Numerator of fraction (a)- the number located above the fraction line and showing the number of shares into which the unit was divided.

Fraction denominator (b)- the number located under the fraction line and showing how many parts the unit is divided into.

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The main property of a fraction

If ad=bc then two fractions \frac(a)(b) And \frac(c)(d) are considered equal. For example, the 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 equality of fractions it follows that the fractions will be equal \frac(a)(b) And \frac(am)(bm), since a(bm)=b(am) - clear example application of the associative and commutative properties of multiplication natural numbers In action.

Means \frac(a)(b) = \frac(am)(bm)- this is what it looks like main 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.

Reducing a fraction is the process of replacing a fraction in which the new fraction is equal to the original one, but with a smaller numerator and denominator.

It is customary to reduce fractions based on the basic property of the fraction.

For example, \frac(45)(60)=\frac(15)(20)(numerator and denominator are divided by the number 3); the resulting fraction can again be reduced by dividing by 5, that is \frac(15)(20)=\frac 34.

Irreducible fraction is a fraction of the form \frac 34, where the numerator and denominator are mutually prime numbers. The main purpose of reducing a fraction is to make the fraction irreducible.

Reducing 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, we 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 we multiply the numerator and denominator of the fraction \frac(5)(8) by 3. As a result we get: \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) If the denominators are the same, the numerator of the first fraction is added to the numerator of the second fraction, leaving the denominator the same. As you can see in the example:

\frac(a)(b)+\frac(c)(b)=\frac(a+c)(b);

b) For 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).

Subtracting fractions

a) If the denominators are the same, 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 brought to a common denominator, and then the actions are repeated as in point a).

Multiplying common fractions

Multiplying fractions obeys the following rule:

\frac(a)(b) \cdot \frac(c)(d)=\frac(a \cdot c)(b \cdot d),

that is, they 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).

Dividing 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 reciprocal number for the number a.

Example: for the number 9 the reciprocal is \frac(1)(9), because 9\cdot\frac(1)(9)=1, for the number 5 - \frac(1)(5), because 5\cdot\frac(1)(5)=1.

Decimals

Decimal called 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.

Irregular numbers with a denominator of 10^n or mixed numbers are written in the same way.

For example: 5\frac(1)(10)=5.1;\enspace \frac(763)(100)=7\frac(63)(100)=7.63.

Any ordinary fraction with a denominator that is a divisor of a certain power of 10 is represented as a decimal fraction.

Example: 5 is a divisor of 100, so it is a fraction \frac(1)(5)=\frac(1 \cdot 20)(5 \cdot 20)=\frac(20)(100)=0.2.

Arithmetic operations on decimals

Adding Decimals

To add two decimal fractions, you need to arrange them so that there are identical digits under each other and a comma under the comma, and then add the fractions like ordinary numbers.

Subtracting Decimals

It is performed in the same way as addition.

Multiplying Decimals

When multiplying decimal numbers, it is enough to multiply the given numbers, not paying attention to commas (like natural numbers), and in the resulting answer, a comma on the right separates as many digits as there are after the decimal point in both factors in total.

Let's multiply 2.7 by 1.3. We have 27 \cdot 13=351 . We separate two digits on 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 resulting result contains fewer digits than need to be separated by a comma, then the missing zeros are written in front, for example:

To multiply by 10, 100, 1000, you need to move the decimal point 1, 2, 3 digits to the right (if necessary, a certain number of zeros are assigned to the right).

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. The comma in the quotient is placed after the division of the whole part is completed.

If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:

Let's look at dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. First of all, let's multiply the dividend and divisor of the fraction by 100, that is, move the decimal point to the right in the dividend and divisor by as many digits 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:

It happens that the final decimal fraction is not always obtained when dividing one number by another. The result is an infinite decimal fraction. In such cases, we move on 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).

Fraction- a form of representing a number in mathematics. The fraction bar denotes the division operation. Numerator fraction is called the dividend, and denominator- divider. For example, in a fraction the numerator is 5 and the denominator is 7.

Correct A fraction is called in which the modulus of the numerator is greater than the modulus of the denominator. If a fraction is proper, then the modulus of its value is always less than 1. All other fractions are wrong.

The fraction is called mixed, if it is written as an integer and a fraction. This is the same as the sum of this number and the fraction:

The main property of a fraction

If the numerator and denominator of a fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,

Reducing fractions to a common denominator

To bring two fractions to a common denominator, you need:

1. Multiply the numerator of the first fraction by the denominator of the second
2. Multiply the numerator of the second fraction by the denominator of the first
3. Replace the denominators of both fractions with their product

Operations with fractions

Addition. To add two fractions you need

1. Add the new numerators of both fractions and leave the denominator unchanged

Example:

Subtraction. To subtract one fraction from another, you need

1. Reduce fractions to a common denominator
2. Subtract the numerator of the second from the numerator of the first fraction, and leave the denominator unchanged

Example:

Multiplication. To multiply one fraction by another, multiply their numerators and denominators.

Now that we have learned how to add and multiply individual fractions, we can look at more complex structures. For example, what if the same problem involves adding, subtracting, and multiplying fractions?

First of all, you need to convert all fractions to improper ones. Then we perform the required actions sequentially - in the same order as for ordinary numbers. Namely:

1. Exponentiation is done first - get rid of all expressions containing exponents;
2. Then - division and multiplication;
3. The last step is addition and subtraction.

Of course, if there are parentheses in the expression, the order of operations changes - everything that is inside the parentheses must be counted first. And remember about improper fractions: you need to highlight the whole part only when all other actions have already been completed.

Let's convert all the fractions from the first expression to improper ones, and then perform the following steps:

Now let's find the value of the second expression. There are no fractions with an integer part, but there are parentheses, so first we perform addition, and only then division. Note that 14 = 7 · 2. Then:

Finally, consider the third example. There are brackets and a degree here - it is better to count them separately. Considering that 9 = 3 3, we have:

Pay attention to the last example. To raise a fraction to a power, you must separately raise the numerator to this power, and separately, the denominator.

You can decide differently. If we recall the definition of a degree, the problem will be reduced to the usual multiplication of fractions:

Multistory fractions

Until now, we have considered only “pure” fractions, when the numerator and denominator are ordinary numbers. This is quite consistent with the definition of a number fraction given in the very first lesson.

But what if you put a more complex object in the numerator or denominator? For example, another numerical fraction? Such constructions arise quite often, especially when working with long expressions. Here are a couple of examples:

There is only one rule for working with multi-level fractions: you must get rid of them immediately. Removing “extra” floors is quite simple, if you remember that the slash means the standard division operation. Therefore, any fraction can be rewritten as follows:

Using this fact and following the procedure, we can easily reduce any multi-story fraction to an ordinary one. Take a look at the examples:

Task. Convert multistory fractions to ordinary ones:

In each case, we rewrite the main fraction, replacing the dividing line with a division sign. Also remember that any integer can be represented as a fraction with a denominator of 1. That is 12 = 12/1; 3 = 3/1. We get:

IN last example the fractions were canceled before the final multiplication.

Specifics of working with multi-level fractions

There is one subtlety in multi-level fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:

1. The numerator contains the single number 7, and the denominator contains the fraction 12/5;
2. The numerator contains the fraction 7/12, and the denominator contains the separate number 5.

So, for one recording we got two completely different interpretations. If you count, the answers will also be different:

To ensure that the record is always read unambiguously, use a simple rule: the dividing line of the main fraction must be longer than the line of the nested fraction. Preferably several times.

If you follow this rule, then the above fractions should be written as follows:

Yes, it may be unsightly and takes up too much space. But you will count correctly. Finally, a couple of examples where multi-story fractions actually arise:

Task. Find the meanings of the expressions:

So, let's work with the first example. Let's convert all fractions to improper ones, and then perform addition and division operations:

Let's do the same with the second example. Let's convert all fractions to improper ones and perform the required operations. In order not to bore the reader, I will omit some obvious calculations. We have:

Due to the fact that the numerator and denominator of the basic fractions contain sums, the rule for writing multi-story fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fraction form to perform division.

I will also note that in both examples the fraction bar actually replaces the parentheses: first of all, we found the sum, and only then the quotient.

Some will say that the transition to improper fractions in the second example was clearly redundant. Perhaps this is true. But by doing this we insure ourselves against mistakes, because next time the example may turn out to be much more complicated. Choose for yourself what is more important: speed or reliability.

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 the 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 slices. 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 shares. 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 does not hold.

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 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 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, the improper fraction 9/9 essentially gives the 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 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).

Numerator of fraction- this is the number that appears in the notation of an ordinary fraction above the fraction line, that is, on top. The numerator shows the number of shares.

Fraction denominator- this is the number that appears in the notation of a fraction under the fraction line, that is, below. The denominator shows what fractions these are and how many equal parts the unit is divided into.

Fractional bar is a horizontal line in a fraction that separates the numerator and denominator from each other.

Together, the numerator and denominator of a fraction are called members of the fraction.

How to read the notation of common fractions

The writing of ordinary fractions reads like this: first the numerator is called, then the denominator. When reading the numerator, it should always answer the question: how many shares?. For example, one , two , three etc. When reading the denominator, it should always answer one of the questions: which? or which ones?. Which of these questions he must answer depends on the number of shares. If the numerator contains the number 1, then the denominator will answer the question which?, if the number is greater than one, then the question which ones?. If the numerator contains the number 0, then the denominator will always answer the question which ones? .

All ordinary fractions are read using this rule.

Example 1. Read the fraction, name the numerator and denominator.

Solution:

The fraction reads like this: one eighth(how many shares are taken? - one, which one? - eighth). Numerator - one(or unit), denominator - eight .

Example 2. Read the fraction.

Solution:

The fraction reads like this: three sevenths(how many shares are taken? - three, what three? - seventh).