A number that can be written as a fraction. Common fractions

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

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

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

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

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

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

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

Fraction. Numerator and denominator of fraction

Definition 1. A fraction is one or more identical shares (parts) of an object or some quantity.

A fraction is written using two natural numbers, one of which stands above the horizontal line, and the second below it.

Definition 2. The number above the line is called numerator of the fraction. The number below the line is called denominator of the fraction The numerator and denominator are called terms of the fraction.

Denominator fractions shows for how long we have equal shares divide object or quantity, and numerator fractions shows How many such shares taken.

For example, fraction

in which the numerator is equal to 8 and the denominator is equal to 17, means that we divide an object or quantity into 17 equal shares (parts) and take 8 such shares.

Example 1. There are 25 students in the class, some of whom attend the theater club. How many students go to the drama club?

Solution . To solve the example, you need to divide 25 students into 5 parts and take 2 such parts.

Answer . 10 students.

Example 2. On the first day of the hike, the tourist walked the intended route, and on the second day - the remaining 24 kilometers. How many kilometers did the tourist walk in total?

Solution . The entire route is divided into 7 equal parts, 3 of which the tourist passed on the first day (Fig. 1).

1 day

Day 2

1
day

2
day

From Figure 1 it can be seen that 24 kilometers make up 4 of the 7 parts of the route. So 1 part of the route is equal to

24: 4 = 6 (km),

and the whole route is equal

Answer . 42 kilometers.

Comment. If it is not indicated from what object or what size the fraction is taken, then it is considered that the fraction is taken from the number 1.

Term fraction has synonyms: simple fraction, common fraction, rational fraction, a fractional number .

Proper and improper fractions. Mixed numbers

Definition 3. If a fraction has a numerator less than its denominator, it is called proper fraction. Otherwise - improper fraction.

From this definition, in particular, it follows that a proper fraction is less than one, and an improper fraction is greater than one or equal to one.

Example 3. - proper fraction, and - improper fractions.

An improper fraction can always be expressed as the sum of a whole number and a proper fraction. This operation is called highlighting the whole part from an improper fraction and is carried out by dividing with the remainder the numerator of the improper fraction by the denominator.

Example 4.

The number is an example mixed number. The integer 2 and a proper fraction are called an integer and fractional part of a mixed number respectively.

Any mixed number can always be converted into an improper fraction, for example,

Basic properties of fractions, reduction of fractions, irreducible fraction

The main property of a fraction call the following

Statement . A fraction becomes an equal fraction when its numerator and denominator are multiplied or divided by the same number.

Definition 4. An operation in which the numerator and denominator of a fraction are divided by the same number is called reducing a fraction.

Example 4.

We use fractions all the time in life. For example, when we eat cake with friends. The cake can be divided into 8 equal parts or 8 shares. Share- This is an equal part of something whole. Four friends ate a piece of cake. Four taken from eight pieces can be written mathematically in the form common fraction$\frac(4)(8)$, the fraction “four eighths” or “four divided by eight” is read. A common fraction is also called simple fraction.

The fraction bar replaces the division:
$4 \div 8 = \frac(4)(8)$
We wrote down the shares in fractions. In literal form it will be like this:
$\bf m \div n = \frac(m)(n)$

4 – numerator or dividend, is located above the fractional line and shows how many parts or shares were taken from the total.
8 – denominator or divisor, is located below the fraction line and shows the total number of parts or shares.

If we look closely, we will see that the friends ate half the cake or one part of two. Let's write it as an ordinary fraction $\frac(1)(2)$, read “one second”.

Let's look at another example:
There is a square. The square was divided into 5 equal parts. Two parts were painted over. Write down the fraction for the shaded parts? Write down the fraction for the unshaded parts?

Two parts were painted over, and there are five parts in total, so the fraction will look like $\frac(2)(5)$, read as “two-fifths”.
Three parts were not painted over, there are five parts in total, so we write the fraction as $\frac(3)(5)$, the fraction reads “three-fifths”.

Let's divide the square into smaller squares and write down the fractions for the shaded and unshaded parts.

There are 6 painted parts, and there are 25 parts in total. We get the fraction $\frac(6)(25)$, the fraction is read “six twenty-fifths”.
There are 19 parts not painted over, but a total of 25 parts. We get the fraction $\frac(19)(25)$, the fraction read “nineteen twenty-fifths”.

There are 4 parts painted over, and there are 25 parts in total. We get the fraction $\frac(4)(25)$, the fraction read “four twenty-fifths”.
There are 21 parts not painted over, but only 25 parts. We get the fraction $\frac(21)(25)$, the fraction read “twenty-one twenty-fifths”.

Any natural number can be represented as a fraction. For example:

$5 = \frac(5)(1)$
$\bf m = \frac(m)(1)$

Any number is divisible by one, so this number can be represented as a fraction.

Questions on the topic “common fractions”:
What is a share?
Answer: share- This is an equal part of something whole.

What does the denominator show?
Answer: the denominator shows how many parts or shares the total is divided into.

What does the numerator show?
Answer: the numerator shows how many parts or shares were taken.

The road was 100m. Misha walked 31m. Write down the expression as a fraction: how far has Misha walked?
Answer:$\frac(31)(100)$

What is a common fraction?
Answer: A common fraction is the ratio of the numerator to the denominator, where the numerator is less than the denominator. Example, ordinary fractions $\frac(1)(4), \frac(3)(7), \frac(5)(13), \frac(9)(11)…$

How to convert a natural number to a common fraction?
Answer: any number can be written as a fraction, for example, $5 = \frac(5)(1)$

Task #1:
We bought 2kg 700g melon. They cut off $\frac(2)(9)$ melons for Misha. What is the mass of the cut piece? How many grams of melon are left?

Solution:
Let's convert kilograms to grams.
2kg = 2000g
2000g + 700g = 2700g total weight of a melon.

They cut off $\frac(2)(9)$ melons for Misha. The denominator contains the number 9, which means the melon is divided into 9 parts.
2700: 9 =300g weight of one piece.
The numerator contains the number 2, which means you need to give Misha two pieces.
300 + 300 = 600g or 300 ⋅ 2 = 600g is how much melon Misha ate.

To find the mass of melon left, you need to subtract the mass eaten from the total mass of the melon.
2700 - 600 = 2100g melon left.

Definition

A number made up of one or more equal parts of a unit is called ordinary fraction or fraction.

Such fractions are written using two natural numbers and a horizontal line called fraction line. Sometimes it is not a horizontal line, but an oblique line. Fractions are read like this: first the numerator is called, then the denominator.

For example.$\frac(3)(4)=3 / 4$ . Reads: three quarters.

Numerator and denominator of fraction

Definition

Under the line of the fraction, write a number showing how many shares (parts) the unit is divided into. It's called denominator of the fraction.

A number is written above the fractional line indicating how many such parts are taken. This number is called numerator of the fraction.

For example. The fraction $\frac(2)(3)$ (two thirds) has a numerator of 2 and a denominator of 3.

For example. Figure 1 shows the fraction $\frac(3)(4)$ . The denominator of the fraction, which is equal to 4, indicates that the whole was divided into four parts (shares), and the numerator, which is equal to 3, indicates that three of these four parts were taken.

The fraction bar essentially replaces the division sign. That is, the quotient of dividing one number by another is equal to a fraction, the numerator of which is equal to the dividend, and the denominator is equal to the divisor.

For example.$3: 5=\frac(3)(5), \frac(7)(8)=7: 8$

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 line 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.