Definition of a common fraction
Definition 1
Common fractions used to describe the number of shares. Let's look at an example that can be used to define a common fraction.
The apple was divided into $8$ shares. In this case, each share represents one-eighth of a whole apple, i.e. $\frac(1)(8)$. Two shares are denoted by $\frac(2)(8)$, three shares by $\frac(3)(8)$, etc., and $8$ shares by $\frac(8)(8)$ . Each of the entries presented is called ordinary fraction.
Let's give general definition ordinary fraction.
Definition 2
Common fraction is called a notation of the form $\frac(m)(n)$, where $m$ and $n$ are any natural numbers.
You can often find the following notation for a common fraction: $m/n$.
Example 1
Examples of common fractions:
\[(3)/(4), \frac(101)(345),\ \ (23)/(5), \frac(15)(15), (111)/(81).\]
Note 1
Numbers $\frac(\sqrt(2))(3)$, $-\frac(13)(37)$, $\frac(4)(\frac(2)(7))$, $\frac( 2,4)(8,3)$ are not ordinary fractions, because do not fit the above definition.
Numerator and denominator
A common fraction consists of a numerator and a denominator.
Definition 3
Numerator the ordinary fraction $\frac(m)(n)$ is called natural number$m$, which shows the number of equal parts taken from a single whole.
Definition 4
Denominator An ordinary fraction $\frac(m)(n)$ is a natural number $n$, which shows how many equal parts the whole whole is divided into.
Figure 1.
The numerator is located above the fraction line, and the denominator is located below the fraction line. For example, the numerator of the common fraction $\frac(5)(17)$ is the number $5$, and the denominator is the number $17$. The denominator shows that the item is divided into $17$ shares, and the numerator shows that $5$ such shares were taken.
Natural number as a fraction with denominator 1
The denominator of a common fraction can be one. In this case, the object is considered to be indivisible, i.e. represents a single whole. The numerator of such a fraction shows how many whole objects are taken. An ordinary fraction of the form $\frac(m)(1)$ has the meaning of a natural number $m$. Thus, we obtain the well-founded equality $\frac(m)(1)=m$.
If we rewrite the equality in the form $m=\frac(m)(1)$, then it will make it possible to represent any natural number $m$ as an ordinary fraction. For example, the number $5$ can be represented as a fraction $\frac(5)(1)$, the number $123\456$ can be represented as a fraction $\frac(123\456)(1)$.
Thus, any natural number $m$ can be represented as an ordinary fraction with a denominator $1$, and any ordinary fraction of the form $\frac(m)(1)$ can be replaced by a natural number $m$.
Fractional bar as a division sign
Representing an object in the form of $n$ parts is a division into $n$ equal parts. After dividing an item into $n$ shares, it can be divided equally between $n$ people - each will receive one share.
Let there be $m$ identical items, divided into $n$ parts. These $m$ items can be divided equally among $n$ people by giving each person one share of each of the $m$ items. In this case, each person will receive $m$ shares of $\frac(1)(n)$, which give the common fraction $\frac(m)(n)$. We find that the common fraction $\frac(m)(n)$ can be used to denote the division of $m$ items between $n$ people.
The connection between ordinary fractions and division is expressed in the fact that the fraction bar can be understood as a division sign, i.e. $\frac(m)(n)=m:n$.
An ordinary fraction makes it possible to write down the result of dividing two natural numbers for which a whole division is not performed.
Example 2
For example, the result of dividing $7$ apples by $9$ people can be written as $\frac(7)(9)$, i.e. everyone will receive seven-ninths of an apple: $7:9=\frac(7)(9)$.
Equal and unequal fractions, comparison of fractions
The result of comparing two ordinary fractions can be either their equality or their non-equality. When ordinary fractions are equal, they are called equal; otherwise, ordinary fractions are called unequal.
equal, if the equality $a\cdot d=b\cdot c$ is true.
The ordinary fractions $\frac(a)(b)$ and $\frac(c)(d)$ are called unequal, if the equality $a\cdot d=b\cdot c$ does not hold.
Example 3
Find out whether the fractions $\frac(1)(3)$ and $\frac(2)(6)$ are equal.
The equality is satisfied, which means that the fractions $\frac(1)(3)$ and $\frac(2)(6)$ are equal: $\frac(1)(3)=\frac(2)(6)$.
This example can be considered using apples: one of two identical apples is divided into three equal shares, the second into $6$ shares. It can be seen that two-sixths of an apple constitutes a $\frac(1)(3)$ share.
Example 4
Check whether the ordinary fractions $\frac(3)(17)$ and $\frac(4)(13)$ are equal.
Let's check whether the equality $a\cdot d=b\cdot c$ holds:
\ \
The equality does not hold, which means that the fractions $\frac(3)(17)$ and $\frac(4)(13)$ are not equal: $\frac(3)(17)\ne \frac(4)(13) $.
By comparing two common fractions and finding that they are not equal, you can find out which is larger and which is smaller than the other. To do this, use the rule for comparing ordinary fractions: you need to bring the fractions to a common denominator and then compare their numerators. Whichever fraction has a larger numerator, that fraction will be the larger one.
Fractions on a coordinate ray
All fractional numbers that correspond to ordinary fractions can be displayed on a coordinate ray.
To mark a point on a coordinate ray that corresponds to the fraction $\frac(m)(n)$, it is necessary to plot $m$ segments from the origin of coordinates in the positive direction, the length of which is $\frac(1)(n)$ a fraction of a unit segment . Such segments are obtained by dividing a unit segment into $n$ equal parts.
To display a fractional number on a coordinate ray, you need to divide the unit segment into parts.
Figure 2.
Equal fractions are described by the same fractional number, i.e. equal fractions represent the coordinates of the same point on the coordinate ray. For example, the coordinates $\frac(1)(3)$, $\frac(2)(6)$, $\frac(3)(9)$, $\frac(4)(12)$ describe the same the same point on the coordinate ray, since all written fractions are equal.
If a point is described by a coordinate with a larger fraction, then it will be located to the right on a horizontal coordinate ray directed to the right from the point whose coordinate is minor fraction. For example, because fraction $\frac(5)(6)$ more fractions$\frac(2)(6)$, then the point with coordinate $\frac(5)(6)$ is located to the right of the point with coordinate $\frac(2)(6)$.
Likewise, a point with a smaller coordinate will lie to the left of a point with a larger coordinate.
In mathematics, a fraction is a number consisting of one or more parts (fractions) of a unit. According to the form of recording, fractions are divided into ordinary (example \frac(5)(8)) and decimal (for example 123.45).
Definition. Common fraction (or simple fraction)
Ordinary (simple) fraction is called a number of the form \pm\frac(m)(n) where m and n are natural numbers. The number m is called numerator this fraction, and the number n is its denominator.
A horizontal or slash indicates a division sign, that is, \frac(m)(n)=()^m/n=m:n
Common fractions are divided into two types: proper and improper.
Definition. Proper and improper fractions
Correct A fraction whose numerator is less than its denominator is called a fraction. For example, \frac(9)(11) , because 9
Wrong a fraction whose numerator is greater than or equal to equal to modulus denominator. Such a fraction is a rational number, modulo greater than or equal to one. An example would be the fractions \frac(11)(2) , \frac(2)(1) , -\frac(7)(5) , \frac(1)(1)
Along with the improper fraction, there is another representation of the number, which is called mixed fraction(mixed number). This is not an ordinary fraction.
Definition. Mixed fraction (mixed number)
Mixed fraction is a fraction written as a whole number and proper fraction and is understood as the sum of this number and a fraction. For example, 2\frac(5)(7)
(record in the form mixed number) 2\frac(5)(7)=2+\frac(5)(7)=\frac(14)(7)+\frac(5)(7)=\frac(19)(7) (record in the form improper fraction)
A fraction is just a representation of a number. The same number can correspond different fractions, both ordinary and decimal. Let us form a sign for the equality of two ordinary fractions.
Definition. Sign of equality of fractions
The two fractions \frac(a)(b) and \frac(c)(d) are equal, if a\cdot d=b\cdot c . For example, \frac(2)(3)=\frac(8)(12) since 2\cdot12=3\cdot8
From this attribute follows the main property of a fraction.
Property. The main property of a fraction
If the numerator and denominator of a given fraction are multiplied or divided by the same number, not equal to zero, you get a fraction equal to the given one.
\frac(A)(B)=\frac(A\cdot C)(B\cdot C)=\frac(A:K)(B:K);\quad C \ne 0,\quad K \ne 0
Using the basic property, fractions can be replaced given fraction another fraction equal to the given one, but with a smaller numerator and denominator. This replacement is called fraction reduction. For example, \frac(12)(16)=\frac(6)(8)=\frac(3)(4) (here the numerator and denominator were divided first by 2, and then by 2 more). A fraction can be reduced if and only if its numerator and denominator are not mutually exclusive. prime numbers. If the numerator and denominator of a given fraction are mutually prime, then the fraction cannot be reduced, for example, \frac(3)(4) is an irreducible fraction.
Rules for positive fractions:
From two fractions With same denominators The fraction whose numerator is greater is greater. For example, \frac(3)(15)
From two fractions with the same numerators The larger is the fraction whose denominator is smaller. For example, \frac(4)(11)>\frac(4)(13) .
To compare two fractions with different numerators and denominators, you must convert both fractions so that their denominators are the same. This transformation is called reducing fractions to a common denominator.
Common fraction
Quarters
- Orderliness. a And b there is a rule that allows you 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 relationship as two integers and ; two non-positive numbers a And b are related by the same relationship 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. Order adding three rational numbers 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 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 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.
In the process of such a traversal, each new rational number is associated with another natural number. That is, the fractions 1/1 are assigned to the number 1, the fractions 2/1 to the number 2, etc. It should be noted that only irreducible fractions. 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 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 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.
Numerator and denominator of a fraction. Types of fractions. Let's continue looking at fractions. First, a small disclaimer - while we are considering fractions and corresponding examples with them, for now we will only work with its numerical representation. There are also fractional literal expressions(with and without numbers).However, all “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 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 have confusion about what they call it.
Here's a trick on how to forever remember where the numerator is and where the denominator is. This technique is associated with verbal-figurative association. Imagine a jar with muddy water. It is known that as water settles, clean water remains on top, and turbidity (dirt) settles, remember:
CHISS melt water ABOVE (CHISS litel top)
Grya Z33NN water is BELOW (ZNNNN amenator is below)
So, as soon as the need arises to remember where the numerator is and where the denominator is, we immediately visually imagined a jar of settled water with PURE water, and below is dirty water. There are other memory tricks, if they help you, then good.
Examples of common 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 of the action of division. This action is simply recorded in this form. That is, the top number (numerator) is divided by the bottom (denominator):
In addition, there is another form of notation - 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 like this:
The result of division is how this number is known.
We figured it out - THIS IS A FRACTION!!!
As you have already noticed, a common fraction can have a numerator less than the denominator, may be greater than the denominator and may be equal to it. There are many important points, which are intuitively understandable, without any theoretical refinements. For example:
1. Fractions 1 and 3 can be written as 0.5 and 0.01. Let's jump ahead a little - these are decimal fractions, we'll talk about them a little lower.
2. Fractions 4 and 6 result in the integer 45:9=5, 11:1 = 11.
3. The fraction 5 results in one 155:155 = 1.
What conclusions suggest themselves? Next:
1. The numerator when divided by the denominator can give final number. It may not work, divide with a column 7 by 13 or 17 by 11 - no way! You can divide endlessly, but we’ll also talk about this below.
2. A fraction can result in a whole number. 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 integer as a fraction - the number itself is in the numerator, the unit is in the denominator:
3. We can always represent a unit as a fraction with any denominator:
*These points are extremely important for working with fractions during calculations and transformations.
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 its 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 move on!
Decimal fractions.
We have already touched on them above, these are examples (1) and (3), now in more detail. Here are examples of decimal fractions: 0.3 0.89 0.001 5.345.
A fraction whose denominator is a power of 10, such as 10, 100, 1000, etc., is called a decimal. It is not difficult to write the first three indicated fractions in the form of ordinary fractions:
The fourth is a mixed fraction (mixed number):
The decimal fraction has the following form records - fromthe whole part begins, then the separator of the whole 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 in one digit; if thousandths - three; ten thousandths - four, etc.
These fractions can be finite or infinite.
Examples of ending decimal fractions: 0.234; 0.87; 34.00005; 5.765.
The examples are endless. For example, Pi is infinite decimal, more – 0.333333333333…... 0.16666666666…. and others. Also the result of extracting the root of the numbers 3, 5, 7, etc. will be an infinite fraction.
The fractional part can be cyclic (it contains a cycle), the two examples above are exactly like this, and 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.
In the examples below, words such as “turning over” a fraction will sound - this means that the numerator and denominator are swapped. In fact, such a fraction has a name - reciprocal fraction. Examples of reciprocal fractions:
A small summary! Fractions are:
Ordinary (correct and incorrect).
Decimals (finite and infinite).
Mixed (mixed numbers).
That's all!
Best regards, Alexander.
The numerator, and that which is divided by is the denominator.
To write a fraction, first write the numerator, then draw a horizontal line under the number, and write the denominator below the line. The horizontal line that separates the numerator and denominator is called a fraction line. 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 the numerator is different. Write the result in the numerator of the new fractions. After this, multiply the denominators. Enter the total 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 actions, first “flip” the divisor, if it is more convenient for you: the denominator should appear 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 fraction problems
Fractional numbers can be expressed in in different forms exact value quantities. You can do the same with fractions mathematical operations, as with whole numbers: subtraction, addition, multiplication and division. To learn to decide fractions, we must 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
Instructions
Look closely at the numbers. If among the fractions there are decimals and irregular ones, sometimes it is more convenient to first perform operations with decimals, and then convert them to the irregular 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 an integer part is isolated must be converted to the wrong form by multiplying it by the denominator and adding the numerator to the result. Given value will become the new numerator fractions. To select a whole part from an initially incorrect one fractions, you need to divide the numerator by the denominator. Whole result write down from fractions. And the remainder of the division will become the new numerator, denominator fractions it does not change. For fractions with whole 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;
- Summing separately integers and fractional parts terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 +(8/12 + 9/12) = 3 + 12/17 = 3 + 1 5/12 = 4 5 /12.
Rewrite them using the “:” separator and continue regular division.
To receive 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 integers above and below the line.
Please note
Do not perform arithmetic with fractions whose denominators are different. Choose a number such that when you multiply the numerator and denominator of each fraction by it, the result is that the denominators of both fractions are equal.
When recording fractional numbers The dividend is written above the line. This quantity is designated as the numerator of the fraction. The divisor, or denominator, of the fraction is written below the line. For example, one and a half kilograms of rice as a fraction will be written as follows: 1 ½ kg of rice. If the denominator of a fraction is 10, the fraction is called a decimal. 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 ease of calculation, such a fraction can always be written in in the wrong form: 1 2/10 kg potatoes. To simplify, you can reduce the numerator and denominator values by dividing them by one integer. IN in this example may be divided by 2. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to perform arithmetic with are presented in the same form.
