Types of fractional numbers. Shares, ordinary fractions, definitions, notation, examples, actions with fractions
Definition of a common fraction
Definition 1
Ordinary fractions are used to describe the number of shares. Consider an example with which you can define an ordinary fraction.
The apple was divided into $8$ shares. In this case, each share represents one-eighth of the whole apple, i.e. $\frac(1)(8)$. Two beats are $\frac(2)(8)$, three beats are $\frac(3)(8)$, etc., and $8$ beats are $\frac(8)(8)$ . Each of the entries is called common fraction.
Let's bring general definition ordinary fraction.
Definition 2
Common fraction is a notation of the form $\frac(m)(n)$, where $m$ and $n$ are any natural numbers.
Often you can find the following record of an ordinary fraction: $m/n$.
Example 1
Examples of ordinary fractions:
\[(3)/(4), \frac(101)(345),\ \ (23)/(5), \frac(15)(15), (111)/(81).\]
Remark 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 An ordinary fraction $\frac(m)(n)$ is a 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 a single whole is divided into.
Picture 1.
The numerator is above the fractional bar, and the denominator is below the fractional bar. For example, the numerator of the common fraction $\frac(5)(17)$ is $5$ and the denominator is $17$. The denominator shows that the item is divided into $17$ shares, and the numerator shows that $5$ of such shares are taken.
Natural number as a fraction with denominator 1
The denominator of a common fraction can be one. In this case, it is considered that the subject is indivisible, i.e. is a single entity. The numerator of such a fraction shows how many whole items are taken. An ordinary fraction of the form $\frac(m)(1)$ has the meaning of a natural number $m$. Thus, we obtain a justified 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 $123 \ 456$ is a fraction $\frac(123\ 456)(1)$.
Thus, any natural number $m$ can be represented as an ordinary fraction with denominator $1$, and any ordinary fraction of the form $\frac(m)(1)$ can be replaced by a natural number $m$.
Fraction as division sign
Representing an object as $n$ shares is a division by $n$ equal parts. After dividing an item into $n$ shares, it can be divided equally between $n$ people - each gets one share.
Let there be $m$ identical items divided into $n$ shares. These $m$ items can be equally divided among $n$ people by giving each person one share of each of the $m$ items. In addition, each person will receive $m$ shares of $\frac(1)(n)$, which give an ordinary fraction $\frac(m)(n)$. We get that the ordinary fraction $\frac(m)(n)$ can be used to denote the division of $m$ objects between $n$ people.
The connection between ordinary fractions and division is expressed in the fact that the fractional 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 integer 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. each will receive seven ninths of the apple: $7:9=\frac(7)(9)$.
Equal and unequal ordinary fractions, comparison of fractions
The result of comparing two ordinary fractions can either be equal or not equal. When ordinary fractions are equal, they are called equal, otherwise common fractions 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 $a\cdot d=b\cdot c$ is not satisfied.
Example 3
Find out if the fractions $\frac(1)(3)$ and $\frac(2)(6)$ are equal.
The equality holds, so the fractions $\frac(1)(3)$ and $\frac(2)(6)$ are equal: $\frac(1)(3)=\frac(2)(6)$.
This example can be considered on the example of apples: one of the two identical apples is divided into three equal parts, the second - into $6$ parts. It can be seen that two-sixths of an apple is a $\frac(1)(3)$ share.
Example 4
Check if the common fractions $\frac(3)(17)$ and $\frac(4)(13)$ are equal.
Let's check if the equality $a\cdot d=b\cdot c$ is true:
\ \
The equality is not satisfied, so the fractions $\frac(3)(17)$ and $\frac(4)(13)$ are not equal: $\frac(3)(17)\ne \frac(4)(13) $.
When comparing two ordinary fractions, if it turns out that they are not equal, you can find out which of them is greater and which is less 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. Which fraction has a larger numerator, that fraction will be larger.
Fractions on the coordinate beam
All fractional numbers that correspond to ordinary fractions can be displayed on the coordinate beam.
In order to mark a point on the coordinate ray that corresponds to the fraction $\frac(m)(n)$, it is necessary to set aside $m$ segments in the positive direction from the origin of coordinates, the length of which is $\frac(1)(n)$ a fraction of the unit segment . Such segments are obtained by dividing a single segment into $n$ equal parts.
To display a fractional number on the 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 the horizontal coordinate ray directed to the right from the point whose coordinate is a smaller fraction. For example, because fraction $\frac(5)(6)$ is greater than fraction $\frac(2)(6)$, then the point with coordinate $\frac(5)(6)$ is to the right of the point with coordinate $\frac(2) (6)$.
Similarly, a point with a smaller coordinate will lie to the left of a point with a larger coordinate.
Do you want to feel like a sapper? Then this lesson is for you! Because now we will study fractions - these are such simple and harmless mathematical objects that surpass the rest of the algebra course in their ability to “take out the brain”.
The main danger of fractions is that they occur in real life. In this they differ, for example, from polynomials and logarithms, which can be passed and easily forgotten after the exam. Therefore, the material presented in this lesson, without exaggeration can be called explosive.
A numeric fraction (or simply a fraction) is a pair of integers written through a slash or horizontal bar.
Fractions written through a horizontal bar:
The same fractions written with a slash:
5/7; 9/(−30); 64/11; (−1)/4; 12/1.
Usually fractions are written through a horizontal line - it's easier to work with them, and they look better. The number written on top is called the numerator of the fraction, and the number written on the bottom is called the denominator.
Any whole number can be represented as a fraction with a denominator of 1. For example, 12 = 12/1 is the fraction from the above example.
In general, you can put any whole number in the numerator and denominator of a fraction. The only restriction is that the denominator must be different from zero. Remember the good old rule: “You can’t divide by zero!”
If the denominator is still zero, the fraction is called indefinite. Such a record does not make sense and cannot participate in calculations.
Basic property of a fraction
Fractions a /b and c /d are called equal if ad = bc.
From this definition it follows that the same fraction can be written in different ways. For example, 1/2 = 2/4 because 1 4 = 2 2. Of course, there are many fractions that are not equal to each other. For example, 1/3 ≠ 5/4 because 1 4 ≠ 3 5.
A reasonable question arises: how to find all fractions equal to a given one? We give the answer in the form of a definition:
The main property of a fraction is that the numerator and denominator can be multiplied by the same number other than zero. This will result in a fraction equal to the given one.
This is a very important property - remember it. With the help of the basic property of a fraction, many expressions can be simplified and shortened. In the future, it will constantly “emerge” in the form various properties and theorems.
Incorrect fractions. Selection of the whole part
If the numerator is less than the denominator, such a fraction is called proper. Otherwise (that is, when the numerator is greater than or at least equal to the denominator), the fraction is called an improper fraction, and an integer part can be distinguished in it.
The integer part is written as a large number in front of the fraction and looks like this (marked in red):
To highlight the whole part in improper fraction you need to follow three simple steps:
- Find how many times the denominator fits in the numerator. In other words, find the maximum integer that, when multiplied by the denominator, will still be less than the numerator (in the extreme case, equal). This number will be whole part, so we write it in front;
- Multiply the denominator by the integer part found in the previous step, and subtract the result from the numerator. The resulting "stub" is called the remainder of the division, it will always be positive (in extreme cases, zero). We write it down in the numerator of the new fraction;
- We rewrite the denominator unchanged.
Well, is it difficult? At first glance, it may be difficult. But it takes a little practice - and you will do it almost verbally. For now, take a look at the examples:
A task. Select the whole part in the given fractions:
In all examples, the integer part is highlighted in red, and the remainder of the division is in green.
Pay attention to the last fraction, where the remainder of the division turned out to be zero. It turns out that the numerator is completely divided by the denominator. This is quite logical, because 24: 6 = 4 - harsh fact from the multiplication table.
If everything is done correctly, the numerator of the new fraction will necessarily be less than the denominator, i.e. fraction becomes correct. I also note that it is better to highlight the whole part at the very end of the task, before writing the answer. Otherwise, you can significantly complicate the calculations.
Transition to improper fraction
There is also an inverse operation, when we get rid of the whole part. This is called the improper fraction transition and is much more common because improper fractions are much easier to work with.
The transition to an improper fraction is also done in three steps:
- Multiply the integer part by the denominator. The result can be quite large numbers, but we should not be embarrassed;
- Add the resulting number to the numerator of the original fraction. Write the result in the numerator of an improper fraction;
- Rewrite the denominator - again, no change.
Here are specific examples:
A task. Convert to an improper fraction:
For clarity, the integer part is again highlighted in red, and the numerator of the original fraction is in green.
Consider the case when the numerator or denominator of a fraction contains a negative number. For example:
In principle, there is nothing criminal in this. However, working with such fractions can be inconvenient. Therefore, in mathematics it is customary to take out minuses as a fraction sign.
This is very easy to do if you remember the rules:
- Plus times minus equals minus. Therefore, if the numerator is a negative number, and the denominator is positive (or vice versa), feel free to cross out the minus and put it in front of the whole fraction;
- "Two negatives make an affirmative". When the minus is in both the numerator and the denominator, we simply cross them out - no additional action is required.
Of course, these rules can also be applied in the opposite direction, i.e. you can add a minus under the fraction sign (most often - in the numerator).
We deliberately do not consider the case of “plus on plus” - with him, I think, everything is clear anyway. Let's take a look at how these rules work in practice:
A task. Take out the minuses of the four fractions written above.
Pay attention to the last fraction: it already has a minus sign in front of it. However, it is “burned” according to the rule “minus times minus gives plus”.
Also, do not move minuses in fractions with a highlighted integer part. These fractions are first converted to improper ones - and only then they begin to calculate.
Fractions
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
Fractions in high school are not very annoying. For the time being. Until you come across exponents with rational exponents and logarithms. And there…. You press, you press the calculator, and it shows all the full scoreboard of some numbers. You have to think with your head, like in the third grade.
Let's deal with fractions, finally! Well, how much can you get confused in them!? Moreover, it is all simple and logical. So, what are fractions?
Types of fractions. Transformations.
Fractions happen three types.
1. Common fractions , for example:
Sometimes, instead of a horizontal line, they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens ...), tell yourself the phrase with the expression: " Zzzzz remember! Zzzzz denominator - out zzzz u!" Look, everything will be remembered.)
A dash, which is horizontal, which is oblique, means division top number (numerator) to bottom number (denominator). And that's it! Instead of a dash, it is quite possible to put a division sign - two dots.
When the division is possible entirely, it must be done. So, instead of the fraction "32/8" it is much more pleasant to write the number "4". Those. 32 is simply divided by 8.
32/8 = 32: 8 = 4
I'm not talking about the fraction "4/1". Which is also just "4". And if it doesn’t divide completely, we leave it as a fraction. Sometimes you have to do the reverse. Make a fraction from a whole number. But more on that later.
2. Decimals , for example:
It is in this form that it will be necessary to write down the answers to tasks "B".
3. mixed numbers , for example:
Mixed numbers are practically not used in high school. In order to work with them, they must be converted to ordinary fractions. But you definitely need to know how to do it! And then such a number will come across in the puzzle and hang ... From scratch. But we remember this procedure! A little lower.
Most versatile common fractions. Let's start with them. By the way, if there are all sorts of logarithms, sines and other letters in the fraction, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!
Basic property of a fraction.
So let's go! First of all, I will surprise you. The whole variety of fraction transformations is provided by a single property! That's what it's called basic property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction will not change. Those:
It is clear that you can write further, until you are blue in the face. Do not let sines and logarithms confuse you, we will deal with them further. The main thing to understand is that all these various expressions are the same fraction . 2/3.
And we need it, all these transformations? And how! Now you will see for yourself. First, let's use the basic property of a fraction for fraction abbreviations. It would seem that the thing is elementary. We divide the numerator and denominator by the same number and that's it! It's impossible to go wrong! But... man is a creative being. You can make mistakes everywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.
How to reduce fractions correctly and quickly without doing unnecessary work can be found in special Section 555.
A normal student does not bother dividing the numerator and denominator by the same number (or expression)! He just crosses out everything the same from above and below! This is where it hides typical mistake, blooper if you want.
For example, you need to simplify the expression:
There is nothing to think about, we cross out the letter "a" from above and the deuce from below! We get:
Everything is correct. But really you shared the whole numerator and the whole denominator "a". If you are used to just cross out, then, in a hurry, you can cross out the "a" in the expression
and get again
Which would be categorically wrong. Because here the whole numerator on "a" already not shared! This fraction cannot be reduced. By the way, such an abbreviation is, um ... a serious challenge to the teacher. This is not forgiven! Remember? When reducing, it is necessary to divide the whole numerator and the whole denominator!
Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. And how to work with her now? Without a calculator? Multiply, say, add, square!? And if you are not too lazy, but carefully reduce by five, and even by five, and even ... while it is being reduced, in short. We get 3/8! Much nicer, right?
The basic property of a fraction allows you to convert ordinary fractions to decimals and vice versa without calculator! This is important for the exam, right?
How to convert fractions from one form to another.
It's easy with decimals. As it is heard, so it is written! Let's say 0.25. It's zero point, twenty-five hundredths. So we write: 25/100. We reduce (divide the numerator and denominator by 25), we get the usual fraction: 1/4. Everything. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.
What if integers are non-zero? It's OK. Write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three whole, seventeen hundredths. We write 317 in the numerator, and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary Watson! From all of the above, a useful conclusion: any decimal fraction can be converted to a common fraction .
But the reverse conversion, ordinary to decimal, some cannot do without a calculator. But you must! How will you write down the answer on the exam!? We carefully read and master this process.
What is a decimal fraction? She has in the denominator always is worth 10 or 100 or 1000 or 10000 and so on. If your usual fraction has such a denominator, there is no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. And if in the answer to the task of section "B" it turned out 1/2? What will we write in response? Decimals are required...
We remember basic property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. For anyone, by the way! Except zero, of course. Let's use this feature to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But, then the numerator must also be multiplied by 5. This is already maths demands! We get 1/2 \u003d 1x5 / 2x5 \u003d 5/10 \u003d 0.5. That's all.
However, all sorts of denominators come across. For example, the fraction 3/16 will fall. Try it, figure out what to multiply 16 by to get 100, or 1000... Doesn't work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide in a corner, on a piece of paper, as they taught in elementary grades. We get 0.1875.
And there are some very bad denominators. For example, the fraction 1/3 cannot be turned into a good decimal. Both on a calculator and on a piece of paper, we get 0.3333333 ... This means that 1/3 into an exact decimal fraction does not translate. Just like 1/7, 5/6 and so on. Many of them are untranslatable. Hence another useful conclusion. Not every common fraction converts to a decimal. !
By the way, this useful information for self-test. In section "B" in response, you need to write down a decimal fraction. And you got, for example, 4/3. This fraction is not converted to decimal. This means that somewhere along the way you made a mistake! Come back, check the solution.
So, with ordinary and decimal fractions sorted out. It remains to deal with mixed numbers. To work with them, they all need to be converted to ordinary fractions. How to do it? You can catch a sixth grader and ask him. But not always a sixth grader will be at hand ... We will have to do it ourselves. This is not difficult. Multiply the denominator of the fractional part by the integer part and add the numerator of the fractional part. This will be the numerator of a common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but it's actually quite simple. Let's see an example.
Let in the problem you saw with horror the number:
Calmly, without panic, we understand. The whole part is 1. One. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of an ordinary fraction. That's all. It looks even simpler in mathematical notation:
Clearly? Then secure your success! Convert to common fractions. You should get 10/7, 7/2, 23/10 and 21/4.
The inverse operation - converting an improper fraction to mixed number Rarely required in high school. Well, if... And if you - not in high school - you can look into the special Section 555. In the same place, by the way, you will learn about improper fractions.
Well, almost everything. You remembered the types of fractions and understood how convert them from one type to another. The question remains: why do it? Where and when to apply this deep knowledge?
I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed into a bunch, we translate everything into ordinary fractions. It can always be done. Well, if something like 0.8 + 0.3 is written, then we think so, without any translation. Why do we need extra work? We choose the solution that is convenient us !
If the task is entirely decimals, but um... some evil ones, go to the ordinary ones, try it! Look, everything will be fine. For example, you have to square the number 0.125. Not so easy if you have not lost the habit of the calculator! Not only do you need to multiply the numbers in a column, but also think about where to insert the comma! It certainly doesn't work in my mind! And if you go to an ordinary fraction?
0.125 = 125/1000. We reduce by 5 (this is for starters). We get 25/200. Once again on 5. We get 5/40. Oh, it's shrinking! Back to 5! We get 1/8. Easily square (in your mind!) and get 1/64. Everything!
Let's summarize this lesson.
1. There are three types of fractions. Ordinary, decimal and mixed numbers.
2. Decimals and mixed numbers always can be converted to common fractions. Reverse Translation not always available.
3. The choice of the type of fractions for working with the task depends on this very task. If there are different types of fractions in one task, the most reliable thing is to switch to ordinary fractions.
Now you can practice. First, convert these decimal fractions to ordinary ones:
3,8; 0,75; 0,15; 1,4; 0,725; 0,012
You should get answers like this (in a mess!):
On this we will finish. In this lesson, we refreshed our memory key points by fractions. It happens, however, that there is nothing special to refresh ...) If someone has completely forgotten, or has not mastered it yet ... Those can go to a special Section 555. All the basics are detailed there. Many suddenly understand everything are starting. And they solve fractions on the fly).
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