Very often in the school mathematics curriculum, children are faced with the problem of how to convert a regular fraction to a decimal. In order to convert a common fraction to a decimal, let us first remember what a common fraction and a decimal are. An ordinary fraction is a fraction of the form m/n, where m is the numerator and n is the denominator. Example: 8/13; 6/7, etc. Fractions are divided into regular, improper and mixed numbers. A proper fraction is when the numerator is less than the denominator: m/n, where m 3. An improper fraction can always be represented as a mixed number, namely: 4/3 = 1 and 1/3;
Converting a fraction to a decimal
Now let's look at how to convert a mixed fraction to a decimal. Any ordinary fraction, whether proper or improper, can be converted to a decimal. To do this, you need to divide the numerator by the denominator. Example: simple fraction (proper) 1/2. Divide numerator 1 by denominator 2 to get 0.5. Let's take the example of 45/12; it is immediately clear that this is an irregular fraction. Here the denominator is less than the numerator. Converting an improper fraction to a decimal: 45: 12 = 3.75.
Converting mixed numbers to decimals
Example: 25/8. First we turn the mixed number into an improper fraction: 25/8 = 3x8+1/8 = 3 and 1/8; then divide the numerator equal to 1 by the denominator equal to 8, using a column or on a calculator and get a decimal fraction equal to 0.125. The article provides the easiest examples of conversion to decimal fractions. Having understood the translation technique using simple examples, you can easily solve the most complex ones.
Already in elementary school, students are exposed to fractions. And then they appear in every topic. You cannot forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are not complicated, the main thing is to understand everything in order.
Why are fractions needed?
The world around us consists of entire objects. Therefore, there is no need for shares. But everyday life constantly pushes people to work with parts of objects and things.
For example, chocolate consists of several pieces. Consider a situation where his tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It can easily be divided into three. But it will not be possible to give five people a whole number of chocolate slices.
By the way, these slices are already fractions. And their further division leads to the appearance of more complex numbers.
What is a "fraction"?
This is a number made up of parts of a unit. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written at the top (left) is called the numerator. What is at the bottom (right) is the denominator.
Essentially, the slash turns out to be a division sign. That is, the numerator can be called the dividend, and the denominator can be called the divisor.
What fractions are there?
In mathematics there are only two types: ordinary and decimal fractions. Schoolchildren become acquainted with the first ones in elementary school, calling them simply “fractions.” The latter will be learned in 5th grade. That's when these names appear.
Common fractions are all those that are written as two numbers separated by a line. For example, 4/7. A decimal is a number in which the fractional part has a positional notation and is separated from the whole number by a comma. For example, 4.7. Students need to clearly understand that the two examples given are completely different numbers.
Every simple fraction can be written as a decimal. This statement is almost always true in reverse. There are rules that allow you to write a decimal fraction as a common fraction.
What subtypes do these types of fractions have?
It is better to start in chronological order, as they are studied. Common fractions come first. Among them, 5 subspecies can be distinguished.
Correct. Its numerator is always less than its denominator.
Wrong. Its numerator is greater than or equal to its denominator.
Reducible/irreducible. It may turn out to be either right or wrong. Another important thing is whether the numerator and denominator have common factors. If there are, then it is necessary to divide both parts of the fraction by them, that is, reduce it.
Mixed. An integer is assigned to its usual regular (irregular) fractional part. Moreover, it is always on the left.
Composite. It is formed from two fractions divided by each other. That is, it contains three fractional lines at once.
Decimal fractions have only two subtypes:
finite, that is, one whose fractional part is limited (has an end);
infinite - a number whose digits after the decimal point do not end (they can be written endlessly).
How to convert a decimal fraction to a common fraction?
If this is a finite number, then an association is applied based on the rule - as I hear, so I write. That is, you need to read it correctly and write it down, but without a comma, but with a fractional bar.
As a hint about the required denominator, you need to remember that it is always one and several zeros. You need to write as many of the latter as there are digits in the fractional part of the number in question.
How to convert decimal fractions into ordinary fractions if their integer part is missing, that is, equal to zero? For example, 0.9 or 0.05. After applying the specified rule, it turns out that you need to write zero integers. But it is not indicated. All that remains is to write down the fractional parts. The first number will have a denominator of 10, the second will have a denominator of 100. That is, the given examples will have the following numbers as answers: 9/10, 5/100. Moreover, it turns out that the latter can be reduced by 5. Therefore, the result for it needs to be written as 1/20.
How can you convert a decimal fraction into an ordinary fraction if its integer part is different from zero? For example, 5.23 or 13.00108. In both examples, the whole part is read and its value is written. In the first case it is 5, in the second it is 13. Then you need to move on to the fractional part. The same operation is supposed to be carried out with them. The first number appears 23/100, the second - 108/100000. The second value needs to be reduced again. The answer gives the following mixed fractions: 5 23/100 and 13 27/25000.
How to convert an infinite decimal fraction to an ordinary fraction?
If it is non-periodic, then such an operation will not be possible. This fact is due to the fact that each decimal fraction is always converted to either a finite or a periodic fraction.
The only thing you can do with such a fraction is round it. But then the decimal will be approximately equal to that infinite. It can already be turned into an ordinary one. But the reverse process: converting to decimal will never give the initial value. That is, infinite non-periodic fractions are not converted into ordinary fractions. This needs to be remembered.
How to write an infinite periodic fraction as an ordinary fraction?
In these numbers, there are always one or more digits after the decimal point that are repeated. They are called a period. For example, 0.3(3). Here "3" is in the period. They are classified as rational because they can be converted into ordinary fractions.
Those who have encountered periodic fractions know that they can be pure or mixed. In the first case, the period starts immediately from the comma. In the second, the fractional part begins with some numbers, and then the repetition begins.
The rule by which you need to write an infinite decimal as an ordinary fraction will be different for the two types of numbers indicated. It is quite easy to write pure periodic fractions as ordinary fractions. As with finite ones, they need to be converted: write down the period in the numerator, and the denominator will be the number 9, repeated as many times as the number of digits the period contains.
For example, 0,(5). The number does not have an integer part, so you need to immediately start with the fractional part. Write 5 as the numerator and 9 as the denominator. That is, the answer will be the fraction 5/9.
The rule on how to write an ordinary decimal periodic fraction that is mixed.
Look at the length of the period. That's how many 9s the denominator will have.
Write down the denominator: first nines, then zeros.
To determine the numerator, you need to write down the difference of two numbers. All numbers after the decimal point will be minified, along with the period. Deductible - it is without a period.
For example, 0.5(8) - write the periodic decimal fraction as a common fraction. The fractional part before the period contains one digit. So there will be one zero. There is also only one number in the period - 8. That is, there is only one nine. That is, you need to write 90 in the denominator.
To determine the numerator, you need to subtract 5 from 58. It turns out 53. For example, the answer would have to be written as 53/90.
How are fractions converted to decimals?
The simplest option is a number whose denominator is the number 10, 100, etc. Then the denominator is simply discarded, and a comma is placed between the fractional and integer parts.
There are situations when the denominator easily turns into 10, 100, etc. For example, the numbers 5, 20, 25. It is enough to multiply them by 2, 5 and 4, respectively. You just need to multiply not only the denominator, but also the numerator by the same number.
For all other cases, a simple rule is useful: divide the numerator by the denominator. In this case, you may get two possible answers: a finite or a periodic decimal fraction.
Operations with ordinary fractions
Addition and subtraction
Students become acquainted with them earlier than others. Moreover, at first the fractions have the same denominators, and then they have different ones. General rules can be reduced to this plan.
Find the least common multiple of the denominators.
Write additional factors for all ordinary fractions.
Multiply the numerators and denominators by the factors specified for them.
Add (subtract) the numerators of the fractions and leave the common denominator unchanged.
If the numerator of the minuend is less than the subtrahend, then we need to find out whether we have a mixed number or a proper fraction.
In the first case, you need to borrow one from the whole part. Add the denominator to the numerator of the fraction. And then do the subtraction.
In the second, it is necessary to apply the rule of subtracting a larger number from a smaller number. That is, from the module of the subtrahend, subtract the module of the minuend, and in response put a “-” sign.
Look carefully at the result of addition (subtraction). If you get an improper fraction, then you need to select the whole part. That is, divide the numerator by the denominator.
Multiplication and division
To perform them, fractions do not need to be reduced to a common denominator. This makes it easier to perform actions. But they still require you to follow the rules.
When multiplying fractions, you need to look at the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.
Multiply the numerators.
Multiply the denominators.
If the result is a reducible fraction, then it must be simplified again.
When dividing, you must first replace division with multiplication, and the divisor (second fraction) with the reciprocal fraction (swap the numerator and denominator).
Then proceed as with multiplication (starting from point 1).
In tasks where you need to multiply (divide) by a whole number, the latter should be written as an improper fraction. That is, with a denominator of 1. Then act as described above.
Operations with decimals
Addition and subtraction
Of course, you can always convert a decimal into a fraction. And act according to the plan already described. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction will be exactly the same.
Equalize the number of digits in the fractional part of the number, that is, after the decimal point. Add the missing number of zeros to it.
Write the fractions so that the comma is below the comma.
Add (subtract) like natural numbers.
Remove the comma.
Multiplication and division
It is important that you do not need to add zeros here. Fractions should be left as they are given in the example. And then go according to plan.
To multiply, you need to write the fractions one below the other, ignoring the commas.
Multiply like natural numbers.
Place a comma in the answer, counting from the right end of the answer as many digits as they are in the fractional parts of both factors.
To divide, you must first transform the divisor: make it a natural number. That is, multiply it by 10, 100, etc., depending on how many digits are in the fractional part of the divisor.
Multiply the dividend by the same number.
Divide a decimal fraction by a natural number.
Place a comma in your answer at the moment when the division of the whole part ends.
What if one example contains both types of fractions?
Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. In such tasks there are two possible solutions. You need to objectively weigh the numbers and choose the optimal one.
First way: represent ordinary decimals
It is suitable if division or translation results in finite fractions. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you don’t like working with ordinary fractions, you will have to count them.
Second way: write decimal fractions as ordinary
This technique turns out to be convenient if the part after the decimal point contains 1-2 digits. If there are more of them, you may end up with a very large common fraction and decimal notation will make the task faster and easier to calculate. Therefore, you always need to soberly evaluate the task and choose the simplest solution method.
It happens that for convenience of calculations you need to convert an ordinary fraction to a decimal and vice versa. We will talk about how to do this in this article. Let's look at the rules for converting ordinary fractions to decimals and vice versa, and also give examples.
Yandex.RTB R-A-339285-1
We will consider converting ordinary fractions to decimals, following a certain sequence. First, let's look at how ordinary fractions with a denominator that is a multiple of 10 are converted into decimals: 10, 100, 1000, etc. Fractions with such denominators are, in fact, a more cumbersome notation of decimal fractions.
Next, we will look at how to convert ordinary fractions with any denominator, not just a multiple of 10, into decimal fractions. Note that when converting ordinary fractions to decimals, not only finite decimals are obtained, but also infinite periodic decimal fractions.
Let's get started!
Translation of ordinary fractions with denominators 10, 100, 1000, etc. to decimals
First of all, let's say that some fractions require some preparation before converting to decimal form. What is it? Before the number in the numerator, you need to add so many zeros so that the number of digits in the numerator becomes equal to the number of zeros in the denominator. For example, for the fraction 3100, the number 0 must be added once to the left of the 3 in the numerator. Fraction 610, according to the rule stated above, does not need modification.
Let's look at one more example, after which we will formulate a rule that is especially convenient to use at first, while there is not much experience in converting fractions. So, the fraction 1610000 after adding zeros in the numerator will look like 001510000.
How to convert a common fraction with a denominator of 10, 100, 1000, etc. to decimal?
Rule for converting ordinary proper fractions to decimals
- Write down 0 and put a comma after it.
- We write down the number from the numerator that was obtained after adding zeros.
Now let's move on to examples.
Example 1: Converting fractions to decimals
Let's convert the fraction 39,100 to a decimal.
First, we look at the fraction and see that there is no need to carry out any preparatory actions - the number of digits in the numerator coincides with the number of zeros in the denominator.
Following the rule, we write 0, put a decimal point after it and write the number from the numerator. We get the decimal fraction 0.39.
Let's look at the solution to another example on this topic.
Example 2: Converting fractions to decimals
Let's write the fraction 105 10000000 as a decimal.
The number of zeros in the denominator is 7, and the numerator has only three digits. Let's add 4 more zeros before the number in the numerator:
0000105 10000000
Now we write down 0, put a decimal point after it and write down the number from the numerator. We get the decimal fraction 0.0000105.
The fractions considered in all examples are ordinary proper fractions. But how do you convert an improper fraction to a decimal? Let us say right away that there is no need for preparation with adding zeros for such fractions. Let's formulate a rule.
Rule for converting ordinary improper fractions to decimals
- Write down the number that is in the numerator.
- We use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.
Below is an example of how to use this rule.
Example 3. Converting fractions to decimals
Let's convert the fraction 56888038009 100000 from an ordinary irregular fraction to a decimal.
First, let's write down the number from the numerator:
Now, on the right, we separate five digits with a decimal point (the number of zeros in the denominator is five). We get:
The next question that naturally arises is: how to convert a mixed number into a decimal fraction if the denominator of its fractional part is the number 10, 100, 1000, etc. To convert such a number to a decimal fraction, you can use the following rule.
Rule for converting mixed numbers to decimals
- We prepare the fractional part of the number, if necessary.
- We write down the whole part of the original number and put a comma after it.
- We write down the number from the numerator of the fractional part along with the added zeros.
Let's look at an example.
Example 4: Converting mixed numbers to decimals
Let's convert the mixed number 23 17 10000 to a decimal fraction.
In the fractional part we have the expression 17 10000. Let's prepare it and add two more zeros to the left of the numerator. We get: 0017 10000.
Now we write down the whole part of the number and put a comma after it: 23, . .
After the decimal point, write down the number from the numerator along with zeros. We get the result:
23 17 10000 = 23 , 0017
Converting ordinary fractions to finite and infinite periodic fractions
Of course, you can convert to decimals and ordinary fractions with a denominator not equal to 10, 100, 1000, etc.
Often a fraction can be easily reduced to a new denominator, and then use the rule set out in the first paragraph of this article. For example, it is enough to multiply the numerator and denominator of the fraction 25 by 2, and we get the fraction 410, which is easily converted to the decimal form 0.4.
However, this method of converting a fraction to a decimal cannot always be used. Below we will consider what to do if it is impossible to apply the considered method.
A fundamentally new way to convert a fraction to a decimal is to divide the numerator by the denominator with a column. This operation is very similar to dividing natural numbers with a column, but has its own characteristics.
When dividing, the numerator is represented as a decimal fraction - a comma is placed to the right of the last digit of the numerator and zeros are added. In the resulting quotient, a decimal point is placed when the division of the integer part of the numerator ends. How exactly this method works will become clear after looking at the examples.
Example 5. Converting fractions to decimals
Let's convert the common fraction 621 4 to decimal form.
Let's represent the number 621 from the numerator as a decimal fraction, adding a few zeros after the decimal point. 621 = 621.00
Now let's divide 621.00 by 4 using a column. The first three steps of division will be the same as when dividing natural numbers, and we will get.
When we reach the decimal point in the dividend, and the remainder is different from zero, we put a decimal point in the quotient and continue dividing, no longer paying attention to the comma in the dividend.
As a result, we get the decimal fraction 155, 25, which is the result of reversing the common fraction 621 4
621 4 = 155 , 25
Let's look at another example to reinforce the material.
Example 6. Converting fractions to decimals
Let's reverse the common fraction 21 800.
To do this, divide the fraction 21,000 into a column by 800. The division of the whole part will end at the first step, so immediately after it we put a decimal point in the quotient and continue the division, not paying attention to the comma in the dividend until we get a remainder equal to zero.
As a result, we got: 21,800 = 0.02625.
But what if, when dividing, we still do not get a remainder of 0. In such cases, the division can be continued indefinitely. However, starting from a certain step, the residues will be repeated periodically. Accordingly, the numbers in the quotient will be repeated. This means that an ordinary fraction is converted into a decimal infinite periodic fraction. Let us illustrate this with an example.
Example 7. Converting fractions to decimals
Let's convert the common fraction 19 44 to a decimal. To do this, we perform division by column.
We see that during division, residues 8 and 36 are repeated. In this case, the numbers 1 and 8 are repeated in the quotient. This is the period in decimal fraction. When recording, these numbers are placed in brackets.
Thus, the original ordinary fraction is converted into an infinite periodic decimal fraction.
19 44 = 0 , 43 (18) .
Let us have an irreducible ordinary fraction. What form will it take? Which ordinary fractions are converted to finite decimals, and which ones are converted to infinite periodic ones?
First, let's say that if a fraction can be reduced to one of the denominators 10, 100, 1000..., then it will have the form of a final decimal fraction. In order for a fraction to be reduced to one of these denominators, its denominator must be a divisor of at least one of the numbers 10, 100, 1000, etc. From the rules for factoring numbers into prime factors it follows that the divisor of numbers is 10, 100, 1000, etc. must, when factored into prime factors, contain only the numbers 2 and 5.
Let's summarize what has been said:
- A common fraction can be reduced to a final decimal if its denominator can be factored into prime factors of 2 and 5.
- If, in addition to the numbers 2 and 5, other prime numbers are present in the expansion of the denominator, the fraction is reduced to the form of an infinite periodic decimal fraction.
Let's give an example.
Example 8. Converting fractions to decimals
Which of these fractions 47 20, 7 12, 21 56, 31 17 is converted into a final decimal fraction, and which one - only into a periodic one. Let's answer this question without directly converting a fraction to a decimal.
The fraction 47 20, as is easy to see, by multiplying the numerator and denominator by 5 is reduced to a new denominator 100.
47 20 = 235 100. From this we conclude that this fraction is converted to a final decimal fraction.
Factoring the denominator of the fraction 7 12 gives 12 = 2 · 2 · 3. Since the prime factor 3 is different from 2 and 5, this fraction cannot be represented as a finite decimal fraction, but will have the form of an infinite periodic fraction.
The fraction 21 56, firstly, needs to be reduced. After reduction by 7, we obtain the irreducible fraction 3 8, the denominator of which is factorized to give 8 = 2 · 2 · 2. Therefore, it is a finite decimal fraction.
In the case of the fraction 31 17, factoring the denominator is the prime number 17 itself. Accordingly, this fraction can be converted into an infinite periodic decimal fraction.
An ordinary fraction cannot be converted into an infinite and non-periodic decimal fraction
Above we talked only about finite and infinite periodic fractions. But can any ordinary fraction be converted into an infinite non-periodic fraction?
We answer: no!
Important!
When converting an infinite fraction to a decimal, the result is either a finite decimal or an infinite periodic decimal.
The remainder of a division is always less than the divisor. In other words, according to the divisibility theorem, if we divide some natural number by the number q, then the remainder of the division in any case cannot be greater than q-1. After the division is completed, one of the following situations is possible:
- We get a remainder of 0, and this is where the division ends.
- We get a remainder, which is repeated upon subsequent division, resulting in an infinite periodic fraction.
There cannot be any other options when converting a fraction to a decimal. Let's also say that the length of the period (number of digits) in an infinite periodic fraction is always less than the number of digits in the denominator of the corresponding ordinary fraction.
Converting decimals to fractions
Now it's time to look at the reverse process of converting a decimal fraction into a common fraction. Let us formulate a translation rule that includes three stages. How to convert a decimal fraction to a common fraction?
Rule for converting decimal fractions to ordinary fractions
- In the numerator we write the number from the original decimal fraction, discarding the comma and all zeros on the left, if any.
- In the denominator we write one followed by as many zeros as there are digits after the decimal point in the original decimal fraction.
- If necessary, reduce the resulting ordinary fraction.
Let's look at the application of this rule using examples.
Example 8. Converting decimal fractions to ordinary fractions
Let's imagine the number 3.025 as an ordinary fraction.
- We write the decimal fraction itself into the numerator, discarding the comma: 3025.
- In the denominator we write one, and after it three zeros - this is exactly how many digits are contained in the original fraction after the decimal point: 3025 1000.
- The resulting fraction 3025 1000 can be reduced by 25, resulting in: 3025 1000 = 121 40.
Example 9. Converting decimal fractions to ordinary fractions
Let's convert the fraction 0.0017 from decimal to ordinary.
- In the numerator we write the fraction 0, 0017, discarding the comma and zeros on the left. It will turn out to be 17.
- We write one in the denominator, and after it we write four zeros: 17 10000. This fraction is irreducible.
If a decimal fraction has an integer part, then such a fraction can be immediately converted to a mixed number. How to do this?
Let's formulate one more rule.
Rule for converting decimal fractions to mixed numbers.
- The number before the decimal point in the fraction is written as the integer part of the mixed number.
- In the numerator we write the number after the decimal point in the fraction, discarding the zeros on the left if there are any.
- In the denominator of the fractional part we add one and as many zeros as there are digits after the decimal point in the fractional part.
Let's take an example
Example 10: Converting a decimal to a mixed number
Let's imagine the fraction 155, 06005 as a mixed number.
- We write the number 155 as an integer part.
- In the numerator we write the numbers after the decimal point, discarding the zero.
- We write one and five zeros in the denominator
Let's learn a mixed number: 155 6005 100000
The fractional part can be reduced by 5. We shorten it and get the final result:
155 , 06005 = 155 1201 20000
Converting infinite periodic decimals to fractions
Let's look at examples of how to convert periodic decimal fractions into ordinary fractions. Before we begin, let's clarify: any periodic decimal fraction can be converted to an ordinary fraction.
The simplest case is when the period of the fraction is zero. A periodic fraction with a zero period is replaced by a final decimal fraction, and the process of reversing such a fraction is reduced to reversing the final decimal fraction.
Example 11. Converting a periodic decimal fraction to a common fraction
Let us invert the periodic fraction 3, 75 (0).
Eliminating the zeros on the right, we get the final decimal fraction 3.75.
Converting this fraction to an ordinary fraction using the algorithm discussed in the previous paragraphs, we obtain:
3 , 75 (0) = 3 , 75 = 375 100 = 15 4 .
What if the period of the fraction is different from zero? The periodic part should be considered as the sum of the terms of a geometric progression, which decreases. Let's explain this with an example:
0 , (74) = 0 , 74 + 0 , 0074 + 0 , 000074 + 0 , 00000074 + . .
There is a formula for the sum of terms of an infinite decreasing geometric progression. If the first term of the progression is b and the denominator q is such that 0< q < 1 , то сумма равна b 1 - q .
Let's look at a few examples using this formula.
Example 12. Converting a periodic decimal fraction to a common fraction
Let us have a periodic fraction 0, (8) and we need to convert it to an ordinary fraction.
0 , (8) = 0 , 8 + 0 , 08 + 0 , 008 + . .
Here we have an infinite decreasing geometric progression with the first term 0, 8 and the denominator 0, 1.
Let's apply the formula:
0 , (8) = 0 , 8 + 0 , 08 + 0 , 008 + . . = 0 , 8 1 - 0 , 1 = 0 , 8 0 , 9 = 8 9
This is the required ordinary fraction.
To consolidate the material, consider another example.
Example 13. Converting a periodic decimal fraction to a common fraction
Let's reverse the fraction 0, 43 (18).
First we write the fraction as an infinite sum:
0 , 43 (18) = 0 , 43 + (0 , 0018 + 0 , 000018 + 0 , 00000018 . .)
Let's look at the terms in brackets. This geometric progression can be represented as follows:
0 , 0018 + 0 , 000018 + 0 , 00000018 . . = 0 , 0018 1 - 0 , 01 = 0 , 0018 0 , 99 = 18 9900 .
We add the result to the final fraction 0, 43 = 43 100 and get the result:
0 , 43 (18) = 43 100 + 18 9900
After adding these fractions and reducing, we get the final answer:
0 , 43 (18) = 19 44
To conclude this article, we will say that non-periodic infinite decimal fractions cannot be converted into ordinary fractions.
If you notice an error in the text, please highlight it and press Ctrl+Enter
All fractions are divided into two types: ordinary and decimal. Fractions of this type are called ordinary: 9/8.3/4.1/2.1 3/4. They have a top number (numerator) and a bottom number (denominator). When the numerator is less than the denominator, the fraction is called proper; otherwise, the fraction is called improper. Fractions such as 1 7/8 consist of an integer part (1) and a fractional part (7/8) and are called mixed.
So, fractions are:
- Ordinary
- Correct
- Wrong
- Mixed
- Decimal
How to make a decimal from a fraction
A basic school mathematics course teaches how to convert a fraction to a decimal. Everything is extremely simple: you need to divide the numerator by the denominator “manually” or, if you’re really lazy, then using a microcalculator. Here's an example: 2/5=0.4;3/4=0.75; 1/2=0.5. It's not much harder to convert an improper fraction to a decimal. Example: 1 3/4= 7/4= 1.75. The last result can be obtained without division, if we take into account that 3/4 = 0.75 and add one: 1 + 0.75 = 1.75.
However, not all ordinary fractions are so simple. For example, let's try to convert 1/3 from ordinary fractions to decimals. Even someone who had a C in mathematics (using a five-point system) will notice that no matter how long the division continues, after zero and a comma there will be an infinite number of triples 1/3 = 0.3333…. . It is customary to read this way: zero point, three in period. It is written accordingly as follows: 1/3=0,(3). A similar situation will occur if you try to convert 5/6 into a decimal fraction: 5/6=0.8(3). Such fractions are called infinite periodic. Here is an example for the fraction 3/7: 3/7= 0.42857142857142857142857142857143…, that is, 3/7=0.(428571).
So, as a result of converting a common fraction into a decimal, you can get:
- non-periodic decimal fraction;
- periodic decimal fraction.
It should be noted that there are also infinite non-periodic fractions that are obtained by performing the following actions: taking the nth root, logarithm, potentiation. For example, √3= 1.732050807568877… . The famous number π≈ 3.1415926535897932384626433832795…. .
Let's now multiply 3 by 0,(3): 3×0,(3)=0,(9)=1. It turns out that 0,(9) is another form of writing unit. Likewise, 9=9/9.16=16.0, etc.
The question opposite to that given in the title of this article is also legitimate: “how to convert a decimal fraction into a regular one.” The answer to this question is given by an example: 0.5= 5/10=1/2. In the last example, we reduced the numerator and denominator of the fraction 5/10 by 5. That is, to turn a decimal into a common fraction, you need to represent it as a fraction with a denominator of 10.
It will be interesting to watch this video about what fractions are:
To learn how to convert a decimal fraction to a common fraction, see here:
It would seem that converting a decimal fraction into a regular fraction is an elementary topic, but many students do not understand it! Therefore, today we will take a detailed look at several algorithms at once, with the help of which you will understand any fractions in just a second.
Let me remind you that there are at least two forms of writing the same fraction: common and decimal. Decimal fractions are all kinds of constructions of the form 0.75; 1.33; and even −7.41. Here are examples of ordinary fractions that express the same numbers:
Now let's figure it out: how to move from decimal notation to regular notation? And most importantly: how to do this as quickly as possible?
Basic algorithm
In fact, there are at least two algorithms. And we'll look at both now. Let's start with the first one - the simplest and most understandable.
To convert a decimal to a fraction, you need to follow three steps:
An important note about negative numbers. If in the original example there is a minus sign in front of the decimal fraction, then in the output there should also be a minus sign in front of the common fraction. Here are some more examples:
Examples of transition from decimal notation of fractions to ordinary ones I would like to pay special attention to the last example. As you can see, the fraction 0.0025 contains many zeros after the decimal point. Because of this, you have to multiply the numerator and denominator by 10 as many as four times. Is it possible to somehow simplify the algorithm in this case?
Of course you can. And now we will look at an alternative algorithm - it is a little more difficult to understand, but after a little practice it works much faster than the standard one.
Faster way
This algorithm also has 3 steps. To get a fraction from a decimal, do the following:
- Count how many digits are after the decimal point. For example, the fraction 1.75 has two such digits, and 0.0025 has four. Let's denote this quantity by the letter $n$.
- Rewrite the original number as a fraction of the form $\frac(a)(((10)^(n)))$, where $a$ are all the digits of the original fraction (without the “starting” zeros on the left, if any), and $n$ is the same number of digits after the decimal point that we calculated in the first step. In other words, you need to divide the digits of the original fraction by one followed by $n$ zeros.
- If possible, reduce the resulting fraction.
That's it! At first glance, this scheme is more complicated than the previous one. But in fact it is both simpler and faster. Judge for yourself:
As you can see, in the fraction 0.64 there are two digits after the decimal point - 6 and 4. Therefore $n=2$. If we remove the comma and zeros on the left (in this case, just one zero), we get the number 64. Let’s move on to the second step: $((10)^(n))=((10)^(2))=100$, Therefore, the denominator is exactly one hundred. Well, then all that remains is to reduce the numerator and denominator. :)
Another example:
Here everything is a little more complicated. Firstly, there are already 3 numbers after the decimal point, i.e. $n=3$, so you have to divide by $((10)^(n))=((10)^(3))=1000$. Secondly, if we remove the comma from the decimal notation, we get this: 0.004 → 0004. Remember that the zeros on the left must be removed, so in fact we have the number 4. Then everything is simple: divide, reduce and get the answer.
Finally, the last example:
The peculiarity of this fraction is the presence of a whole part. Therefore, the output we get is an improper fraction of 47/25. You can, of course, try to divide 47 by 25 with a remainder and thus again isolate the whole part. But why complicate your life if this can be done at the stage of transformation? Well, let's figure it out.
What to do with the whole part
In fact, everything is very simple: if we want to get a proper fraction, then we need to remove the whole part from it during the transformation, and then, when we get the result, add it again to the right before the fraction line.
For example, consider the same number: 1.88. Let's score by one (the whole part) and look at the fraction 0.88. It can be easily converted:
Then we remember about the “lost” unit and add it to the front:
\[\frac(22)(25)\to 1\frac(22)(25)\]
That's it! The answer turned out to be the same as after selecting the whole part last time. A couple more examples:
\[\begin(align)& 2.15\to 0.15=\frac(15)(100)=\frac(3)(20)\to 2\frac(3)(20); \\& 13.8\to 0.8=\frac(8)(10)=\frac(4)(5)\to 13\frac(4)(5). \\\end(align)\]
This is the beauty of mathematics: no matter which way you go, if all the calculations are done correctly, the answer will always be the same. :)
In conclusion, I would like to consider one more technique that helps many.
Transformations "by ear"
Let's think about what a decimal even is. More precisely, how we read it. For example, the number 0.64 - we read it as "zero point 64 hundredths", right? Well, or just “64 hundredths”. The key word here is “hundredths”, i.e. number 100.
What about 0.004? This is “zero point 4 thousandths” or simply “four thousandths”. One way or another, the key word is “thousands”, i.e. 1000.
So what's the big deal? And the fact is that it is these numbers that ultimately “pop up” in the denominators at the second stage of the algorithm. Those. 0.004 is “four thousandths” or “4 divided by 1000”:
Try to practice yourself - it's very simple. The main thing is to read the original fraction correctly. For example, 2.5 is “2 whole, 5 tenths”, so
And some 1.125 is “1 whole, 125 thousandths”, so
In the last example, of course, someone will object that it is not obvious to every student that 1000 is divisible by 125. But here you need to remember that 1000 = 10 3, and 10 = 2 ∙ 5, therefore
\[\begin(align)& 1000=10\cdot 10\cdot 10=2\cdot 5\cdot 2\cdot 5\cdot 2\cdot 5= \\& =2\cdot 2\cdot 2\cdot 5\ cdot 5\cdot 5=8\cdot 125\end(align)\]
Thus, any power of ten can be decomposed only into factors 2 and 5 - it is these factors that need to be looked for in the numerator so that in the end everything is reduced.
This concludes the lesson. Let's move on to a more complex reverse operation - see "
