This would seem to be the translation decimal to normal - elementary topic, but many students don’t 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. Decimals- these are all kinds of designs 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 decimal notation go to normal? 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:
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 ordinary 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 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 obtain ordinary fraction from decimal, you need to 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 all! 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 you remove the comma and zeros on the left (in in this case— only one zero), then we get the number 64. Let’s move on to the second step: $((10)^(n))=((10)^(2))=100$, so the denominator is exactly one hundred. Well, then all that remains is to reduce the numerator and denominator. :)
One more 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 improper fraction 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 correct fraction, then it is necessary to remove the whole part from it for the duration of the transformations, and then, when we get the result, add it again to the right before the fractional 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 all! The answer turned out to be the same as after selecting the whole part in 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”. Anyway, keyword- “thousandths”, 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 is 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 "
Decimal concept
Fractions in which the denominator is a power of 10 are often written in a simpler form, without a denominator, separating the whole and fractional parts with a comma (it is believed that whole part the proper fraction is 0).
For example,
Fractions written in this form are called in decimals. So there are 2.7 different forms of writing the same number: the first is in the form of an ordinary fraction, the second is in the form of a decimal fraction. For now we will only consider positive decimals.
The decimal form of notating fractions allows you to write, compare and perform with them arithmetic operations according to rules very similar to the rules for writing, comparing and operating with natural numbers.
Let us recall that in decimal system In notation, the meaning of each digit depends on the digit (position) in which it is written. In this case, the units of adjacent digits differ by 10 times. For example, ten is 10 times less than a hundred, one is 10 times less than ten.
The first place after the decimal point is called tenth place.
For example, the number 2.7 consists of 2 point seven, read “two point seven.”
The second place after the decimal point is called hundredths place.
For example, the number 0.35 consists of 0 whole, 3 tenths and 5 hundredths - read “zero point thirty-five hundredths”.
To better understand the rules for writing and reading decimal fractions, consider the table of digits and the examples of writing numbers given in it.
To write a number in decimal form, you need to take into account that
So the recording of a number contains 1 thousandth and 9 ten-thousandths and does not contain whole units, tenths, hundredths - in the decimal fraction, zeros are written in the corresponding digits.
It must be remembered that after the decimal point there must be as many digits after the decimal point as there are zeros in the denominator of this fraction.
fractional number.
Decimal notation of a fractional number is a set of two or more digits from $0$ to $9$, between which there is a so-called \textit (decimal point).
Example 1
For example, $35.02$; $100.7$; $123\456.5$; $54.89$.
The leftmost digit in the decimal notation of a number cannot be zero, the only exception being when the decimal point is immediately after the first digit $0$.
Example 2
For example, $0.357$; $0.064$.
Often the decimal point is replaced with a decimal point. For example, $35.02$; $100.7$; $123\456.5$; $54.89$.
Decimal definition
Definition 1
Decimals-- these are fractional numbers that are represented in decimal notation.
For example, $121.05; $67.9$; $345.6700$.
Decimals are used to more compactly write proper fractions, the denominators of which are the numbers $10$, $100$, $1\000$, etc. And mixed numbers, the denominators of the fractional part of which are the numbers $10$, $100$, $1\000$, etc.
For example, common fraction$\frac(8)(10)$ can be written as a decimal $0.8$, and the mixed number $405\frac(8)(100)$ can be written as a decimal $405.08$.
Reading Decimals
Decimal fractions, which correspond to regular fractions, are read the same as ordinary fractions, only the phrase “zero integer” is added in front. For example, the common fraction $\frac(25)(100)$ (read “twenty-five hundredths”) corresponds to the decimal fraction $0.25$ (read “zero point twenty-five hundredths”).
Decimal fractions that correspond to mixed numbers are read the same way as mixed numbers. For example, the mixed number $43\frac(15)(1000)$ corresponds to the decimal fraction $43.015$ (read “forty-three point fifteen thousandths”).
Places in decimals
In writing a decimal fraction, the meaning of each digit depends on its position. Those. in decimal fractions the concept also applies category.
Places in decimal fractions up to decimal point are called the same as the digits in natural numbers. The decimal places after the decimal point are listed in the table:
Picture 1.
Example 3
For example, in the decimal fraction $56.328$, the digit $5$ is in the tens place, $6$ is in the units place, $3$ is in the tenths place, $2$ is in the hundredths place, $8$ is in the thousandths place.
Places in decimal fractions are distinguished by precedence. When reading a decimal fraction, move from left to right - from senior rank to younger.
Example 4
For example, in the decimal fraction $56.328$, the most significant (highest) place is the tens place, and the low (lowest) place is the thousandths place.
A decimal fraction can be expanded into digits similar to the digit decomposition of a natural number.
Example 5
For example, let's break down the decimal fraction $37.851$ into digits:
$37,851=30+7+0,8+0,05+0,001$
Ending decimals
Definition 2
Ending decimals are called decimal fractions whose records contain final number characters (digits).
For example, $0.138$; $5.34$; $56.123456$; $350,972.54.
Any finite decimal fraction can be converted to a fraction or a mixed number.
Example 6
For example, the final decimal fraction $7.39$ answers a fractional number$7\frac(39)(100)$, and the final decimal fraction $0.5$ corresponds to the proper common fraction $\frac(5)(10)$ (or any fraction that is equal to it, for example, $\frac(1) (2)$ or $\frac(10)(20)$.
Converting a fraction to a decimal
Converting fractions with denominators $10, 100, \dots$ to decimals
Before converting some proper fractions to decimals, they must first be “prepared.” The result of such preparation should be the same number of digits in the numerator and the same number of zeros in the denominator.
The essence of " preliminary preparation» converting regular fractions to decimals - adding such a number of zeros to the left in the numerator so that the total number of digits becomes equal to the number of zeros in the denominator.
Example 7
For example, let's prepare the fraction $\frac(43)(1000)$ for conversion to a decimal and get $\frac(043)(1000)$. And the ordinary fraction $\frac(83)(100)$ does not need any preparation.
Let's formulate rule for converting a proper common fraction with a denominator of $10$, or $100$, or $1\000$, $\dots$ into a decimal fraction:
write $0$;
after it put a decimal point;
write down the number from the numerator (along with added zeros after preparation, if necessary).
Example 8
Convert the proper fraction $\frac(23)(100)$ to a decimal.
Solution.
The denominator contains the number $100$, which contains $2$ and two zeros. The numerator contains the number $23$, which is written with $2$.digits. This means that there is no need to prepare this fraction for conversion to a decimal.
Let's write $0$, put a decimal point and write down the number $23$ from the numerator. We get the decimal fraction $0.23$.
Answer: $0,23$.
Example 9
Write the proper fraction $\frac(351)(100000)$ as a decimal.
Solution.
The numerator of this fraction contains $3$ digits, and the number of zeros in the denominator is $5$, so this ordinary fraction must be prepared for conversion to a decimal. To do this, you need to add $5-3=2$ zeros to the left in the numerator: $\frac(00351)(100000)$.
Now we can form the desired decimal fraction. To do this, write down $0$, then add a comma and write down the number from the numerator. We get the decimal fraction $0.00351$.
Answer: $0,00351$.
Let's formulate rule for converting improper fractions with denominators $10$, $100$, $\dots$ into decimal fractions:
write down the number from the numerator;
Use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.
Example 10
Convert the improper fraction $\frac(12756)(100)$ to a decimal.
Solution.
Let's write down the number from the numerator $12756$, then separate the $2$ digits on the right with a decimal point, because the denominator of the original fraction $2$ is zero. We get the decimal fraction $127.56$.
To rational number m/n is written as a decimal fraction; you need to divide the numerator by the denominator. In this case, the quotient is written as a finite or infinite decimal fraction.
Write down given number as a decimal fraction.
Solution. Divide the numerator of each fraction into a column by its denominator: A) divide 6 by 25; b) divide 2 by 3; V) divide 1 by 2, and then add the resulting fraction to one - the integer part of this mixed number.
Irreducible ordinary fractions whose denominators do not contain prime factors other than 2 And 5 , are written as a final decimal fraction.
IN example 1 when A) denominator 25=5·5; when V) the denominator is 2, so we get the final decimals 0.24 and 1.5. When b) the denominator is 3, so the result cannot be written as a finite decimal.
Is it possible, without long division, to convert into a decimal fraction such an ordinary fraction, the denominator of which does not contain other divisors other than 2 and 5? Let's figure it out! What fraction is called a decimal and is written without a fraction bar? Answer: fraction with denominator 10; 100; 1000, etc. And each of these numbers is a product equal number of twos and fives. In fact: 10=2 ·5 ; 100=2 ·5 ·2 ·5 ; 1000=2 ·5 ·2 ·5 ·2 ·5 etc.
Consequently, the denominator of an irreducible ordinary fraction will need to be represented as the product of “twos” and “fives”, and then multiplied by 2 and (or) 5 so that the “twos” and “fives” become equal. Then the denominator of the fraction will be equal to 10 or 100 or 1000, etc. To ensure that the value of the fraction does not change, we multiply the numerator of the fraction by the same number by which we multiplied the denominator.
Express the following common fractions as decimals:
Solution. Each of these fractions is irreducible. Let's factor the denominator of each fraction into prime factors.
20=2·2·5. Conclusion: one “A” is missing.
8=2·2·2. Conclusion: three “A”s are missing.
25=5·5. Conclusion: two “twos” are missing.
Comment. In practice, they often do not use factorization of the denominator, but simply ask the question: by how much should the denominator be multiplied so that the result is one with zeros (10 or 100 or 1000, etc.). And then the numerator is multiplied by the same number.
So, in case A)(example 2) from the number 20 you can get 100 by multiplying by 5, therefore, you need to multiply the numerator and denominator by 5.
When b)(example 2) from the number 8 the number 100 will not be obtained, but the number 1000 will be obtained by multiplying by 125. Both the numerator (3) and the denominator (8) of the fraction are multiplied by 125.
When V)(example 2) from 25 you get 100 if you multiply by 4. This means that the numerator 8 must be multiplied by 4.
An infinite decimal fraction in which one or more digits invariably repeat in the same sequence is called periodic as a decimal. The set of repeating digits is called the period of this fraction. For brevity, the period of a fraction is written once, enclosed in parentheses.
When b)(example 1) there is only one repeating digit and is equal to 6. Therefore, our result 0.66... will be written like this: 0,(6) . They read: zero point, six in period.
If there are one or more non-repeating digits between the decimal point and the first period, then such a periodic fraction is called a mixed periodic fraction.
An irreducible common fraction whose denominator is together with others multiplier contains multiplier 2 or 5 , becomes mixed periodic fraction.
Write the numbers as a decimal fraction:
Any rational number can be written as an infinite periodic decimal fraction.
Write it as infinite periodic fraction numbers.
We have already said that there are fractions ordinary And decimal. On this moment We've studied fractions a little. We learned that there are regular and improper fractions. We also learned that common fractions can be reduced, added, subtracted, multiplied and divided. And we also learned that there are so-called mixed numbers, which consist of an integer and a fractional part.
We haven't fully explored common fractions yet. There are many subtleties and details that should be talked about, but today we will begin to study decimal fractions, since ordinary and decimal fractions often have to be combined. That is, when solving problems you have to use both types of fractions.
This lesson may seem complicated and confusing. It's quite normal. These kinds of lessons require that they be studied, and not skimmed superficially.
Lesson contentExpressing quantities in fractional form
Sometimes it is convenient to show something in fractional form. For example, one tenth of a decimeter is written like this:
This expression means that one decimeter was divided into ten parts, and from these ten parts one part was taken:
As you can see in the figure, one tenth of a decimeter is one centimeter.
Consider the following example. Show 6 cm and another 3 mm in centimeters in fractional form.
So, you need to express 6 cm and 3 mm in centimeters, but in fractional form. We already have 6 whole centimeters:
but there are still 3 millimeters left. How to show these 3 millimeters, and in centimeters? Fractions come to the rescue. 3 millimeters is the third part of a centimeter. And the third part of a centimeter is written as cm
A fraction means that one centimeter was divided by ten equal parts, and from these ten parts they took three parts (three out of ten).
As a result, we have six whole centimeters and three tenths of a centimeter:
In this case, 6 shows the number of whole centimeters, and the fraction shows the number of fractional centimeters. This fraction is read as "six point three centimeters".
Fractions whose denominator contains the numbers 10, 100, 1000 can be written without a denominator. First write the whole part, and then the numerator of the fractional part. The integer part is separated from the numerator of the fractional part by a comma.
For example, let's write it without a denominator. To do this, let's first write down the whole part. The integer part is the number 6. First we write down this number:
The whole part is recorded. Immediately after writing the whole part we put a comma:
And now we write down the numerator of the fractional part. In a mixed number, the numerator of the fractional part is the number 3. We write a three after the decimal point:
Any number that is represented in this form is called decimal.
Therefore, you can show 6 cm and another 3 mm in centimeters using a decimal fraction:
6.3 cm
It will look like this:
In fact, decimals are the same as ordinary fractions and mixed numbers. The peculiarity of such fractions is that the denominator of their fractional part contains the numbers 10, 100, 1000 or 10000.
Like a mixed number, a decimal fraction has an integer part and a fractional part. For example, in a mixed number the integer part is 6, and the fractional part is .
In the decimal fraction 6.3, the integer part is the number 6, and the fractional part is the numerator of the fraction, that is, the number 3.
It also happens that ordinary fractions in the denominator of which the numbers 10, 100, 1000 are given without an integer part. For example, a fraction is given without a whole part. To write such a fraction as a decimal, first write 0, then put a comma and write the numerator of the fraction. A fraction without a denominator will be written as follows:
Reads like "zero point five".
Converting mixed numbers to decimals
When we write mixed numbers without a denominator, we thereby convert them to decimal fractions. When converting fractions to decimals, there are a few things you need to know, which we'll talk about now.
After the whole part is written down, it is necessary to count the number of zeros in the denominator of the fractional part, since the number of zeros of the fractional part and the number of digits after the decimal point in the decimal fraction must be the same. What does it mean? Consider the following example:
At first
And you could immediately write down the numerator of the fractional part and the decimal fraction is ready, but you definitely need to count the number of zeros in the denominator of the fractional part.
So, we count the number of zeros in the fractional part of a mixed number. The denominator of the fractional part has one zero. This means that in a decimal fraction there will be one digit after the decimal point and this digit will be the numerator of the fractional part of the mixed number, that is, the number 2
Thus, when converted to a decimal fraction, a mixed number becomes 3.2.
This decimal fraction reads like this:
"Three point two"
“Tenths” because the number 10 is in the fractional part of a mixed number.
Example 2. Convert a mixed number to a decimal.
Write down the whole part and put a comma:
And you could immediately write down the numerator of the fractional part and get the decimal fraction 5.3, but the rule says that after the decimal point there should be as many digits as there are zeros in the denominator of the fractional part of the mixed number. And we see that the denominator of the fractional part has two zeros. This means that our decimal fraction must have two digits after the decimal point, not one.
In such cases, the numerator of the fractional part needs to be slightly modified: add a zero before the numerator, that is, before the number 3
Now you can convert this mixed number to a decimal fraction. Write down the whole part and put a comma:
And write down the numerator of the fractional part:
The decimal fraction 5.03 is read as follows:
"Five point three"
“Hundreds” because the denominator of the fractional part of a mixed number contains the number 100.
Example 3. Convert a mixed number to a decimal.
From previous examples, we learned that to successfully convert a mixed number to a decimal, the number of digits in the numerator of the fraction and the number of zeros in the denominator of the fraction must be the same.
Before converting a mixed number to a decimal fraction, its fractional part needs to be slightly modified, namely, to make sure that the number of digits in the numerator of the fractional part and the number of zeros in the denominator of the fractional part are the same.
First of all, we look at the number of zeros in the denominator of the fractional part. We see that there are three zeros:
Our task is to organize three digits in the numerator of the fractional part. We already have one digit - this is the number 2. It remains to add two more digits. They will be two zeros. Add them before the number 2. As a result, the number of zeros in the denominator and the number of digits in the numerator will be the same:
Now you can start converting this mixed number to a decimal fraction. First we write down the whole part and put a comma:
and immediately write down the numerator of the fractional part
3,002
We see that the number of digits after the decimal point and the number of zeros in the denominator of the fractional part of the mixed number are the same.
The decimal fraction 3.002 is read as follows:
"Three point two thousandths"
“Thousandths” because the denominator of the fractional part of the mixed number contains the number 1000.
Converting fractions to decimals
Common fractions with denominators of 10, 100, 1000, or 10000 can also be converted to decimals. Since an ordinary fraction does not have an integer part, first write down 0, then put a comma and write down the numerator of the fractional part.
Here also the number of zeros in the denominator and the number of digits in the numerator must be the same. Therefore, you should be careful.
Example 1.
The whole part is missing, so first we write 0 and put a comma:
Now we look at the number of zeros in the denominator. We see that there is one zero. And the numerator has one digit. This means you can safely continue the decimal fraction by writing the number 5 after the decimal point
In the resulting decimal fraction 0.5, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
The decimal fraction 0.5 is read as follows:
"Zero point five"
Example 2. Convert a fraction to a decimal.
A whole part is missing. First we write 0 and put a comma:
Now we look at the number of zeros in the denominator. We see that there are two zeros. And the numerator has only one digit. To make the number of digits and the number of zeros the same, add one zero in the numerator before the number 2. Then the fraction will take the form . Now the number of zeros in the denominator and the number of digits in the numerator are the same. So you can continue the decimal fraction:
In the resulting decimal fraction 0.02, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
The decimal fraction 0.02 is read as follows:
“Zero point two.”
Example 3. Convert a fraction to a decimal.
Write 0 and put a comma:
Now we count the number of zeros in the denominator of the fraction. We see that there are five zeros, and there is only one digit in the numerator. To make the number of zeros in the denominator and the number of digits in the numerator the same, you need to add four zeros in the numerator before the number 5:
Now the number of zeros in the denominator and the number of digits in the numerator are the same. So we can continue with the decimal fraction. Write the numerator of the fraction after the decimal point
In the resulting decimal fraction 0.00005, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
The decimal fraction 0.00005 is read as follows:
“Zero point five hundred thousandths.”
Converting improper fractions to decimals
An improper fraction is a fraction in which the numerator is greater than the denominator. There are improper fractions in which the denominator contains the numbers 10, 100, 1000 or 10000. Such fractions can be converted to decimals. But before converting to a decimal fraction, such fractions must be separated into the whole part.
Example 1.
The fraction is an improper fraction. To convert such a fraction to a decimal fraction, you must first select the whole part of it. Let's remember how to isolate the whole part of improper fractions. If you have forgotten, we advise you to return to and study it.
So, let's highlight the whole part in the improper fraction. Recall that a fraction means division - in this case, dividing the number 112 by the number 10
Let's look at this picture and assemble a new mixed number, like a children's construction set. The number 11 will be the integer part, the number 2 will be the numerator of the fractional part, and the number 10 will be the denominator of the fractional part.
We got a mixed number. Let's convert it to a decimal fraction. And we already know how to convert such numbers into decimal fractions. First, write down the whole part and put a comma:
Now we count the number of zeros in the denominator of the fractional part. We see that there is one zero. And the numerator of the fractional part has one digit. This means that the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part are the same. This gives us the opportunity to immediately write down the numerator of the fractional part after the decimal point:
In the resulting decimal fraction 11.2, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
This means that an improper fraction becomes 11.2 when converted to a decimal.
The decimal fraction 11.2 is read as follows:
"Eleven point two."
Example 2. Convert improper fraction to decimal.
It is an improper fraction because the numerator is greater than the denominator. But it can be converted to a decimal fraction, since the denominator contains the number 100.
First of all, let's select the whole part of this fraction. To do this, divide 450 by 100 with a corner:
Let's collect a new mixed number - we get . And we already know how to convert mixed numbers into decimal fractions.
Write down the whole part and put a comma:
Now we count the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part. We see that the number of zeros in the denominator and the number of digits in the numerator are the same. This gives us the opportunity to immediately write down the numerator of the fractional part after the decimal point:
In the resulting decimal fraction 4.50, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. This means the fraction is translated correctly.
This means that an improper fraction becomes 4.50 when converted to a decimal.
When solving problems, if there are zeros at the end of the decimal fraction, they can be discarded. Let's also drop the zero in our answer. Then we get 4.5
This is one of interesting features decimal fractions. It lies in the fact that the zeros that appear at the end of a fraction do not give this fraction any weight. In other words, the decimals 4.50 and 4.5 are equal. Let's put an equal sign between them:
4,50 = 4,5
The question arises: why does this happen? After all, it looks like 4.50 and 4.5 different fractions. The whole secret lies in the basic property of fractions, which we studied earlier. We will try to prove why the decimal fractions 4.50 and 4.5 are equal, but after studying next topic, which is called “converting a decimal to a mixed number.”
Converting a decimal to a mixed number
Any decimal fraction can be converted back to a mixed number. To do this, it is enough to be able to read decimal fractions. For example, let's convert 6.3 to a mixed number. 6.3 is six point three. First we write down six integers:
and next to three tenths:
Example 2. Convert decimal 3.002 to mixed number
3.002 is three whole and two thousandths. First we write down three integers
and next to it we write two thousandths:
Example 3. Convert decimal 4.50 to mixed number
4.50 is four point fifty. Write down four integers
and next fifty hundredths:
By the way, let's remember the last example from the previous topic. We said that the decimals 4.50 and 4.5 are equal. We also said that the zero can be discarded. Let's try to prove that the decimals 4.50 and 4.5 are equal. To do this, we convert both decimal fractions into mixed numbers.
When converted to a mixed number, the decimal 4.50 becomes , and the decimal 4.5 becomes
We have two mixed numbers and . Let's convert these mixed numbers to improper fractions:
Now we have two fractions and . It's time to remember the basic property of a fraction, which says that when you multiply (or divide) the numerator and denominator of a fraction by the same number, the value of the fraction does not change.
Let's divide the first fraction by 10
We got , and this is the second fraction. This means that both are equal to each other and equal to the same value:
Try using a calculator to divide first 450 by 100, and then 45 by 10. It will be a funny thing.
Converting a decimal fraction to a fraction
Any decimal fraction can be converted back to a fraction. To do this, again, it is enough to be able to read decimal fractions. For example, let's convert 0.3 to a common fraction. 0.3 is zero point three. First we write down zero integers:
and next to three tenths 0. Zero is traditionally not written down, so the final answer will not be 0, but simply .
Example 2. Convert the decimal fraction 0.02 to a fraction.
0.02 is zero point two. We don’t write down zero, so we immediately write down two hundredths
Example 3. Convert 0.00005 to fraction
0.00005 is zero point five. We don’t write down zero, so we immediately write down five hundred thousandths
Did you like the lesson?
Join our new group VKontakte and start receiving notifications about new lessons