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From regular to decimal. Converting a fraction into a understandable number

If we need to divide 497 by 4, then when dividing we will see that 497 is not evenly divisible by 4, i.e. the remainder of the division remains. In such cases it is said that it is completed division with remainder, and the solution is written as follows:
497: 4 = 124 (1 remainder).

The division components on the left side of the equality are called the same as in division without a remainder: 497 - dividend, 4 - divider. The result of division when divided with a remainder is called incomplete private. In our case, this is the number 124. And finally, the last component, which is not in ordinary division, is remainder. In cases where there is no remainder, one number is said to be divided by another without a trace, or completely. It is believed that with such a division the remainder is zero. In our case, the remainder is 1.

The remainder is always less than the divisor.

Division can be checked by multiplication. If, for example, there is an equality 64: 32 = 2, then the check can be done like this: 64 = 32 * 2.

Often in cases where division with a remainder is performed, it is convenient to use the equality
a = b * n + r,
where a is the dividend, b is the divisor, n is the partial quotient, r is the remainder.

The quotient of natural numbers can be written as a fraction.

The numerator of a fraction is the dividend, and the denominator is the divisor.

Since the numerator of a fraction is the dividend and the denominator is the divisor, believe that the line of a fraction means the action of division. Sometimes it is convenient to write division as a fraction without using the ":" sign.

The quotient of the division of natural numbers m and n can be written as a fraction $\frac(m)(n)$, where the numerator m is the dividend, and the denominator n is the divisor:
$m:n = \frac(m)(n) $

The following rules are true:

To get the fraction $\frac(m)(n)$, you need to divide the unit into n equal parts (shares) and take m such parts.

To get the fraction $\frac(m)(n)$, you need to divide the number m by the number n.

To find a part of a whole, you need to divide the number corresponding to the whole by the denominator and multiply the result by the numerator of the fraction that expresses this part.

To find a whole from its part, you need to divide the number corresponding to this part by the numerator and multiply the result by the denominator of the fraction that expresses this part.

If both the numerator and denominator of a fraction are multiplied by the same number (except zero), the value of the fraction will not change:
$\large \frac(a)(b) = \frac(a \cdot n)(b \cdot n) $

If both the numerator and denominator of a fraction are divided by the same number (except zero), the value of the fraction will not change:
$\large \frac(a)(b) = \frac(a: m)(b: m) $
This property is called main property of a fraction.

The last two transformations are called reducing a fraction.

If fractions need to be represented as fractions with the same denominator, then this action is called reducing fractions to a common denominator.

Proper and improper fractions. Mixed numbers

You already know that a fraction can be obtained by dividing a whole into equal parts and taking several such parts. For example, the fraction $\frac(3)(4)$ means three-quarters of one. In many of the problems in the previous paragraph, fractions were used to represent parts of a whole. Common sense suggests that the part should always be less than the whole, but then what about fractions such as, for example, $\frac(5)(5)$ or $\frac(8)(5)$? It is clear that this is no longer part of the unit. This is probably why fractions whose numerator is greater than or equal to the denominator are called improper fractions. The remaining fractions, i.e. fractions whose numerator is less than the denominator, are called correct fractions.

As you know, any common fraction, both proper and improper, can be thought of as the result of dividing the numerator by the denominator. Therefore, in mathematics, unlike ordinary language, the term “improper fraction” does not mean that we did something wrong, but only that the numerator of this fraction is greater than or equal to the denominator.

If a number consists of an integer part and a fraction, then such fractions are called mixed.

For example:
$5:3 = 1\frac(2)(3)$ : 1 - whole part, and $\frac(2)(3)$ is the fractional part.

If the numerator of the fraction $\frac(a)(b)$ is divisible by a natural number n, then in order to divide this fraction by n, its numerator must be divided by this number:
$\large \frac(a)(b) : n = \frac(a:n)(b) $

If the numerator of the fraction $\frac(a)(b)$ is not divisible by a natural number n, then to divide this fraction by n, you need to multiply its denominator by this number:
$\large \frac(a)(b) : n = \frac(a)(bn) $

Note that the second rule is also true when the numerator is divisible by n. Therefore, we can use it when it is difficult to determine at first glance whether the numerator of a fraction is divisible by n or not.

Actions with fractions. Adding fractions.

You can perform arithmetic operations with fractional numbers, just like with natural numbers. Let's look at adding fractions first. Easily add fractions with same denominators. Let us find, for example, the sum of $\frac(2)(7)$ and $\frac(3)(7)$. It is easy to understand that $\frac(2)(7) + \frac(2)(7) = \frac(5)(7) $

To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.

Using letters, the rule for adding fractions with like denominators can be written as follows:
$\large \frac(a)(c) + \frac(b)(c) = \frac(a+b)(c) $

If you need to add fractions with different denominators, then they must first be brought to a common denominator. For example:
$\large \frac(2)(3)+\frac(4)(5) = \frac(2\cdot 5)(3\cdot 5)+\frac(4\cdot 3)(5\cdot 3 ) = \frac(10)(15)+\frac(12)(15) = \frac(10+12)(15) = \frac(22)(15) $

For fractions, as for natural numbers, the commutative and associative properties of addition are valid.

Adding mixed fractions

Notations such as $2\frac(2)(3)$ are called mixed fractions. In this case, the number 2 is called whole part mixed fraction, and the number $\frac(2)(3)$ is its fractional part. The entry $2\frac(2)(3)$ is read as follows: “two and two thirds.”

When dividing the number 8 by the number 3, you can get two answers: $\frac(8)(3)$ and $2\frac(2)(3)$. They express the same fractional number, i.e. $\frac(8)(3) = 2 \frac(2)(3)$

Thus, the improper fraction $\frac(8)(3)$ is represented as a mixed fraction $2\frac(2)(3)$. In such cases it is said that improper fraction highlighted the whole part.

Subtracting fractions (fractional numbers)

Subtraction fractional numbers, like natural numbers, is determined on the basis of the action of addition: subtracting another from one number means finding a number that, when added to the second, gives the first. For example:
$\frac(8)(9)-\frac(1)(9) = \frac(7)(9) $ since $\frac(7)(9)+\frac(1)(9 ) = \frac(8)(9)$

The rule for subtracting fractions with like denominators is similar to the rule for adding such fractions:
To find the difference between fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and leave the denominator the same.

Using letters, this rule is written like this:
$\large \frac(a)(c)-\frac(b)(c) = \frac(a-b)(c) $

Multiplying fractions

To multiply a fraction by a fraction, you need to multiply their numerators and denominators and write the first product as the numerator, and the second as the denominator.

Using letters, the rule for multiplying fractions can be written as follows:
$\large \frac(a)(b) \cdot \frac(c)(d) = \frac(a \cdot c)(b \cdot d) $

Using the formulated rule, you can multiply a fraction by a natural number, by a mixed fraction, and also multiply mixed fractions. To do this, you need to write a natural number as a fraction with a denominator of 1, a mixed fraction - as an improper fraction.

The result of multiplication should be simplified (if possible) by reducing the fraction and isolating the whole part of the improper fraction.

For fractions, as for natural numbers, the commutative and combinative properties of multiplication, as well as the distributive property of multiplication relative to addition, are valid.

Division of fractions

Let's take the fraction $\frac(2)(3)$ and “flip” it, swapping the numerator and denominator. We get the fraction $\frac(3)(2)$. This fraction is called reverse fractions $\frac(2)(3)$.

If we now “reverse” the fraction $\frac(3)(2)$, we will get the original fraction $\frac(2)(3)$. Therefore, fractions such as $\frac(2)(3)$ and $\frac(3)(2)$ are called mutually inverse.

For example, the fractions $\frac(6)(5) $ and $\frac(5)(6) $, $\frac(7)(18) $ and $\frac (18)(7)$.

Using letters, reciprocal fractions can be written as follows: $\frac(a)(b) $ and $\frac(b)(a) $

It is clear that the product of reciprocal fractions is equal to 1. For example: $\frac(2)(3) \cdot \frac(3)(2) =1 $

Using reciprocal fractions, you can reduce division of fractions to multiplication.

The rule for dividing a fraction by a fraction is:
To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor.

Using letters, the rule for dividing fractions can be written as follows:
$\large \frac(a)(b) : \frac(c)(d) = \frac(a)(b) \cdot \frac(d)(c) $

If the dividend or divisor is a natural number or a mixed fraction, then in order to use the rule for dividing fractions, it must first be represented as an improper fraction.

Author on Youtube: Anastasia Ivanova

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Convert decimal to normal

Every decimal fraction can be represented as a regular fraction. Just write using the denominator to do this.

The basic rule for converting a decimal to a regular fraction is to read the decimal, but it is usually written. For example:

2,3 - two points out of three tens

Since the fraction is complete, it can be converted to a mixed number or irregular fraction:

Converting a correct fraction to a decimal

A non-traditional fraction can be converted to a decimal, just as for conventional decimal notation, the denominator must be followed by one or more zeros, such as 10, 100, 1000, and so on.

How to convert total fraction to decimal

If we expand such a denominator with the primary factors, we get the same number of doublings and five:

100 = 10 10 = 2 5 2.5

1000 = 10 10 10 = 2 5 2 5 2 5

There are no other prime factors, so these extensions do not contain, so:

A regular fraction can be represented as a decimal only if its denominator contains no factors other than 2 and 5.

Let's take part:

When the denominator is extended to the main factors, the result is a product of 2 2:

If you multiply it by two fours, equate the number five to two, you will get one of the required denominators - 100.

To get a passage equal to this, the counter must be multiplied by the product of two five:

Let's look at another faction:

When the denominator is extended to the main factors, the product is 2.7, containing the number 7:

A factor of 7 will be present in the denominator to multiply it or the integers, so that a product containing only two and five will never occur.

Therefore, this fraction cannot be reduced to any of the necessary denominators: 10, 100, 1000, etc. This means that it cannot be represented as a decimal number.

A regular incompatible fraction cannot be represented as a decimal if its denominator contains at least one major factor from one to two.

Note that the rule only talks about irreversible fractions, since some fractions can be represented as decimal abbreviations.

Let's look at two parts:

Now all that's left is to multiply as phrasal fractions by 5 to get 10 in the denominator, and you can convert the fraction to a decimal:

How to convert a decimal fraction to a common fraction

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 translate decimal As usual, you need to complete three steps:

Rewrite the original fraction as a new fraction: the original decimal fraction will remain in the numerator, and you need to put one in the denominator. In this case, the sign of the original number is also placed in the numerator.
For example:
Multiply the numerator and denominator of the resulting fraction by 10 until the decimal point disappears from the numerator. Let me remind you: for each multiplication by 10, the decimal point is shifted to the right by one place. Of course, since the denominator is also multiplied, instead of the number 1 there will appear 10, 100, etc.
Finally, we reduce the resulting fraction by standard scheme: divide the numerator and denominator by the numbers to which they are multiples. For example, in the first example 0.75=75/100, and both 75 and 100 are divisible by 25.
Therefore, we get $0.75=\frac(75)(100)=\frac(3\cdot 25)(4\cdot 25)=\frac(3)(4)$ - that’s the whole answer. :)

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.

Converting a fraction to a decimal

Here are some more examples:

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 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 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 all! 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).

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 = 103, and 10 = 2 ∙ 5, so

\[\begin(align)& 1000=10\cdot 10\cdot 10=2\cdot 5\cdot 2\cdot 5\cdot 2\cdot 5= \\& =2\cdot 2\cdot 2\cdot 5\ cdot 5\cdot 5=8\cdot 125\end(align)\]

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 "Transition from an ordinary fraction to a decimal."

Then press the buttons and the task is completed. The result will be either a whole number or a decimal fraction. A decimal fraction may have a long remainder after . In this case, the fraction must be rounded to the specific digit you need, using rounding (numbers up to 5 are rounded down, from 5 inclusive and more - up).

If you don't have a calculator at hand, you will have to. Write the numerator of the fraction with the denominator, with a corner between them indicating . For example, convert the fraction 10/6 to a number. First, divide 10 by 6. You get 1. Write the result in a corner. Multiply 1 by 6, you get 6. Subtract 6 from 10. You get a remainder of 4. The remainder must be divided by 6 again. Add the number 0 to 4, and divide 40 by 6. You get 6. Write 6 in the result, after the decimal point. Multiply 6 by 6. You get 36. Subtract 36 from 40. The remainder is again 4. You don’t need to continue further, since it becomes obvious that the result will be the number 1.66(6). Round this fraction to the digit you need. For example, 1.67. This is the final result.

Sources:

converting fractions with whole numbers

Fractions are used to represent numbers that consist of one or more parts of a unit. The term "fraction" comes from the Latin fractura, which means "to crush, break." There are differences between ordinary and decimal fractions. Moreover, in ordinary fractions, a unit can be divided into any number of parts, and in a decimal, this quantity must be a multiple of 10. Any fraction can be either ordinary or decimal.

You will need

To calculate the result you will need a calculator or a piece of paper and a pen.

Instructions

So, first, take a common fraction and divide it into parts. For example, 2 1\8, in which 2 is an integer part, and 1\8 is a fraction. From it you can see that the number was divided by 8, but only one was taken. The part taken is the numerator, and the number of parts divided by is the denominator.

note

There are often fractions that cannot be completely converted to decimals. In this case, rounding comes to the rescue. If you want to round to the nearest thousand, look at the fourth decimal place. If it is less than 5, then write down the answer, the first three digits after the decimal point without changing, otherwise you must add one to the last digit of the three. For example, 0.89643123 can be written as 0.896, but 0.89663123 is 0.897.

Helpful advice

If you are calculating the result manually, then before dividing the fraction it is better to reduce it as much as possible, and also separate whole parts from it.

Sources:

how to convert fractions

Fraction is one of the elements of formulas for entering in the Word word processor there is a Microsoft Equation tool. Using it, you can enter any complex mathematical or physical formulas, equations and other elements that include special characters.

Instructions

To launch the Microsoft Equation tool, you need to go to: “Insert” -> “Object”, in the dialog box that opens, on the first tab from the list you need to select Microsoft Equation and click “Ok” or double-click on the selected item. After launching the editor, a toolbar will open in front of you and an input field will be displayed: a dotted rectangle. The toolbar is divided into sections, each of which contains a set of action symbols or expressions. When you click on one of the sections, a list of tools located in it will expand. From the list that opens, select the desired symbol and click on it. Once selected, the specified symbol will appear in the selected rectangle in the document.

The section containing elements for writing fractions is located in the second line of the toolbar. When you hover your mouse over it, you will see the tooltip “Patterns of Fractions and Radicals”. Click the section once and expand the list. The drop-down menu contains templates for horizontal and oblique fractions. From the options that appear, you can choose the one that suits your task. Click on the desired option. After clicking, a fraction symbol and places for entering the numerator and denominator, framed by a dotted line, will appear in the input field that opens in the document. The default cursor is automatically placed in the numerator input field. Enter the numerator. In addition to numbers, you can also enter symbols, letters or action signs. They can be entered either from the keyboard or from the corresponding sections of the Microsoft Equation toolbar. After the numerator, press the TAB key to move to the denominator. You can also go by clicking in the field to enter the denominator. Once written, click the mouse pointer anywhere in the document, the toolbar will close, and entering the fraction will be completed. To edit, double-click on it with the left mouse button.

If, when you open the “Insert” -> “Object” menu, you do not find the Microsoft Equation tool in the list, you need to install it. Run installation disk, disk image or Word distribution file. In the installer window that appears, select “Add or remove components. Add or remove individual components" and click "Next". In the next window, check the “Advanced application settings” option. Click Next. In the next window, find the “Office Tools” list item and click on the plus sign on the left. In the expanded list, we are interested in the “Formula Editor” item. Click on the icon next to “Formula Editor” and, in the menu that opens, click “Run from Computer”. After that, click “Update” and wait until the required component is installed.

Converting a Fraction to a Decimal

Let's say we want to convert the fraction 11/4 to a decimal. The easiest way to do it is this:


						2∙2∙5∙5

We succeeded because in this case the decomposition of the denominator into prime factors consists only of twos. We supplemented this expansion with two more fives, took advantage of the fact that 10 = 2∙5, and got a decimal fraction. Such a procedure is obviously possible if and only if the decomposition of the denominator into prime factors contains nothing but twos and fives. If any other prime number is present in the expansion of the denominator, then such a fraction cannot be converted to a decimal. Nevertheless, we will try to do this, but only in a different way, which we will get acquainted with using the example of the same fraction 11/4. Let's divide 11 by 4 using the “corner”:

In the response line we received the whole part (2), and we also have the remainder (3). Previously, we ended the division here, but now we know that we can add a comma and several zeros to the right of the dividend (11), which we will now mentally do. After the decimal point comes the tenths place. The zero that appears at the dividend in this digit will be added to the resulting remainder (3):

Now the division can continue as if nothing had happened. You just need to remember to put a comma after the whole part in the answer line:

Now we add a zero to the remainder (2), which is in the hundredths place of the dividend, and complete the division:

As a result, we get, as before,

Let's now try to calculate in exactly the same way what the fraction 27/11 is equal to:

We received the number 2.45 in the answer line, and the number 5 in the remainder line. But we have already encountered such a remnant before. Therefore, we can immediately say that if we continue our division with a “corner”, then the next number in the answer line will be 4, then the number 5 will come, then again 4 and again 5, and so on, ad infinitum:

27 / 11 = 2,454545454545...

We got the so-called periodic a decimal fraction with a period of 45. For such fractions, a more compact notation is used, in which the period is written only once, but it is enclosed in parentheses:

2,454545454545... = 2,(45).

Generally speaking, if we divide one natural number by another with a “corner”, writing the answer in the form of a decimal fraction, then only two outcomes are possible: (1) either sooner or later we will get zero in the remainder line, (2) or there will be such a remainder there, which we have already encountered before (the set of possible remainders is limited, since all of them are obviously smaller than the divisor). In the first case, the result of division is a finite decimal fraction, in the second case - a periodic one.

Convert periodic decimal to fraction

Let us be given a positive periodic decimal fraction with a zero integer part, for example:

a = 0,2(45).

How can I convert this fraction back to a common fraction?

Let's multiply it by 10 k, Where k is the number of digits between the decimal point and the opening parenthesis indicating the beginning of the period. In this case k= 1 and 10 k = 10:

a∙ 10 k = 2,(45).

Multiply the result by 10 n, Where n- the “length” of the period, that is, the number of digits enclosed between parentheses. In this case n= 2 and 10 n = 100:

a∙ 10 k ∙ 10 n = 245,(45).

Now let's calculate the difference

a∙ 10 k ∙ 10 n − a∙ 10 k = 245,(45) − 2,(45).

Since the fractional parts of the minuend and the subtrahend are the same, then the fractional part of the difference is equal to zero, and we come to simple equation relatively a:

a∙ 10 k ∙ (10 n − 1) = 245 − 2.

This equation is solved using the following transformations:

a∙ 10 ∙ (100 − 1) = 245 − 2.

a∙ 10 ∙ 99 = 245 − 2.

	245 − 2
	10 ∙ 99

We deliberately do not complete the calculations yet, so that it is clearly visible how this result can be immediately written down, omitting intermediate arguments. The minuend in the numerator (245) is the fractional part of the number

a = 0,2(45)

if you erase the brackets in her entry. The subtrahend in the numerator (2) is the non-periodic part of the number A, located between the comma and the opening parenthesis. The first factor in the denominator (10) is a unit, to which as many zeros are assigned as there are digits in the non-periodic part ( k). The second factor in the denominator (99) is as many nines as there are digits in the period ( n).

Now our calculations can be completed:

Here the numerator contains the period, and the denominator contains as many nines as there are digits in the period. After reduction by 9, the resulting fraction is equal to

In the same way,

In this article we will look at how converting fractions to decimals, and also consider the reverse process - converting decimal fractions into ordinary fractions. Here we will outline the rules for converting fractions and provide detailed solutions to typical examples.

Page navigation.

Converting fractions to decimals

Let us denote the sequence in which we will deal with converting fractions to decimals.

First, we'll look at how to represent fractions with denominators 10, 100, 1,000, ... as decimals. This is explained by the fact that decimal fractions are essentially a compact form of writing ordinary fractions with denominators 10, 100, ....

After that, we will go further and show how to write any ordinary fraction (not just those with denominators 10, 100, ...) as a decimal fraction. When ordinary fractions are treated in this way, both finite decimal fractions and infinite periodic decimal fractions are obtained.

Now let's talk about everything in order.

Converting common fractions with denominators 10, 100, ... to decimals

Some proper fractions require "preliminary preparation" before being converted to decimals. This applies to ordinary fractions, the number of digits in the numerator of which is less than the number of zeros in the denominator. For example, the common fraction 2/100 must first be prepared for conversion to a decimal fraction, but the fraction 9/10 does not need any preparation.

“Preliminary preparation” of proper ordinary fractions for conversion to decimal fractions consists of adding so many zeros to the left in the numerator that the total number of digits there becomes equal to the number of zeros in the denominator. For example, a fraction after adding zeros will look like .

Once you have a proper fraction prepared, you can begin converting it to a decimal.

Let's give rule for converting a proper common fraction with a denominator of 10, or 100, or 1,000, ... into a decimal fraction. It consists of three steps:

write 0;
after it we put a decimal point;
We write down the number from the numerator (along with added zeros, if we added them).

Let's consider the application of this rule when solving examples.

Example.

Convert the proper fraction 37/100 to a decimal.

Solution.

The denominator contains the number 100, which has two zeros. The numerator contains the number 37, its notation has two digits, therefore, this fraction does not need to be prepared for conversion to a decimal fraction.

Now we write 0, put a decimal point, and write the number 37 from the numerator, and we get the decimal fraction 0.37.

Answer:

0,37 .

To strengthen the skills of converting proper ordinary fractions with numerators 10, 100, ... into decimal fractions, we will analyze the solution to another example.

Example.

Write the proper fraction 107/10,000,000 as a decimal.

Solution.

The number of digits in the numerator is 3, and the number of zeros in the denominator is 7, so this common fraction needs to be prepared for conversion to a decimal. We need to add 7-3=4 zeros to the left in the numerator so that the total number of digits there becomes equal to the number of zeros in the denominator. We get.

All that remains is to create the required decimal fraction. To do this, firstly, we write 0, secondly, we put a comma, thirdly, we write the number from the numerator together with zeros 0000107, as a result we have a decimal fraction 0.0000107.

Answer:

0,0000107 .

Improper fractions do not require any preparation when converting to decimals. The following should be adhered to rules for converting improper fractions with denominators 10, 100, ... into decimals:

write down the number from the numerator;
separate decimal point there are as many digits on the right as there are zeros in the denominator of the original fraction.

Let's look at the application of this rule when solving an example.

Example.

Convert the improper fraction 56,888,038,009/100,000 to a decimal.

Solution.

Firstly, we write down the number from the numerator 56888038009, and secondly, we separate the 5 digits on the right with a decimal point, since the denominator of the original fraction has 5 zeros. As a result, we have the decimal fraction 568880.38009.

Answer:

568 880,38009 .

To convert a mixed number to a decimal fraction, the denominator of the fractional part of which is the number 10, or 100, or 1,000, ..., you can convert mixed number into an improper fraction, and then convert the resulting fraction to a decimal fraction. But you can also use the following the rule for converting mixed numbers with a fractional denominator of 10, or 100, or 1,000, ... into decimal fractions:

if necessary, we perform “preliminary preparation” of the fractional part of the original mixed number by adding required amount zeros on the left in the numerator;
write down the integer part of the original mixed number;
put a decimal point;
We write down the number from the numerator along with the added zeros.

Let's look at an example in which we complete all the necessary steps to represent a mixed number as a decimal fraction.

Example.

Convert the mixed number to a decimal.

Solution.

The denominator of the fractional part has 4 zeros, and the numerator contains the number 17, consisting of 2 digits, therefore, we need to add two zeros to the left in the numerator so that the number of digits there becomes equal to the number of zeros in the denominator. Having done this, the numerator will be 0017.

Now we write down the integer part of the original number, that is, the number 23, put a decimal point, after which we write the number from the numerator along with the added zeros, that is, 0017, and we get the desired decimal fraction 23.0017.

Let's write down the whole solution briefly: .

Of course, it was possible to first represent the mixed number as an improper fraction and then convert it to a decimal fraction. With this approach, the solution looks like this: .

Answer:

23,0017 .

Converting fractions to finite and infinite periodic decimals

You can convert not only ordinary fractions with denominators 10, 100, ... into a decimal fraction, but also ordinary fractions with other denominators. Now we will figure out how this is done.

In some cases, the original ordinary fraction is easily reduced to one of the denominators 10, or 100, or 1,000, ... (see bringing an ordinary fraction to a new denominator), after which it is not difficult to represent the resulting fraction as a decimal fraction. For example, it is obvious that the fraction 2/5 can be reduced to a fraction with a denominator 10, for this you need to multiply the numerator and denominator by 2, which will give the fraction 4/10, which, according to the rules discussed in the previous paragraph, is easily converted to the decimal fraction 0, 4 .

In other cases, you have to use another method of converting an ordinary fraction to a decimal, which we now move on to consider.

To convert an ordinary fraction to a decimal fraction, the numerator of the fraction is divided by the denominator, the numerator is first replaced by an equal decimal fraction with any number of zeros after the decimal point (we talked about this in the section equal and unequal decimal fractions). In this case, division is performed in the same way as division by a column of natural numbers, and in the quotient a decimal point is placed when the division of the whole part of the dividend ends. All this will become clear from the solutions to the examples given below.

Example.

Convert the fraction 621/4 to a decimal.

Solution.

Let's represent the number in the numerator 621 as a decimal fraction, adding a decimal point and several zeros after it. First, let's add 2 digits 0, later, if necessary, we can always add more zeros. So, we have 621.00.

Now let's divide the number 621,000 by 4 with a column. The first three steps are no different from dividing natural numbers by a column, after which we arrive at the following picture:

This is how we get to the decimal point in the dividend, and the remainder is different from zero. In this case, we put a decimal point in the quotient and continue dividing in a column, not paying attention to the commas:

This completes the division, and as a result we get the decimal fraction 155.25, which corresponds to the original ordinary fraction.

Answer:

155,25 .

To consolidate the material, consider the solution to another example.

Example.

Convert the fraction 21/800 to a decimal.

Solution.

To convert this common fraction to a decimal, we divide with a column of the decimal fraction 21,000... by 800. After the first step, we will have to put a decimal point in the quotient, and then continue the division:

Finally, we got the remainder 0, this completes the conversion of the common fraction 21/400 to a decimal fraction, and we arrived at the decimal fraction 0.02625.

Answer:

0,02625 .

It may happen that when dividing the numerator by the denominator of an ordinary fraction, we still do not get a remainder of 0. In these cases, division can be continued indefinitely. However, starting from a certain step, the remainders begin to repeat periodically, and the numbers in the quotient also repeat. This means that the original fraction is converted to an infinite periodic decimal fraction. Let's show this with an example.

Example.

Write the fraction 19/44 as a decimal.

Solution.

To convert an ordinary fraction to a decimal, perform division by column:

It is already clear that during division the residues 8 and 36 began to be repeated, while in the quotient the numbers 1 and 8 are repeated. Thus, the original common fraction 19/44 is converted into a periodic decimal fraction 0.43181818...=0.43(18).

Answer:

0,43(18) .

To conclude this point, we will figure out which ordinary fractions can be converted into finite decimal fractions, and which ones can only be converted into periodic ones.

Let us have an irreducible ordinary fraction in front of us (if the fraction is reducible, then we first reduce the fraction), and we need to find out which decimal fraction it can be converted into - finite or periodic.

It is clear that if an ordinary fraction can be reduced to one of the denominators 10, 100, 1,000, ..., then the resulting fraction can be easily converted into a final decimal fraction according to the rules discussed in the previous paragraph. But to the denominators 10, 100, 1,000, etc. Not all ordinary fractions are given. Only fractions whose denominators are at least one of the numbers 10, 100, ... can be reduced to such denominators. And what numbers can be divisors of 10, 100, ...? The numbers 10, 100, ... will allow us to answer this question, and they are as follows: 10 = 2 5, 100 = 2 2 5 5, 1,000 = 2 2 2 5 5 5, .... It follows that the divisors are 10, 100, 1,000, etc. There can only be numbers whose decompositions into prime factors contain only the numbers 2 and (or) 5.

Now we can make a general conclusion about converting ordinary fractions to decimals:

if in the decomposition of the denominator into prime factors only the numbers 2 and (or) 5 are present, then this fraction can be converted into a final decimal fraction;
if, in addition to twos and fives, there are other prime numbers in the expansion of the denominator, then this fraction is converted to an infinite decimal periodic fraction.

Example.

Without converting ordinary fractions to decimals, tell me which of the fractions 47/20, 7/12, 21/56, 31/17 can be converted into a final decimal fraction, and which ones can only be converted into a periodic fraction.

Solution.

The denominator of the fraction 47/20 is factorized into prime factors as 20=2·2·5. In this expansion there are only twos and fives, so this fraction can be reduced to one of the denominators 10, 100, 1,000, ... (in this example, to the denominator 100), therefore, can be converted to a final decimal fraction.

The decomposition of the denominator of the fraction 7/12 into prime factors has the form 12=2·2·3. Since it contains a prime factor of 3, different from 2 and 5, this fraction cannot be represented as a finite decimal, but can be converted into a periodic decimal.

Fraction 21/56 – contractile, after contraction it takes the form 3/8. Factoring the denominator into prime factors contains three factors equal to 2, therefore, the common fraction 3/8, and therefore the equal fraction 21/56, can be converted into a final decimal fraction.

Finally, the expansion of the denominator of the fraction 31/17 is 17 itself, therefore this fraction cannot be converted into a finite decimal fraction, but can be converted into an infinite periodic fraction.

Answer:

47/20 and 21/56 can be converted to a finite decimal fraction, but 7/12 and 31/17 can only be converted to a periodic fraction.

Ordinary fractions do not convert to infinite non-periodic decimals

The information in the previous paragraph gives rise to the question: “Can dividing the numerator of a fraction by the denominator result in an infinite non-periodic fraction?”

Answer: no. When converting a common fraction, the result can be either a finite decimal fraction or an infinite periodic decimal fraction. Let us explain why this is so.

From the theorem on divisibility with a remainder, it is clear that the remainder is always less than the divisor, that is, if we divide some integer by an integer q, then the remainder can only be one of the numbers 0, 1, 2, ..., q−1. It follows that after the column has completed dividing the integer part of the numerator of an ordinary fraction by the denominator q, in no more than q steps one of the following two situations will arise:

or we will get a remainder of 0, this will end the division, and we will get the final decimal fraction;
or we will get a remainder that has already appeared before, after which the remainders will begin to repeat as in the previous example (since when dividing equal numbers equal remainders are obtained on q, which follows from the already mentioned divisibility theorem), this will result in an infinite periodic decimal fraction.

There cannot be any other options, therefore, when converting an ordinary fraction to a decimal fraction, an infinite non-periodic decimal fraction cannot be obtained.

From the reasoning given in this paragraph it also follows that the length of the period of a decimal fraction is always less than the value of the denominator of the corresponding ordinary fraction.

Converting decimals to fractions

Now let's figure out how to convert a decimal fraction into an ordinary fraction. Let's start by converting final decimal fractions to ordinary fractions. After this, we will consider a method for inverting infinite periodic decimal fractions. In conclusion, let's say about the impossibility of converting infinite non-periodic decimal fractions into ordinary fractions.

Converting trailing decimals to fractions

Obtaining a fraction that is written as a final decimal is quite simple. The rule for converting a final decimal fraction to a common fraction consists of three steps:

firstly, write the given decimal fraction into the numerator, having previously discarded the decimal point and all zeros on the left, if any;
secondly, write one into the denominator and add as many zeros to it as there are digits after the decimal point in the original decimal fraction;
thirdly, if necessary, reduce the resulting fraction.

Let's look at the solutions to the examples.

Example.

Convert the decimal 3.025 to a fraction.

Solution.

If we remove the decimal point from the original decimal fraction, we get the number 3,025. There are no zeros on the left that we would discard. So, we write 3,025 in the numerator of the desired fraction.

We write the number 1 into the denominator and add 3 zeros to the right of it, since in the original decimal fraction there are 3 digits after the decimal point.

So we got the common fraction 3,025/1,000. This fraction can be reduced by 25, we get .

Answer:

Example.

Convert the decimal fraction 0.0017 to a fraction.

Solution.

Without a decimal point, the original decimal fraction looks like 00017, discarding the zeros on the left we get the number 17, which is the numerator of the desired ordinary fraction.

We write one with four zeros in the denominator, since the original decimal fraction has 4 digits after the decimal point.

As a result, we have an ordinary fraction 17/10,000. This fraction is irreducible, and the conversion of a decimal fraction to an ordinary fraction is complete.

Answer:

When the integer part of the original final decimal fraction is non-zero, it can be immediately converted to a mixed number, bypassing the common fraction. Let's give rule for converting a final decimal fraction to a mixed number:

the number before the decimal point must be written as an integer part of the desired mixed number;
in the numerator of the fractional part you need to write the number obtained from the fractional part of the original decimal fraction after discarding all the zeros on the left;
in the denominator of the fractional part you need to write down the number 1, to which add as many zeros to the right as there are digits after the decimal point in the original decimal fraction;
if necessary, reduce the fractional part of the resulting mixed number.

Let's look at an example of converting a decimal fraction to a mixed number.

Example.

Express the decimal fraction 152.06005 as a mixed number

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