Convert the given number to a decimal. Converting decimal numbers to common fractions

Converting a fraction to a decimal

Let's say we want to convert the common 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 factorization 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 factorization 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 on the example of the same fraction 11/4. Let's divide 11 by 4 "corner":

In the response line, we got the integer part ( 2 ), and we also have the remainder ( 3 ). Previously, we ended the division on this, but now we know that a comma and several zeros can be attributed to the dividend ( 11 ) on the right, which we will mentally do now. After the decimal point comes the tenth place. Zero, which stands for the dividend in this category, we will attribute 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 integer part in the answer line:

Now we attribute to the remainder ( 2 ) zero, which stands for the dividend in the hundredths place and bring the division to the end:

As a result, we get, as before,

Now let's 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 seen such a remnant before. Therefore, we can immediately say that if we continue our division by the “corner”, then the next digit in the answer line will be 4, then the number 5 will go, then again 4 and again 5, and so on, ad infinitum:

27 / 11 = 2,454545454545...

We have received the so-called periodical a decimal fraction with a period of 45. For such fractions, a more compact notation is used, in which the period is written out only once, but at the same time it is enclosed in parentheses:

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

Generally speaking, if we divide one natural number by a “corner” by another, writing the answer as 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, which we have already met before (the set of possible residues is limited, since they are all obviously less than the divisor). In the first case, the result of division is a final decimal fraction, in the second case, a periodic one.

Converting a Periodic Decimal to a Common 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 comma and the opening parenthesis that indicates 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- "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 zero, and we arrive at a simple equation for 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 bring the calculations to the end yet, so that it can be clearly seen how this result can be written out immediately, omitting intermediate arguments. Decreasing in the numerator ( 245 ) is the fractional part of the number

a = 0,2(45)

if you delete 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 one, 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 there is a period in the numerator, and as many nines in the denominator as there are digits in the period. After reducing by 9, the resulting fraction is equal to

In the same way,

A fraction is a number that consists of one or more fractions of a unit. There are three types of fractions in mathematics: common, mixed, and decimal.

Common fractions

An ordinary fraction is written as a ratio in which the numerator reflects how many parts of the number are taken, and the denominator shows how many parts the unit is divided into. If the numerator is less than the denominator, then we have a proper fraction. For example: ½, 3/5, 8/9.

If the numerator is equal to or greater than the denominator, then we are dealing with an improper fraction. For example: 5/5, 9/4, 5/2 Dividing the numerator can result in a finite number. For example, 40/8 \u003d 5. Therefore, any integer can be written as an ordinary improper fraction or a series of such fractions. Consider writing the same number as a series of different .

mixed fractions

In general, a mixed fraction can be represented by the formula:

Thus, a mixed fraction is written as an integer and an ordinary proper fraction, and such a record is understood as the sum of a whole and its fractional part.

Decimals

A decimal is a special kind of fraction in which the denominator can be represented as a power of 10. There are infinite and finite decimals. When writing this type of fraction, the integer part is first indicated, then the fractional part is fixed through the separator (dot or comma).

The record of the fractional part is always determined by its dimension. The decimal entry looks like this:

Translation rules between different types of fractions

Converting a mixed fraction to a common fraction

A mixed fraction can only be converted to an improper fraction. For translation, it is necessary to bring the whole part to the same denominator as the fractional part. In general, it will look like this:
Consider the use of this rule on specific examples:

Converting an ordinary fraction to a mixed one

An improper common fraction can be converted into a mixed fraction by simple division, which results in an integer part and a remainder (fractional part).

For example, let's translate the fraction 439/31 into a mixed one:

Translation of an ordinary fraction

In some cases, converting a fraction to a decimal is quite simple. In this case, the basic property of a fraction is applied, the numerator and denominator are multiplied by the same number, in order to bring the divisor to the power of 10.

For example:

In some cases, you may need to find the quotient by dividing by a corner or using a calculator. And some fractions cannot be reduced to a final decimal fraction. For example, the fraction 1/3 will never give the final result when divided.

An improper fraction is one of the formats for writing an ordinary fraction. Like any ordinary fraction, it has a number above the line (numerator) and below it - the denominator. If the numerator is greater than the denominator, this is the hallmark of the wrong fraction. In this form, you can convert a mixed ordinary fraction. The decimal can also be represented in the wrong ordinary notation, but only if the separating comma is preceded by a number other than zero.

Instruction

In mixed fraction format, the numerator and denominator are separated from the integer part by a space. To convert such an entry to , first multiply its integer part (the number before the space) by the denominator of the fractional part. Add the resulting value to the numerator. The value calculated in this way will be the numerator of an improper fraction, and put the denominator of the mixed fraction in its denominator without any changes. For example, 5 7/11 in regular irregular format can be written like this: (5*11+7)/11 = 62/11.

To convert a decimal fraction to an incorrect ordinary notation, determine the number of digits after the decimal point separating the integer part from the fractional - it is equal to the number of digits to the right of this comma. Use the resulting number as an indicator of the power to which you need to raise ten in order to calculate the denominator of an improper fraction. The numerator is obtained without any calculations - just remove the comma from the decimal fraction. For example, if the original decimal is 12.585, the numerator of the corresponding wrong number should be 10³ = 1000, and the denominator should be 12585: 12.585 = 12585/1000.

Like any ordinary fraction, it can and should be reduced. To do this, after obtaining the result in the ways described in the previous two steps, try to find the greatest common divisor for the numerator and denominator. If you can do this, divide by what you found on both sides of the solid bar. For the example from the second step, this divisor will be the number 5, so the improper fraction can be reduced: 12.585 = 12585/1000 = 2517/200. And for the example from the first step, there is no common divisor, so there is no need to reduce the resulting improper fraction.

Related videos

Decimal fractions are more convenient for automated calculations than natural ones. Any natural fraction can be converted into natural numbers either without loss of accuracy, or with an accuracy of up to a given number of decimal places, depending on the ratio between the numerator and denominator.

Instruction

If necessary, round the result to the required number of decimal places. The rounding rules are as follows: if the highest of the deleted digits contains a digit from 0 to 4, then the next highest digit (which is not deleted) does not change, and if the digit is from 5 to 9, it increases by one. If the last of these operations is subjected to a digit with the number 9, the unit is transferred to another, even more senior digit, like a column. Please note that rounding up to the available number of character spaces does not always perform this operation. Sometimes there are hidden digits in his memory that are not displayed on the indicator. Logarithmic, having low accuracy (up to two decimal places), often at the same time copes with rounding in the right direction better.

If you find that a certain sequence of digits is repeated after the decimal point, place this sequence in brackets. They say about her that she is "", because she repeats periodically. For example, number 53.7854785478547854... can be written as 53,(7854).

A proper fraction, the value of which is greater than one, consists of two parts: a whole and a fraction. First, divide the numerator of the fractional part by its denominator. Then add the result of the division to the integer part. After that, if necessary, round the result to the required number of decimal places, or find the frequency and highlight it in brackets.

Decimals are easy to handle. They are recognized by calculators and many computer programs. But sometimes it is necessary, for example, to draw up a proportion. To do this, you have to convert the decimal fraction to a common fraction. It will not be difficult if you make a short digression into the school curriculum.

Instruction

Reduce the fractional part of the resulting . To do this, the numerator and denominator of the fraction must be divided by the same divisor. In this case, it is the number "5". So "5/10" is converted to "1/2".

Choose a number so that the result of its multiplication by the denominator is 10. Reasoning from the reverse: is it possible to turn the number 4 into 10? Answer: no, because 10 is not divisible by 4. Then 100? Yes, 100 is divisible by 4 without a remainder, the result is 25. Multiply the numerator and denominator by 25 and write the answer in decimal form:
¼ = 25/100 = 0.25.

It is not always possible to use the selection method, there are two more ways. Their principle is almost the same, only the recording differs. One of them is the gradual allocation of decimal places. Example: translate the fraction 1/8.

Very often in the school mathematics curriculum, children are faced with the problem of how to convert a common fraction to a decimal. In order to convert a common fraction to a decimal, let's first recall what a common fraction and a decimal fraction are. A common 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 \u003d 1 and 1/3;

Converting an ordinary fraction to a decimal

Now let's look at how to convert a mixed fraction to a decimal. Any ordinary fraction, whether it is correct or incorrect, can be converted to a decimal. To do this, you need to divide the numerator by the denominator. Example: simple fraction (proper) 1/2. We divide the numerator 1 by the denominator 2, we get 0.5. Take the example of 45/12, it is immediately clear that this is an improper fraction. Here the denominator is less than the numerator. We turn the improper fraction into a decimal: 45: 12 \u003d 3.75.

Convert 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 we divide the numerator equal to 1 by the denominator equal to 8, in a column or on a calculator, and we get a decimal fraction equal to 0.125. The article provides the easiest examples of converting to decimal fractions. Having understood the translation technique using simple examples, you can easily solve the most complex ones.

A decimal has two parts separated by commas. The first part is an integer unit, the second part is tens (if the number after the decimal point is one), hundreds (two numbers after the decimal point, like two zeros in a hundred), thousandths, etc. Let's look at examples of decimals: 0, 2; 7, 54; 235.448; 5.1; 6.32; 0.5. These are all decimals. How do you convert a decimal to a common fraction?

Example one

We have a fraction, for example, 0.5. As mentioned above, it consists of two parts. The first number, 0, shows how many integer units the fraction has. In our case, they are not. The second number shows tens. The fraction even reads zero point five tenths. Decimal number convert to fraction now it will not be difficult, we write 5/10. If you see that the numbers have a common divisor, you can reduce the fraction. We have this number 5, dividing both parts of the fraction by 5, we get - 1/2.

Example two

Let's take a more complex fraction - 2.25. It is read like this - two whole and twenty-five hundredths. Pay attention - hundredths, since there are two numbers after the decimal point. Now you can convert to a common fraction. We write down - 2 25/100. The integer part is 2, the fractional part is 25/100. As in the first example, this part can be shortened. The common divisor for 25 and 100 is 25. Note that we always choose the greatest common divisor. Dividing both parts of the fraction by GCD, we got 1/4. So 2, 25 is 2 1/4.

Example three

And to consolidate the material, let's take the decimal fraction 4.112 - four whole and one hundred and twelve thousandths. Why thousandths, I think, is clear. Now we write down 4 112/1000. According to the algorithm, we find the GCD of the numbers 112 and 1000. In our case, this is the number 6. We get 4 14/125.

Conclusion

We break the fraction into integer and fractional parts.
We look at how many digits after the decimal point. If one is tens, two is hundreds, three is thousandths, etc.
We write the fraction in the usual form.
We reduce the numerator and denominator of the fraction.
Write down the resulting fraction.
We perform a check, divide the upper part of the fraction by the lower one. If there is an integer part, add to the resulting decimal fraction. It turned out the original version - great, so you did everything right.

Using examples, I showed how you can convert a decimal fraction to an ordinary one. As you can see, it is very easy and simple to do this.