Division by decimal fraction. Decimals, definitions, notation, examples, operations with decimals

Mathematical-Calculator-Online v.1.0

The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, root extraction, exponentiation, percentage calculations and other operations.

Solution:

How to use a math calculator

Key	Designation	Explanation
5	numbers 0-9	Arabic numerals. Entering natural integers, zero. To get a negative integer, you must press the +/- key
.	period (comma)	Separator to indicate a decimal fraction. If there is no number before the point (comma), the calculator will automatically substitute a zero before the point. For example: .5 - 0.5 will be written
+	plus sign	Adding numbers (integers, decimals)
-	minus sign	Subtracting numbers (integers, decimals)
÷	division sign	Dividing numbers (integers, decimals)
X	multiplication sign	Multiplying numbers (integers, decimals)
√	root	Extracting the root of a number. When you press the “root” button again, the root of the result is calculated. For example: root of 16 = 4; root of 4 = 2
x 2	squaring	Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16
1/x	fraction	Output in decimal fractions. The numerator is 1, the denominator is the entered number
%	percent	Getting a percentage of a number. To work, you need to enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button
(	open parenthesis	An open parenthesis to specify the calculation priority. A closed parenthesis is required. Example: (2+3)*2=10
)	closed parenthesis	A closed parenthesis to specify the calculation priority. An open parenthesis is required
±	plus minus	Reverses sign
=	equals	Displays the result of the solution. Also above the calculator, in the “Solution” field, intermediate calculations and the result are displayed.
←	deleting a character	Removes the last character
WITH	reset	Reset button. Completely resets the calculator to position "0"

Algorithm of the online calculator using examples

Addition.

Addition of natural integers (5 + 7 = 12)

Addition of whole natural and negative numbers { 5 + (-2) = 3 }

Adding decimals fractional numbers { 0,3 + 5,2 = 5,5 }

Subtraction.

Subtracting natural integers ( 7 - 5 = 2 )

Subtracting natural and negative integers ( 5 - (-2) = 7 )

Subtracting decimal fractions ( 6.5 - 1.2 = 4.3 )

Multiplication.

Product of natural integers (3 * 7 = 21)

Product of natural and negative integers ( 5 * (-3) = -15 )

Product of decimal fractions ( 0.5 * 0.6 = 0.3 )

Division.

Division of natural integers (27 / 3 = 9)

Division of natural and negative integers (15 / (-3) = -5)

Division of decimal fractions (6.2 / 2 = 3.1)

Extracting the root of a number.

Extracting the root of an integer ( root(9) = 3)

Extracting the root of decimal fractions (root(2.5) = 1.58)

Extracting the root of a sum of numbers ( root(56 + 25) = 9)

Extracting the root of the difference between numbers (root (32 – 7) = 5)

Squaring a number.

Squaring an integer ( (3) 2 = 9 )

Squaring decimals ((2,2)2 = 4.84)

Conversion to decimal fractions.

Calculating percentages of a number

Increase the number 230 by 15% ( 230 + 230 * 0.15 = 264.5 )

Reduce the number 510 by 35% ( 510 – 510 * 0.35 = 331.5 )

18% of the number 140 is (140 * 0.18 = 25.2)

This article is about decimals. Here we will deal with decimal notation fractional numbers, introduce the concept of a decimal fraction and give examples of decimal fractions. Next we’ll talk about the digits of decimal fractions and give the names of the digits. After this, we will focus on infinite decimal fractions, let's talk about periodic and non-periodic fractions. Next we list the basic operations with decimal fractions. In conclusion, let us establish the position of decimal fractions on the coordinate beam.

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Decimal notation of a fractional number

Reading Decimals

Let's say a few words about the rules for reading decimal fractions.

Decimals, which correspond to the correct common fractions, are read the same way as these ordinary fractions, only “zero integers” are added first. For example, the decimal fraction 0.12 corresponds to the common fraction 12/100 (read “twelve hundredths”), therefore, 0.12 is read as “zero point twelve hundredths”.

Decimal fractions that correspond to mixed numbers, are read exactly the same as these mixed numbers. For example, the decimal fraction 56.002 corresponds to a mixed number, so the decimal fraction 56.002 is read as “fifty-six point two thousandths.”

Places in decimals

In writing decimal fractions, as well as in writing natural numbers, the meaning of each digit depends on its position. Indeed, the number 3 in the decimal fraction 0.3 means three tenths, in the decimal fraction 0.0003 - three ten thousandths, and in the decimal fraction 30,000.152 - three ten thousandths. So we can talk about decimal places, as well as about the digits in natural numbers.

Names of decimal places up to decimal point completely coincide with the names of the digits in natural numbers. And the names of the decimal places after the decimal point can be seen from the following table.

For example, in the decimal fraction 37.051, the digit 3 is in the tens place, 7 is in the units place, 0 is in the tenths place, 5 is in the hundredths place, and 1 is in the thousandths place.

Places in decimal fractions also differ in precedence. If in writing a decimal fraction we move from digit to digit from left to right, then we will move from seniors To junior ranks. For example, the hundreds place is older than the tenths place, and the millionth place is lower than the hundredths place. In a given final decimal fraction we can talk about the major and minor digits. For example, in decimal fraction 604.9387 senior (highest) the place is the hundreds place, and junior (lowest)- ten-thousandths digit.

For decimal fractions, expansion into digits takes place. It is similar to expansion by digits of natural numbers. For example, the expansion into decimal places of 45.6072 is as follows: 45.6072=40+5+0.6+0.007+0.0002. And the properties of addition from the decomposition of a decimal fraction into digits allow you to move on to other representations of this decimal fraction, for example, 45.6072=45+0.6072, or 45.6072=40.6+5.007+0.0002, or 45.6072= 45.0072+0.6.

Ending decimals

Up to this point, we have only talked about decimal fractions, in the notation of which there is a finite number of digits after the decimal point. Such fractions are called finite decimals.

Definition.

Ending decimals- These are decimal fractions, the records of which contain a finite number of characters (digits).

Here are some examples of final decimal fractions: 0.317, 3.5, 51.1020304958, 230,032.45.

However, not every fraction can be represented as a final decimal. For example, the fraction 5/13 cannot be replaced by an equal fraction with one of the denominators 10, 100, ..., therefore, cannot be converted into a final decimal fraction. We will talk more about this in the theory section, converting ordinary fractions to decimals.

Infinite Decimals: Periodic Fractions and Non-Periodic Fractions

In writing a decimal fraction after the decimal point, it is possible to allow for the possibility of an infinite number of digits. In this case, we will come to consider the so-called infinite decimal fractions.

Definition.

Infinite decimals- These are decimal fractions, which contain an infinite number of digits.

It is clear that we cannot write down infinite decimal fractions in full form, so in their writing we limit ourselves to only a certain finite number of digits after the decimal point and put an ellipsis indicating an infinitely continuing sequence of digits. Here are some examples of infinite decimal fractions: 0.143940932…, 3.1415935432…, 153.02003004005…, 2.111111111…, 69.74152152152….

If you look closely at the last two infinite decimal fractions, then in the fraction 2.111111111... the endlessly repeating number 1 is clearly visible, and in the fraction 69.74152152152..., starting from the third decimal place, a repeating group of numbers 1, 5 and 2 is clearly visible. Such infinite decimal fractions are called periodic.

Definition.

Periodic decimals(or just periodic fractions) are endless decimal fractions, in the recording of which, starting from a certain decimal place, some number or group of numbers is endlessly repeated, which is called period of the fraction.

For example, the period of the periodic fraction 2.111111111... is the digit 1, and the period of the fraction 69.74152152152... is a group of digits of the form 152.

For infinite periodic decimal fractions, a special form of notation is adopted. For brevity, we agreed to write down the period once, enclosing it in parentheses. For example, the periodic fraction 2.111111111... is written as 2,(1) , and the periodic fraction 69.74152152152... is written as 69.74(152) .

It is worth noting that for the same periodic decimal fraction you can specify different periods. For example, the periodic decimal fraction 0.73333... can be considered as a fraction 0.7(3) with a period of 3, and also as a fraction 0.7(33) with a period of 33, and so on 0.7(333), 0.7 (3333), ... You can also look at the periodic fraction 0.73333 ... like this: 0.733(3), or like this 0.73(333), etc. Here, in order to avoid ambiguity and discrepancies, we agree to consider as the period of a decimal fraction the shortest of all possible sequences of repeating digits, and starting from the closest position to the decimal point. That is, the period of the decimal fraction 0.73333... will be considered a sequence of one digit 3, and the periodicity starts from the second position after the decimal point, that is, 0.73333...=0.7(3). Another example: the periodic fraction 4.7412121212... has a period of 12, the periodicity starts from the third digit after the decimal point, that is, 4.7412121212...=4.74(12).

Infinite decimal periodic fractions are obtained by converting into decimal fractions ordinary fractions whose denominators contain prime factors, different from 2 and 5.

Here it is worth mentioning periodic fractions with a period of 9. Let us give examples of such fractions: 6.43(9) , 27,(9) . These fractions are another notation periodic fractions with period 0, and they are usually replaced by periodic fractions with period 0. To do this, period 9 is replaced by period 0, and the value of the next highest digit is increased by one. For example, a fraction with period 9 of the form 7.24(9) is replaced by a periodic fraction with period 0 of the form 7.25(0) or an equal final decimal fraction 7.25. Another example: 4,(9)=5,(0)=5. The equality of a fraction with period 9 and its corresponding fraction with period 0 is easily established after replacing these decimal fractions with equal ordinary fractions.

Finally, let's take a closer look at infinite decimal fractions, which do not contain an endlessly repeating sequence of digits. They are called non-periodic.

Definition.

Non-recurring decimals(or just non-periodic fractions) are infinite decimal fractions that have no period.

Sometimes non-periodic fractions have a form similar to that of periodic fractions, for example, 8.02002000200002... is a non-periodic fraction. In these cases, you should be especially careful to notice the difference.

Note that non-periodic fractions do not convert to ordinary fractions; infinite non-periodic decimal fractions represent irrational numbers.

Operations with decimals

One of the operations with decimal fractions is comparison, and the four basic arithmetic functions are also defined operations with decimals: addition, subtraction, multiplication and division. Let's consider separately each of the actions with decimal fractions.

Comparison of decimals essentially based on comparison of ordinary fractions corresponding to the decimal fractions being compared. However, converting decimal fractions into ordinary fractions is a rather labor-intensive process, and infinite non-periodic fractions cannot be represented as an ordinary fraction, so it is convenient to use a place-wise comparison of decimal fractions. Place-wise comparison of decimal fractions is similar to comparison of natural numbers. For more detailed information, we recommend studying the article: comparison of decimal fractions, rules, examples, solutions.

Let's move on to the next step - multiplying decimals. Multiplication of finite decimal fractions is carried out similarly to subtraction of decimal fractions, rules, examples, solutions to multiplication by a column of natural numbers. In the case of periodic fractions, multiplication can be reduced to multiplication of ordinary fractions. In turn, the multiplication of infinite non-periodic decimal fractions after rounding them is reduced to the multiplication of finite decimal fractions. We recommend for further study the material in the article: multiplication of decimal fractions, rules, examples, solutions.

Decimals on a coordinate ray

There is a one-to-one correspondence between points and decimals.

Let's figure out how points on the coordinate ray are constructed that correspond to a given decimal fraction.

We can replace finite decimal fractions and infinite periodic decimal fractions with equal ordinary fractions, and then construct the corresponding ordinary fractions on the coordinate ray. For example, the decimal fraction 1.4 corresponds to the common fraction 14/10, so the point with coordinate 1.4 is removed from the origin in the positive direction by 14 segments equal to a tenth of a unit segment.

Decimal fractions can be marked on a coordinate ray, starting from the decomposition of a given decimal fraction into digits. For example, let us need to build a point with coordinate 16.3007, since 16.3007=16+0.3+0.0007, then in this point you can get there by sequentially laying off from the origin 16 unit segments, 3 segments whose length is equal to a tenth of a unit segment, and 7 segments whose length is equal to a ten-thousandth of a unit segment.

This way of building decimal numbers on the coordinate ray allows you to get as close as you like to the point corresponding to the infinite decimal fraction.

Sometimes it is possible to accurately plot the point corresponding to an infinite decimal fraction. For example, , then this infinite decimal fraction 1.41421... corresponds to a point on the coordinate ray, distant from the origin of coordinates by the length of the diagonal of a square with a side of 1 unit segment.

The reverse process of obtaining the decimal fraction corresponding to a given point on a coordinate ray is the so-called decimal measurement of a segment. Let's figure out how it is done.

Let our task be to get from the origin to a given point on the coordinate line (or to infinitely approach it if we can’t get to it). With the decimal measurement of a segment, we can sequentially lay off from the origin any number of unit segments, then segments whose length is equal to a tenth of a unit, then segments whose length is equal to a hundredth of a unit, etc. By recording the number of segments of each length laid aside, we obtain the decimal fraction corresponding to a given point on the coordinate ray.

For example, to get to point M in the above figure, you need to set aside 1 unit segment and 4 segments, the length of which is equal to a tenth of a unit. Thus, point M corresponds to the decimal fraction 1.4.

It is clear that the points of the coordinate ray that cannot be reached in the process of decimal measurement correspond to infinite decimal fractions.

References.

Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
Mathematics. 6th grade: educational. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Of the many fractions found in arithmetic, those that have 10, 100, 1000 in the denominator - in general, any power of ten - deserve special attention. These fractions have a special name and notation.

A decimal is any number fraction whose denominator is a power of ten.

Examples of decimal fractions:

Why was it necessary to separate out such fractions at all? Why do they need their own recording form? There are at least three reasons for this:

Decimals are much easier to compare. Remember: to compare ordinary fractions, you need to subtract them from each other and, in particular, reduce the fractions to common denominator. In decimals nothing like this is required;
Reduce computation. Decimals add and multiply according to their own rules, and with a little practice you'll be able to work with them much faster than with regular fractions;
Ease of recording. Unlike ordinary fractions, decimals are written on one line without loss of clarity.

Most calculators also give answers in decimals. In some cases, a different recording format may cause problems. For example, what if you ask for change in the store in the amount of 2/3 of a ruble :)

Rules for writing decimal fractions

The main advantage of decimal fractions is convenient and visual notation. Namely:

Decimal notation is a form of writing decimal fractions, where whole part separated from a fraction by a regular period or comma. In this case, the separator itself (period or comma) is called a decimal point.

For example, 0.3 (read: “zero pointers, 3 tenths”); 7.25 (7 whole, 25 hundredths); 3.049 (3 whole, 49 thousandths). All examples are taken from the previous definition.

In writing, a comma is usually used as a decimal point. Here and further throughout the site, the comma will also be used.

To write an arbitrary decimal fraction in this form, you need to follow three simple steps:

Write out the numerator separately;
Shift the decimal point to the left by as many places as there are zeros in the denominator. Assume that initially the decimal point is to the right of all digits;
If the decimal point has moved, and after it there are zeros at the end of the entry, they must be crossed out.

It happens that in the second step the numerator does not have enough digits to complete the shift. In this case, the missing positions are filled with zeros. And in general, to the left of any number you can assign any number of zeros without harm to your health. It's ugly, but sometimes useful.

At first glance, this algorithm may seem quite complicated. In fact, everything is very, very simple - you just need to practice a little. Take a look at the examples:

Task. For each fraction, indicate its decimal notation:

The numerator of the first fraction is: 73. We shift the decimal point by one place (since the denominator is 10) - we get 7.3.

Numerator of the second fraction: 9. We shift the decimal point by two places (since the denominator is 100) - we get 0.09. I had to add one zero after the decimal point and one more before it, so as not to leave a strange entry like “.09”.

The numerator of the third fraction: 10029. We shift the decimal point by three places (since the denominator is 1000) - we get 10.029.

The numerator of the last fraction: 10500. Again we shift the point by three digits - we get 10,500. There are extra zeros at the end of the number. Cross them out and we get 10.5.

Pay attention to the last two examples: the numbers 10.029 and 10.5. According to the rules, the zeros on the right must be crossed out, as was done in the last example. However, you should never do this with zeros inside a number (which are surrounded by other numbers). That's why we got 10.029 and 10.5, and not 1.29 and 1.5.

So, we figured out the definition and form of writing decimal fractions. Now let's find out how to convert ordinary fractions to decimals - and vice versa.

Conversion from fractions to decimals

Let's consider a simple numerical fraction of the form a /b. You can use the basic property of a fraction and multiply the numerator and denominator by such a number that the bottom turns out to be a power of ten. But before you do, read the following:

There are denominators that cannot be reduced to powers of ten. Learn to recognize such fractions, because they cannot be worked with using the algorithm described below.

That's how things are. Well, how do you understand whether the denominator is reduced to a power of ten or not?

The answer is simple: factor the denominator into prime factors. If the expansion contains only factors 2 and 5, this number can be reduced to a power of ten. If there are other numbers (3, 7, 11 - whatever), you can forget about the power of ten.

Task. Check whether the indicated fractions can be represented as decimals:

Let us write out and factor the denominators of these fractions:

20 = 4 · 5 = 2 2 · 5 - only the numbers 2 and 5 are present. Therefore, the fraction can be represented as a decimal.

12 = 4 · 3 = 2 2 · 3 - there is a “forbidden” factor 3. The fraction cannot be represented as a decimal.

640 = 8 · 8 · 10 = 2 3 · 2 3 · 2 · 5 = 2 7 · 5. Everything is in order: there is nothing except the numbers 2 and 5. A fraction can be represented as a decimal.

48 = 6 · 8 = 2 · 3 · 2 3 = 2 4 · 3. The factor 3 “surfaced” again. It cannot be represented as a decimal fraction.

So, we’ve sorted out the denominator - now let’s look at the entire algorithm for moving to decimal fractions:

Factor the denominator of the original fraction and make sure that it is generally representable as a decimal. Those. check that only factors 2 and 5 are present in the expansion. Otherwise, the algorithm does not work;
Count how many twos and fives are present in the expansion (there will be no other numbers there, remember?). Choose an additional factor such that the number of twos and fives is equal.
Actually, multiply the numerator and denominator of the original fraction by this factor - we get the desired representation, i.e. the denominator will be a power of ten.

Of course, the additional factor will also be decomposed only into twos and fives. At the same time, in order not to complicate your life, you should choose the smallest multiplier of all possible ones.

And one more thing: if the original fraction contains an integer part, be sure to convert this fraction to an improper fraction - and only then apply the described algorithm.

Task. Convert these numerical fractions to decimals:

Let's factorize the denominator of the first fraction: 4 = 2 · 2 = 2 2 . Therefore, the fraction can be represented as a decimal. The expansion contains two twos and not a single five, so the additional factor is 5 2 = 25. With it, the number of twos and fives will be equal. We have:

Now let's look at the second fraction. To do this, note that 24 = 3 · 8 = 3 · 2 3 - there is a triple in the expansion, so the fraction cannot be represented as a decimal.

The last two fractions have denominators 5 (prime number) and 20 = 4 · 5 = 2 2 · 5 respectively - only twos and fives are present everywhere. Moreover, in the first case, “for complete happiness” a factor of 2 is not enough, and in the second - 5. We get:

Conversion from decimals to common fractions

The reverse conversion - from decimal to regular notation - is much simpler. There are no restrictions or special checks here, so you can always convert a decimal fraction to the classic “two-story” fraction.

The translation algorithm is as follows:

Cross out all the zeros on the left side of the decimal, as well as the decimal point. This will be the numerator of the desired fraction. The main thing is not to overdo it and do not cross out the inner zeros surrounded by other numbers;
Count how many decimal places there are after the decimal point. Take the number 1 and add as many zeros to the right as there are characters you count. This will be the denominator;
Actually, write down the fraction whose numerator and denominator we just found. If possible, reduce it. If the original fraction contained an integer part, we will now get an improper fraction, which is very convenient for further calculations.

Task. Convert decimal fractions to ordinary fractions: 0.008; 3.107; 2.25; 7,2008.

Cross out the zeros on the left and the commas - we get the following numbers (these will be the numerators): 8; 3107; 225; 72008.

In the first and second fractions there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; 100; 10000.

Finally, let's combine the numerators and denominators into ordinary fractions:

As can be seen from the examples, the resulting fraction can very often be reduced. Let me note once again that any decimal fraction can be represented as an ordinary fraction. The reverse conversion may not always be possible.

I. To divide a number by a decimal fraction, you need to move the decimal places in the dividend and divisor as many digits to the right as there are after the decimal point in the divisor, and then divide by the natural number.

Primery.

Perform division: 1) 16,38: 0,7; 2) 15,6: 0,15; 3) 3,114: 4,5; 4) 53,84: 0,1.

Solution.

Example 1) 16,38: 0,7.

In the divider 0,7 there is one digit after the decimal point, so let’s move the commas in the dividend and divisor one digit to the right.

Then we will need to divide 163,8 on 7 .

We divide as they divide natural numbers. How to remove the number 8 - the first digit after the decimal point (i.e. the digit in the tenths place), so immediately put a comma in the quotient and continue dividing.

Answer: 23.4.

Example 2) 15,6: 0,15.

We move commas in the dividend ( 15,6 ) and divisor ( 0,15 ) two digits to the right, since in the divisor 0,15 there are two digits after the decimal point.

We remember that you can add as many zeros as you like to the decimal fraction on the right, and this will not change the decimal fraction.

15,6:0,15=1560:15.

We perform division of natural numbers.

Answer: 104.

Example 3) 3,114: 4,5.

Move the commas in the dividend and divisor one digit to the right and divide 31,14 on 45 By

3,114:4,5=31,14:45.

In the quotient we put a comma as soon as we remove the number 1 in the tenth place. Then we continue dividing.

To complete the division we had to assign zero to the number 9 - differences between numbers 414 And 405 . (we know that zeros can be added to the right side of a decimal fraction)

Answer: 0.692.

Example 4) 53,84: 0,1.

Move the commas in the dividend and divisor to 1 number to the right.

We get: 538,4:1=538,4.

Let's analyze the equality: 53,84:0,1=538,4. Pay attention to the comma in the dividend in in this example and a comma in the resulting quotient. We notice that the comma in the dividend has been moved to 1 number to the right, as if we were multiplying 53,84 on 10. (Watch video “Multiplying a decimal by 10, 100, 1000, etc..") Hence the rule for dividing a decimal fraction by 0,1; 0,01; 0,001 etc.

II. To divide a decimal by 0.1; 0.01; 0.001, etc., you need to move the decimal point to the right by 1, 2, 3, etc. digits. (Dividing a decimal by 0.1, 0.01, 0.001, etc. is the same as multiplying that decimal by 10, 100, 1000, etc.)

Examples.

Perform division: 1) 617,35: 0,1; 2) 0,235: 0,01; 3) 2,7845: 0,001; 4) 26,397: 0,0001.

Solution.

Example 1) 617,35: 0,1.

According to the rule IIdivision by 0,1 is equivalent to multiplying by 10 , and move the comma in the dividend 1 digit to the right:

1) 617,35:0,1=6173,5.

Example 2) 0,235: 0,01.

Division by 0,01 is equivalent to multiplying by 100 , which means we move the comma in the dividend on 2 digits to the right:

2) 0,235:0,01=23,5.

Example 3) 2,7845: 0,001.

Because division by 0,001 is equivalent to multiplying by 1000 , then move the comma 3 digits to the right:

3) 2,7845:0,001=2784,5.

Example 4) 26,397: 0,0001.

Divide a decimal by 0,0001 - it's the same as multiplying it by 10000 (move the comma by 4 digits right). We get:

II. To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to the left by 1, 2, 3, etc. digits.

Examples.

Perform division: 1) 41,56: 10; 2) 123,45: 100; 3) 0,47: 100; 4) 8,5: 1000; 5) 631,2: 10000.

Solution.

Moving the decimal point to the left depends on how many zeros after the one are in the divisor. So, when dividing a decimal fraction by 10 we will carry over in the dividend comma to the left one digit; when divided by 100 - move the comma left two digits; when divided by 1000 convert to this decimal fraction comma three digits to the left.

In examples 3) and 4) we had to add zeros before the decimal fraction to make it easier to move the comma. However, you can assign zeros mentally, and you will do this when you learn to apply the rule well II to divide a decimal fraction by 10, 100, 1000, etc.

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Fractions

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

Fractions are not much of a nuisance in high school. For the time being. Until you come across powers with rational exponents and logarithms. And there... You press and press the calculator, and it shows a full display of some numbers. You have to think with your head like in the third grade.

Let's finally figure out fractions! Well, how much can you get confused in them!? Moreover, it’s all simple and logical. So, what are the types of fractions?

Types of fractions. Transformations.

There are fractions three types.

1. Common fractions , For example:

Sometimes instead of a horizontal line they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens...), say to yourself the phrase: " Zzzzz remember! Zzzzz denominator - look zzzzz uh!" Look, everything will be zzzz remembered.)

The dash, either horizontal or inclined, means division the top number (numerator) to the bottom (denominator). That's all! Instead of a dash, it is quite possible to put a division sign - two dots.

When complete division is possible, this must be done. So, instead of the fraction “32/8” it is much more pleasant to write the number “4”. Those. 32 is simply divided by 8.

32/8 = 32: 8 = 4

I'm not even talking about the fraction "4/1". Which is also just "4". And if it’s not completely divisible, we leave it as a fraction. Sometimes you have to do the opposite operation. Convert a whole number into a fraction. But more on that later.

2. Decimals , For example:

It is in this form that you will need to write down the answers to tasks “B”.

3. Mixed numbers , For example:

Mixed numbers are practically not used in high school. In order to work with them, they must be converted into ordinary fractions. But you definitely need to be able to do this! Otherwise you will come across such a number in a problem and freeze... Out of nowhere. But we will remember this procedure! A little lower.

Most versatile common fractions. Let's start with them. By the way, if a fraction contains all sorts of logarithms, sines and other letters, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!

The main property of a fraction.

So, let's go! To begin with, I will surprise you. The whole variety of fraction transformations is provided by one single property! That's what it's called main property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction does not change. Those:

It is clear that you can continue to write until you are blue in the face. Don’t let sines and logarithms confuse you, we’ll deal with them further. The main thing is to understand that all these various expressions are the same fraction . 2/3.

Do we need it, all these transformations? Yes! Now you will see for yourself. To begin with, let's use the basic property of a fraction for reducing fractions. It would seem like an elementary thing. Divide the numerator and denominator by the same number and that's it! It's impossible to make a mistake! But... man is a creative being. You can make a mistake anywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.

How to correctly and quickly reduce fractions without doing extra work can be read in the special Section 555.

A normal student doesn't bother dividing the numerator and denominator by the same number (or expression)! He simply crosses out everything that is the same above and below! This is where it lurks typical mistake, a blooper, if you will.

For example, you need to simplify the expression:

There’s nothing to think about here, cross out the letter “a” on top and the “2” on the bottom! We get:

Everything is correct. But really you divided all numerator and all the denominator is "a". If you are used to just crossing out, then, in a hurry, you can cross out the “a” in the expression

and get it again

Which would be categorically false. Because here all the numerator on "a" is already not shared! This fraction cannot be reduced. By the way, such a reduction is, um... a serious challenge for the teacher. This is not forgiven! Do you remember? When reducing, you need to divide all numerator and all denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. How can I continue to work with her now? Without a calculator? Multiply, say, add, square!? And if you’re not too lazy, and carefully cut it down by five, and by another five, and even... while it’s being shortened, in short. Let's get 3/8! Much nicer, right?

The main property of a fraction allows you to convert ordinary fractions to decimals and vice versa without a calculator! This is important for the Unified State Exam, right?

How to convert fractions from one type to another.

With decimal fractions everything is simple. As it is heard, so it is written! Let's say 0.25. This is zero point twenty five hundredths. So we write: 25/100. We reduce (we divide the numerator and denominator by 25), we get the usual fraction: 1/4. All. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.

What if the integers are not zero? It's OK. We write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three point seventeen hundredths. We write 317 in the numerator and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary, Watson! From all that has been said, a useful conclusion: any decimal fraction can be converted to a common fraction .

But some people cannot do the reverse conversion from ordinary to decimal without a calculator. And it is necessary! How will you write down the answer on the Unified State Exam!? Read carefully and master this process.

What is the characteristic of a decimal fraction? Her denominator is Always costs 10, or 100, or 1000, or 10000 and so on. If your common fraction has a denominator like this, there's no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. What if the answer to the task in section “B” turned out to be 1/2? What will we write in response? Decimals are required...

Let's remember main property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. Anything, by the way! Except zero, of course. So let’s use this property to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? At 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But then the numerator must also be multiplied by 5. This is already mathematics demands! We get 1/2 = 1x5/2x5 = 5/10 = 0.5. That's it.

However, all sorts of denominators come across. You will come across, for example, the fraction 3/16. Try and figure out what to multiply 16 by to make 100, or 1000... Doesn’t it work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide with a corner, on a piece of paper, as they taught in elementary school. We get 0.1875.

And there are also very bad denominators. For example, there is no way to turn the fraction 1/3 into a good decimal. Both on the calculator and on a piece of paper, we get 0.3333333... This means that 1/3 is an exact decimal fraction not translated. Same as 1/7, 5/6 and so on. There are many of them, untranslatable. This brings us to another useful conclusion. Not every fraction can be converted to a decimal !

By the way, this useful information for self-test. In section "B" you must write down a decimal fraction in your answer. And you got, for example, 4/3. This fraction does not convert to a decimal. This means you made a mistake somewhere along the way! Go back and check the solution.

So, we figured out ordinary and decimal fractions. It remains to deal with mixed numbers. To work with them, they must be converted into ordinary fractions. How to do this? You can catch a sixth grader and ask him. But a sixth grader won’t always be at hand... You’ll have to do it yourself. It's not difficult. You need to multiply the denominator of the fractional part by the whole part and add the numerator of the fractional part. This will be the numerator common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but in reality everything is simple. Let's look at an example.

Suppose you were horrified to see the number in the problem:

Calmly, without panic, we think. The whole part is 1. Unit. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of a common fraction. That's it. It looks even simpler in mathematical notation:

Is it clear? Then secure your success! Convert to ordinary fractions. You should get 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction to a mixed number - is rarely required in high school. Well, if so... And if you are not in high school, you can look into the special Section 555. There, by the way, about improper fractions you'll find out.

Well, that's practically all. You remembered the types of fractions and understood How transfer them from one type to another. The question remains: For what do this? Where and when to apply this deep knowledge?

I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed together, we convert everything into ordinary fractions. It can always be done. Well, if it says something like 0.8 + 0.3, then we count it that way, without any translation. Why do we need extra work? We choose the solution that is convenient us !

If the task is all decimal fractions, but um... some kind of evil ones, go to ordinary ones and try it! Look, everything will work out. For example, you will have to square the number 0.125. It’s not so easy if you haven’t gotten used to using a calculator! Not only do you have to multiply numbers in a column, you also have to think about where to insert the comma! It definitely won’t work in your head! What if we move on to an ordinary fraction?

0.125 = 125/1000. We reduce it by 5 (this is for starters). We get 25/200. Once again by 5. We get 5/40. Oh, it's still shrinking! Back to 5! We get 1/8. We can easily square it (in our minds!) and get 1/64. All!

Let's summarize this lesson.

1. There are three types of fractions. Common, decimal and mixed numbers.

2. Decimals and mixed numbers Always can be converted to ordinary fractions. Reverse transfer not always possible

3. The choice of the type of fractions to work with a task depends on the task itself. Subject to availability different types fractions in one task, the most reliable thing is to move on to ordinary fractions.

Now you can practice. First, convert these decimal fractions to ordinary fractions:

3,8; 0,75; 0,15; 1,4; 0,725; 0,012

You should get answers like this (in a mess!):

Let's finish here. In this lesson we refreshed our memory key points by fractions. It happens, however, that there is nothing special to refresh...) If someone has completely forgotten, or has not yet mastered it... Then you can go to a special Section 555. All the basics are covered in detail there. Many suddenly understand everything are starting. And they solve fractions on the fly).

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