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Ending and endless fractions. Periodic tens fractions. How to write down an unbroken periodic sequence in a casual manner

In this article, we will understand with you what tens of things are, what is the specialness and power. Let's go! 🙂

The tenth fraction is rounded off with a series of fractional fractions (which have a fraction divisible by 10).

Viznachennya

Decimals are fractions whose signifiers are numbers that are made up of a one and a number of zeros following it. These are fractions with the symbol 10, 100, 1000, etc. Otherwise, the tenth fraction can be characterized as a fraction with the sign of 10 or one of the steps of the ten.

Apply shot:

, ,

The tenth drib is written differently, lower than prime. Operations with these fractions are also subordinate to operations with ordinary fractions. The rules for working with them are very similar to the rules for working with whole numbers. Here, let's face it, this demand has been taken into account in the most practical tasks.

Give the fraction in the tenth notation

The entry for the tenth fraction does not have a sign, but the number of the number is displayed. In the traditional way, writing a tenth fraction follows the following scheme:

de X is a whole part of a shot, Y is a whole part of a shot, “,” is a tenth of a coma.

To correctly identify the primary fraction in the form of a tenth, it is necessary that it be correct, with a whole part (as possible) and a number, which is smaller than the sign. Then, in the tenth record, the whole part is written before the tenth coma (X), and the number of the initial fraction is written after the tenth coma (Y).

If the number book contains a number with many signs, less than the number of zeros in the sign, then in part Y the lack of number of signs in the tenth entry will be filled with zeros in front of the digits of the number book.

Butt:

If the initial value is less than 1, then. There is no whole part, then X in the tenth form is written as 0.

In the shot part (Y), after the remaining significant (except for zero) discharge, a sufficient number of zeros can be entered. It is poured onto the intended fraction. And as a note: all zeros such as fractions and tenths can be omitted.

Reading tens fractions

Part X is read casually like this: “X whole.”

Part Y is read consecutively to the number in the sign. For a banner 10 traces read: Y tens, for a banner 100: Y hundreds, for a banner 1000: Y thousand and so on... 😉

It is correct to take a different approach before reading, the basis for the preparation of several discharges of shot parts. For this reason, it is necessary to understand that shot discharges are scaled up in a mirror image to discharge a whole piece of shot.

The name for correct reading is indicated in the table:

Coming from this, the reading may be hidden on the basis of the named category of the remaining digit of the shot part.

3.5 is read as “three point five tens”
0.016 is read as “zero point sixteen thousandths”

Converting a fraction from a fraction to a tenth

If in the sign of the initial fraction there are 10 or any step of ten, then the transfer of the fraction is calculated as described above. In other situations, additional modifications are required.

There are 2 ways of translation.

The first way to translate

The number and the sign must be multiplied by such a whole number so that the number 10 or one of the tens is subtracted from the sign. And then the fraction is given in the tenth entry.

This method is used for shotguns, the sign of which is laid out only on 2 and 5. So, at the front butt . If the layout has other simple multipliers (for example, ), then you will have to go to method 2.

Another way of translation

The 2nd method involves dividing the number reader into a sign reader on a counter or on a calculator. The whole part, such as it is, does not take part in what has been transformed.

The rule for dividing into a stacker, which results in a tenth fraction, is described below (div. Division of tens fractions).

Converted from tenth to decimal

For this trace, write down the fractional part (the right-hand side of the coma) in the view of the number reader, and the result of reading the fractional part - in the view of the corresponding number in the signifier. Further, as much as possible, it is necessary to cut off friends as quickly as possible.

Kintseviy and endless tenth fraction

The end number is the name given to the tenth fraction, the fraction part of which is formed from the end number of digits.

Most of all, we aim to place the very last tens of fractions. However, no ordinary difference can be imagined in the form of a tenth. Since the 1st method of transfer for this fraction does not get stuck, and the 2nd method demonstrates that the division cannot be completed, which means that only an infinite number of tens of fractions can be removed.

It’s impossible to write down a seemingly endless friendship. At first glance, these fractions can be represented:

as a result of the reduction to almost the same number of discharges after coma;
it looks like a periodic fraction.

A number in which you can see a sequence of digits that repeats endlessly is called periodic.

Other fractions are called non-periodic. For non-periodic fractions, only the 1st method of filing (rounding) is permissible.

Example of a periodic fraction: 0.8888888... Here the number 8 is repeated, which, obviously, is repeated ad infinitum, since there is no reason to allow anything else. This number is called periodically a fraction.

Periodic fractions are either pure or mixed. We clean a dozen units, in which the period begins immediately after the coma. Mixing fractions before the period after the coma contains 1 or more digits.

54.33333… – periodic pure decimal fraction

2.5621212121… – periodic mixing of particles

Attach the notation of uncounted tens fractions:

The 2nd example shows how to correctly format the period of recording a periodic fraction.

Conversion of periodic tens fractions from the original

To convert a pure periodic fraction into a primary period, write in the number book, and write in the sign book the number that adds up to the number of nines, which is equal to the number of digits in the period.

Mixed periodic tenths of fractions are transferred in the following order:

you need to formulate a number that adds up to the number that stands after the first period and the first period;
From the number you remove, add the number that stands after the number before the period. Pouch in stock numberer of the primary fraction;
In the sign you need to enter a number that is equal to the number of nines equal to the number of digits of the period, followed by zeros equal to the number of digits of the number that is equal to the number of digits of the number that is equal to the 1st period.

The division of tens fractions

Tens fractions equalize the whole parts. The one who has the most is the greater part.

As a whole, however, the numbers of the different categories of shot parts are equalized, starting from the first (from tens). The same principle applies here: the greater number of fractions is the one with the larger number of tens; For the equality of numbers to the place of tens, the places of hundreds are equal.

Oskolki

, fragments with equal whole parts and equal tenths in the shot part, the 2nd fraction has a greater figure of hundreds.

Addition and removal of tens of fractions

Tens fractions are added and expressed in the same way as whole numbers, writing down the corresponding digits one by one. For this it is necessary that there be dozens of comas one under one. Either one (tens or so) of the whole part, and at least ten (hundreds or so) of shot appear consistently. Discharges of shot parts to be rejected are filled with zeros. Without middle The process of addition and operation works the same way as for whole numbers.

Multiplication of tens fractions

To multiply tens fractions, you need to write them one by one, following the remaining digit and not losing your respects in the place of multiplying tens. Then you need to multiply the numbers the same way as when multiplying whole numbers. After obtaining the result, re-select the number of digits after the number in both fractions and increase the number of shot discharges as a result. If the discharges do not appear, they will be replaced by zeros.

Multiplying and dividing tens fractions by 10 n

These things simply go on until you suffer a dozen comas. P When the coma is multiplied, it is transferred to the right (the fraction becomes larger) by a number of characters, equal to the number of zeros in 10 n, where n is an additional whole step. Then a bunch of numbers are transferred from the shot part to the whole. When dividing, it is likely that the coma is transferred to the left (the number changes), and one part of the numbers is transferred from the whole part to the other. If there are not enough numbers for transfer, then the remaining digits will be filled with zeros.

Divide the tenth fraction and the whole number into the whole number and into the tenth fraction

The division of the tenth fraction into the whole number is equal to the division of two whole numbers. In addition, the appearance of the tenth coma is required: when a number is added to the category, after any trace of the coma, it is necessary to put the number after the serial number of the line that is being formed. Then you need to continue chewing until it reaches zero. If the signs of a shared spouse are defective, the role of the trace will have zeros.

In the same way, 2 whole numbers are divided into a stacker, since all the digits of the division are included, and the outside division is completed. In which case, after removing the remaining digits of the dividend, a 10 is placed. The coma in the line that is formed, and when the remaining digits are removed, zeros are added. Tobto. The division here, in essence, is represented as a tenth shot with a zero shot part.

To subdecimally a fraction (or a whole number) by a tenth number, you need to multiply the numbers and the dilator by the number 10 n, in which the number of zeros is equal to the number of digits after the tenth in the dilator. With this method, one tenth of a coma is added to the fraction that needs to be divided. Next, the process will be described below.

Graphic display of tens fractions

Graphically, tens fractions are displayed behind an additional coordinate line. For each section, divide it into 10 equal parts, just as centimeters and millimeters are laid out on the line. This will ensure accurate representation of tens of fractions and the ability to objectively adjust them.

To ensure that later cuts on individual sections remain consistent, you should carefully think about the end of the single section itself. It is designed in such a way that the handiness of the accessory section can be ensured.

Do you remember how in the very first lesson about tens fractions I said that there are numerical fractions that are not represented by tens fractions (wonderful lesson “Decimals”)? We have just started to sort out the signifiers of fractions into multipliers in order to verify that there are no numbers there, such as 2 and 5.

So the axis: I came across it. And today we will learn to translate absolutely any number fraction in tens. At the same time, we are familiar with a whole class of fractions with an infinite significant part.

Periodic tenth drib - no matter what tenth drib, for any one:

The significant part consists of the impersonality of numbers;

At song intervals, the numbers of the significant part are repeated.

A set of numbers that are repeated to form a significant part is called a periodic part of a fraction, and the number of digits in this set is called a periodic fraction. A section of a significant part that is not repeated is called a non-periodic part.

The fragments have a lot of meaning, so it’s easy to look at the following fractions:

This fact is most common in affairs. Non-periodic part: 0; periodic part: 3; Dovzhina period: 1.

Non-periodic part: 0.58; periodic part: 3; Dovzhina period: I renew 1.

Non-periodic part: 1; periodic part: 54; dove period: 2.

Non-periodic part: 0; periodic part: 641025; the end of the period: 6. For the clarity of the parts that are repeated, they are reinforced with one kind of gap - in which case it is decided to work in this way not obligatory.

Non-periodic part: 3066; periodic part: 6; Dovzhina period: 1.

As you see, the meaning of the periodic fraction is based on the concept significant part of a number. So if you forgot, then I recommend repeating it - marvelous. lesson " ".

Transition to periodic tenth fraction

Let's take a look at the primary difference in the form a/b. Let's decompose this banner into simple multipliers. There are two options:

The layout has only multipliers 2 and 5. It is easy to adjust fractions to tens - divs. lesson "Decimal fractions". Don’t taunt us like that;
The unfolded one has more than 2 and 5. This type of trib doesn’t look like a tenth one, then from it you can create a periodic tenth one.

To specify the periodic tenth fraction, you need to know its periodic and non-periodic parts. Yak? Convert the numbers from the wrong ones, and then divide the number book into a circle.

In this case the following applies:

Split up now the whole part yakshcho won;
Perhaps there will be a bunch of numbers after the tenth point;
In about an hour the numbers will begin repeat.

That's all! Repeated numbers after the tenth point are indicated by a periodic part, and those before them are designated by a non-periodic part.

Zavdannya. Convert primary fractions to periodic tens:

All fractions without a whole part, so we simply divide the number by the sign with a “bump”:

Like Bachimo, the excesses are repeated. Let’s write the difference in the “correct” way: 1.733...=1.7(3).

Through the war, drib comes out: 0.5833 ... = 0.58(3).

We write in normal form: 4.0909...=4,(09).

Removing dribble: 0.4141...=0,(41).

Transition from periodic tenth fraction to prime

Let's take a look at the periodic tenth fraction X = abc (a 1 b 1 c 1). It is necessary to translate it into the classic “double-top”. For whose sake, here are some simple steps:

Find the fraction period, then. Find out how many digits there are in the periodic part. Let there be a number k;
Find the value of the virus X · 10 k. The tenth point has been completely destroyed for the new period, right-handed - div. lesson “Multiplication and division of tens of fractions”;
The exit date must be taken from the selected date. At which point the part is periodically “burned” and lost extreme friend;
In the extracted form, know X. All tens fractions are translated in simple terms.

Zavdannya. Reduce a number to its original improper fraction:

9,(6);
32,(39);
0,30(5);
0,(2475).

Working with the first fraction: X = 9, (6) = 9.666.

The arms have only one digit, so the period k = 1. Then this number is multiplied by 10 k = 10 1 = 10.

10X = 10 9.6666... = 96.666...

We can see the final result and the corresponding comparison:

10X - X = 96.666 ... - 9.666 ... = 96 - 9 = 87;
9X = 87;
X = 87/9 = 29/3.

Now let's look at another fraction. Otje, X = 32, (39) = 32.393939.

Period k = 2, so we multiply everything by 10 k = 10 2 = 100:

100X = 100 · 32.393939 ... = 3239.3939 ...

Once again, we can see the difference and the likely comparison:

100X - X = 3239.3939 ... - 32.3939 ... = 3239 - 32 = 3207;
99X = 3207;
X = 3207/99 = 1069/33.

Let's move on to the third fraction: X = 0.30(5) = 0.30555... The diagram is the same, so I’ll just add tabs:

Period k = 1 ⇒ everything is multiplied by 10 k = 10 1 = 10;

10X = 10 0.30555... = 3.05555...
10X - X = 3.0555 ... - 0.305555 ... = 2.75 = 11/4;
9X = 11/4;
X = (11/4): 9 = 11/36.

The final result is: X = 0, (2475) = 0.2475 2475 ... Again, for clarity, the periodic parts are reinforced with one type of gaps. Maemo:

k = 4 ⇒ 10 k = 10 4 = 10,000;
10000X = 10000 · 0.2475 2475 = 2475.2475 ...
10,000X - X = 2475.2475 ... - 0.2475 2475 ... = 2475;
9999X = 2475;
X = 2475: 9999 = 25/101.

For the impersonality of fractions that are concentrated in arithmetic, we deserve respect for those that have a sign of 10, 100, 1000 - by the way, be it a step of ten. These fractions have a special notation called the form.

A tenth fraction is any kind of numerical fraction, the sign of which is worth a tenth step.

Butts of tens shot:

Do you ever need to see such fractions? Why do they need a powerful recording form? There are at least three reasons for this:

It is easier to mix up tens fractions. Guess: to equalize the numbered fractions, you need to take one side of the other and, in short, bring the fractions to the next sign. In tens fractions nothing like this is needed;
Calculate quickly. Tens fractions are added and multiplied according to powerful rules, and after a little practice you will practice them more than ever before;
Handling of the recording. In the case of simple fractions, tens are written in one row without precision.

Most calculators also give results in tens fractions. Any other recording format may cause problems. For example, if you can get 2/3 of a ruble in a store :)

Rules for writing tens fractions

The main advantage of tens fractions is manual and physical recording. And to yourself:

The tenth notation is a form of writing tenth fractions, where the whole part is added to the fraction behind an additional point or a comma. In this case, the separator itself (a point or a coma) is called a tenth point.

For example, 0.3 (read: “zero whole, 3 tenths”); 7.25 (7 whole, 25 hundred); 3.049 (3 whole, 49 thousandths). All butts are taken from the original design.

On the sheet, like a ten point, there is a coma. Here and throughout the site the same coma will be discussed.

To write down a number of tenths in a given form, you need to enter three simple units:

Vipisati okremo numerator;
Break the tenth point to the left into as many signs as there are zeros to place the sign. Please note that the first ten point is the right hand in front of all numbers;
As soon as the ten point was destroyed, and after it, the record suddenly lost zeros, they need to be added.

It turns out that on the other hand, the number cruncher does not show the numbers to complete the assignment. In this case, all daily positions will be filled with zeros. That's why the left hand can be attributed to any number of zeros without harm. It’s not garnished, but it’s sometimes nutty.

At first glance, this algorithm can be made more complex. In reality, everything is simpler and simpler - you just need a little practice. Look at the butt:

Zavdannya. For the cutaneous fraction, enter the tenth entry:

Number of the first fraction: 73. We add a ten point to one sign (since there are 10 in the sign) - subtract 7.3.

The numerator of the other fraction is: 9. We subtract the tenth point into two digits (since the decimal place is 100) - we subtract 0.09. I had to add one zero after the tenth point and one more before it, so as not to deprive the wonderful entry of the form “09”.

The number of the third fraction is: 10029. We subtract the tenth point into three signs (since the number in the sign is 1000) - we subtract 10.029.

The number of the remaining fraction is: 10,500. Once again, the fraction is broken into three signs - 10,500 is subtracted. Finally, the numbers turned out to be zeros. Let's fasten it - take off 10.5.

Return respect to two remaining butts: the numbers 10.029 and 10.5. According to the rules, the right hand needs to be secured, as it is divided into the rest of the butt. However, every time you can’t do it with zeros so that they stand in the middle of the number (as indicated by other digits). They themselves calculated 10.029 and 10.5, and not 1.29 and 1.5.

Then, we decided to use the designated form to write tens fractions. Now it’s clear how to convert primary fractions to tenths - and all the same.

Transition from fractions to tens

Let's take a look at a simple numerical fraction of the form a/b. You can quickly use the basic power of the fraction and multiply the number and the sign by such a number so that the tenth step appears at the bottom. Ale persh, nizh tse robiti, read this:

There are signs, so as not to be guided to the level of ten. Learn to recognize such fractions because they cannot be used in an algorithm, which we will describe below.

Do it anyway. Well, how can we understand that the flag should be aimed at the tenth degree?

The idea is simple: divide the banner into simple multiples. Since in the layout there are only multipliers 2 and 5, this number can be raised to the power of ten. If other numbers are found (3, 7, 11 - whatever), you can forget about the tenth step.

Zavdannya. Check that fractions can be represented in the form of tens:

We write and decompose the signifiers of these fractions into multipliers:

20 = 4 · 5 = 2 2 · 5 - present only in numbers 2 and 5. Also, the number can be given in the form of a tenth.

12 = 4 · 3 = 2 2 · 3 - є “fencing” multiplier 3. The fraction cannot be represented in tenth form.

640 = 8 · 8 · 10 = 2 3 · 2 3 · 2 · 5 = 2 7 · 5. Everything is in order: besides the numbers 2 and 5 there is nothing. The fraction is presented in the form of a tenth.

48 = 6 · 8 = 2 · 3 · 2 3 = 2 4 · 3. I repeat the “split” multiplier 3. It is impossible to identify the tenth fraction from the eye.

Well, we’ve settled on the standard – now let’s look at the entire algorithm for moving to tens of fractions:

Spread the banner of the output shot into multipliers and roll over, so that the appearance of the tenth appears. Tobto. Make sure that the layout contains only multipliers 2 and 5. Otherwise, the algorithm does not work;
Find out how many twos and fives there are on the table (there won’t be any other numbers there, remember?). Choose such an additional multiplier so that a number of twos and fives are equal.
Please, multiply the number and the sign of the output fraction by this multiplier - it is impossible to remove the phenomenon, then. There are ten steps standing at the banner.

Obviously, the additional multiplier can only be divided into twos and fives. In this case, in order not to complicate your life, choose the smallest such multiplier from all possible ones.

And also: if the output fraction has a whole part, it is obligatory to convert this fraction to the wrong one - and then set up the algorithm descriptions.

Zavdannya. Convert these number fractions to tens:

We divide the banner of the first fraction into multipliers: 4 = 2 · 2 = 2 2 . Well, the difference is great in the eyes of the tenth. The layout has two twos and one five, so the additional multiplier is equal to 5 2 = 25. With it, a number of twos and fives are equal. Maemo:

Now let's look at another shot. For whom it is important that 24 = 3 · 8 = 3 · 2 3 - the laid out presence has a three, so the difference is not apparent in the appearance of the tenth.

The two remaining fractions are marked 5 (simple number) and 20 = 4 · 5 = 2 2 · 5 clearly - through the presence of two and five. In this case, in the first case “for complete happiness” the multiplier 2 is not shown, and in the other case - 5. Eliminate:

Transition from tens fractions to prime fractions

The reversal - from the tenth form to the prime - is much simpler to construct. There are no boundaries or special revisions here, so you can convert the tenth fraction from the classic “double-overhead” from now on.

Algorithm for shifting attacks:

Cross all the zeros that stand at the tenth fraction of evil, as well as the tenth point. Tse will be a number manager for a shot. Golovne - do not overdo it and do not attach internal zeros marked with other numbers;
Find out how many signs there are in the output tenth after the coma. Take the number 1 and write to the right hand as many zeros as there are as many signs as you have added. There will be a banner;
Vlasna, write down the friend, the number officer and the signer whom we have known well. Make it as short as possible. If the output fraction has a whole part, we immediately remove the wrong fraction, which is very easy for subsequent calculations.

Zavdannya. Convert tens fractions from basic ones: 0.008; 3.107; 2.25; 7,2008.

Let's label the zeros of evil and coma - we remove the following numbers (there will be numerals): 8; 3107; 225; 72008.

The first and the other fractions after each have 3 digits, the other has 2, and the third has as many as 4 digits. Removing banners: 1000; 1000; 100; 10,000.

Let's find out the numbers and the signifiers for the primary fraction:

As can be seen from the butts, the removal of the ribs can often be shortened. I’ll point out once again that there are dozens of different things that look extraordinary. The gate of re-creation can be done again and again.

We already said that fractions fly urgentlyі dozens. For now, we have learned simple fractions for a little while. We discovered that there are regular and irregular fractions. We also discovered that primary fractions can be shortened, folded, multiplied and divided. We also discovered that there are so-called mixed numbers that are made up of whole and fractional parts.

We haven’t fully learned simple fractions yet. There were a lot of fine details to talk about, but now we’ll definitely learn dozens fractions, primary fragments and tens of fractions often have to be added. Then, in a high order, you have to work with both types of shot.

This lesson may seem complicated and foolish. It's completely normal. Such lessons are taught so that they are learned by themselves, and not looked at superficially.

Lesson replacement

Viraz size in shotgun view

Sometimes it becomes obvious that it looks like a different person. For example, one tenth of a decimeter is written like this:

This meaning means that one decimeter is divided into ten equal parts, and from these ten parts one part was taken. And one part of ten in this case is equivalent to one centimeter:

Let's take a look at the offensive butt. Show 6 cm and 3 mm in centimeters in a shotgun view.

So, you need to show 6 cm and 3 mm in centimeters, but in a shotgun view. We already have 6 whole centimeters:

Ale has lost another 3 millimeters. How to show 3 millimeters, what about centimeters? Fractions come to help. One centimeter is equal to ten millimeters. Three millimeters is three parts of ten. And three parts of ten are written as cm

Viraz cm means that one centimeter was divided into ten equal parts, and from these ten parts three parts were taken.

As a result, we have six whole centimeters and three ten centimeters:

In this case, 6 shows the number of whole centimeters, and drib - the number of shotguns. This word reads like “six whole and three ten centimeters”.

Fractions such as the numbers 10, 100, 1000 can be written without a sign. First write the whole part, and then the number of the shot part. The whole part is added to the number of shot parts of the coma.

For example, let’s write it down without a banner. We write down the whole part from the beginning. Whole part 6

The whole part has been written down. Immediately after writing the whole part we put to someone:

And now we write down the number of the shot part. In a mixed number, the number of the shot particle is the number 3. We write it after the three:

Whatever number is represented in this way is called in tens fractions.

Therefore, you can show 6 cm and another 3 mm in centimeters using an additional tenth fraction:

6.3 cm

It will look like this:

In fact, tens fractions are the primary fractions themselves and mixed numbers. The peculiarity of such fractions is that the sign of their fractions has the numbers 10, 100, 1000 and 10000.

As a number is mixed, a tenth fraction equals a whole fraction and a fraction. For example, in a mixed number, the whole part is 6, and the part is .

The tenth fraction has 6.3 and the whole fraction has the number 6, and the fraction has the whole fraction, which is the number 3.

It happens that simple fractions in the sign, such as the numbers 10, 100, 1000, are given without a whole part. For example, the word is given without a whole part. To write down such a fraction as a tenth, first write down 0, then write down the number of the fraction. A fraction without a banner will be written in the following order:

Read like "zero point five ten".

Translation of mixed numbers in tens fractions

When we write mixed numbers without a sign, we convert them into tens fractions. When converting elementary fractions to tens of fractions, it is necessary to know a number of moments, which we will now talk about.

After the whole part is written down, it is necessary to take into account the number of zeros in the sign of the fractional part, so that the number of zeros in the fractional part and the number of digits after the tenth part may be the same. What does this mean? Let's take a look at the attack stock:

From now on

You can immediately write down the number of the shot part and the tenth number of the prepared ones, but you will definitely need to remove the number of zeros from the number of the shot part.

Also, what matters is the number of zeros in the fractional part of the mixed number. The banner of the shot part has one zero. This means that the tenth fraction after which there will be one digit and this digit will be the number of the fractional part of the mixed number, then the number is 2

Thus, the mixed number, when converted to tenths, is converted to 3.2.

This tenth fraction reads like this:

“Three wholes, two tenths”

“Ten” because in the shot part of the mixed number there is the number 10.

butt 2. Convert mixed number to tenth fraction.

We write down the whole part and put it to someone:

And you can immediately write down the number of the fractional part and remove the tenth fraction 5.3, but the rule is that after there are as many digits as there are zeros in the symbol of the fractional part of the mixed number. And we understand that the banner of the shotgun part has two zeros. This means that our tenth fraction after the coma has two digits, not one.

In such situations, the number of the shot part needs to be modified a little: add a zero before the number, then before the number 3

Now you can convert this mixed number into a tenth fraction. We write down the whole part and put it to someone:

I write down the number of the shot part:

The tenth fraction 5.03 reads like this:

"Five whole, three hundred"

“Hundreds” because in the sign of the shot part of the mixed number there is the number 100.

butt 3. Convert mixed number to tenth fraction.

From previous applications, we discovered that in order to successfully convert a mixed number into a tenth fraction, the number of digits in the number unit of the fractional part and the number of zeros in the denominator of the fractional part may be the same.

Before converting a mixed number into a tenth fraction, its fractional part needs to be modified a little, and the fraction itself must be divided so that the number of digits in the numberer of the fractional part and the number of zeros in the denominator of the fractional part are the same.

We were immediately amazed at the number of zeros in the shotgun section. Bachimo, why are there three zeros there:

Our department organizes three digits for the shot number manager. We already have one digit - the number 2. We have lost two more digits. They will be two zeros. Add them before the number 2. The result has a number of zeros for the signifier and a number of digits for the number reader:

Now you can start converting this jumbled number into a tenth fraction. Let’s write down the whole part and put it to someone:

And immediately write down the number of the shot part

3,002

It is important that the number of digits after the number and the number of zeros in the fractional part of the mixed number are the same.

The tenth fraction 3.002 reads like this:

“Three whole, two thousand”

“Thousands” is because the number 1000 appears in the sign of the shot part of the mixed number.

Conversion of primary fractions to tens fractions

Simple fractions, such as the numbers 10, 100, 1000 or 10000, can also be converted into tens fractions. The fragments of the original shot are a whole part of the day, first write down 0, then put someone and write down the number of the shot part.

Here, too, the number of zeros in the signifier and the number of digits in the number manager may be different. Therefore, be respectful.

butt 1.

The whole part of the day means we write 0 first and put it to someone:

Now I’m amazed at how many zeros the sign has. Bachimo, there's only one zero there. There is one digit in the number book. You can now calmly continue tenths by writing down the number 5 after the number.

In the extracted tenth fraction 0.5 there are a number of digits after some and a number of zeros in the sign of the fraction, however. This means the drib was transferred correctly.

The tenth fraction 0.5 is read as follows:

"Zero point five ten"

butt 2. Convert zero drib to tenth drib.

The whole part of the day. We write down 0 and put it to someone:

Now I’m amazed at how many zeros the sign has. Bachimo, there are two zeros there. And the number cruncher only has one digit left. To create a number of digits and a number of zeros, add one zero in front of the number 2 in the number book. I’ll see it soon. Now there are a number of zeros in the znamennik and a number of digits in the number reader, however. You can then continue to live for tens of years:

In the extracted tenth fraction 0.02 there are a number of digits after some and a number of zeros in the denominator of the fraction, however. This means the drib was transferred correctly.

The tenth fraction 0.02 is read as follows:

“Zero point two hundred.”

butt 3. Convert zero drib to tenth drib.

We write 0 and put it to whom:

Now we care about the number of zeros in the fraction. Bachimo, there are five zeros, but there is only one digit in the number book. To get the number of zeros in the sign and the number of digits in the number, however, you need to add several zeros in front of the number 5:

Now there are a number of zeros in the znamennik and a number of digits in the number reader, however. You can then live for tens of years. We write after the numeral number to the fraction

In the extracted tenth fraction 0.00005 there are a number of digits after the number and a number of zeros in the denominator of the fraction, however. This means the drib was transferred correctly.

The tenth fraction 0.00005 is read as follows:

"Zero zero, five hundred thousandths."

Conversion of improper fractions to tens fractions

The wrong kind is the same kind, which has a larger number for the sign. Improper fractions such as the numbers 10, 100, 1000 and 10000 appear near the sign. Such fractions can be converted into tens fractions. Even before translations there are tens of fractions, in such fractions it is necessary to see the whole part.

butt 1.

A fraction is an irregular fraction. In order to translate such a fraction into dozens of fractions, we need to first see the part as a whole. It’s easy to guess how to see most of the improper fractions. If you forgot, please turn around and read it.

Well, apparently the whole part is in the improper fraction. Let’s remember that drib means division - in this case, the division of the number 112 by the number 10

Let's look at this little one and select a new number, similar to a child's construction set. The number 11 will be the whole part, the number 2 will be the number of the shot part, the number 10 will be the sign of the shot part.

We took away the mixed number. Yogo i is converted into a tenth drib. And we already know how to convert such numbers to tens of fractions. Now we write down the whole part and put it to someone:

Now we appreciate how many zeros there are in the shotgun part. Bachimo, there's only one zero there. I have one digit in the shot number. This means that the number of zeros in the sign of the shot part and the number of digits in the number of the shot part are the same. This gives us the opportunity to immediately write down the number of the shot part after the coma:

In the extracted tenth fraction 11.2 there are a number of digits after some and a number of zeros in the denominator of the fraction, however. This means the drib was transferred correctly.

This means that the incorrect fraction, when converted to a tenth fraction, increases to 11.2

The tenth fraction 11.2 reads like this:

"Eleven points, two tens."

butt 2. Convert irregular drib to tenth drib.

This is the wrong thing, the fragments of the number greater than the banner. Ale її can be converted into a tenth fraction, and the fragments in the sign are the number 100.

We can see a whole part of it right in front of us. For this we divide 450 into 100 bundles:

We select a new number and remove it. And we already know how to convert mixed numbers to tens fractions.

We write down the whole part and put it to someone:

Now we are aware of the number of zeros in the sign of the shot part and the number of digits in the number of the shot part. Bachimo, there are a lot of zeros in a znamennik and a number of digits in a number, however. This gives us the opportunity to immediately write down the number of the shot part after the coma:

In the extracted tenth fraction 4.50 there are a number of digits after some and a number of zeros in the denominator of the fraction, however. This means the drib was transferred correctly.

This means that an incorrect fraction, when converted to a tenth fraction, increases to 4.50

When problems are solved, if there are zeros at the end of a tenth, they can be thrown out. Let's add zero to our species. Then we take away 4.5

This is one of the most important features of tens fractions. The point is that zeros, as standing at the end of the shot, do not give any weight to the shot. In other words, tens fractions are 4.50 and 4.5 fractions. Let’s put a sign of fidelity between them:

4,50 = 4,5

The food is coming: why are you so excited? Adje looks like 4.50 and 4.5 different fractions. The whole secret lies in the basic power structure that we have learned before. We will try to explain why tens are equivalent to the fractions 4.50 and 4.5, and then replace them with what is called “converting a tenth to a mixed number.”

Converting a tenth fraction to a mixed number

Any tenth of a fraction can be transferred back into the mixed number. For this purpose it is enough to read tens fractions. For example, we can translate 6.3 into a mixed number. 6.3 is six whole and three tenths. Let’s write down six wholes:

and about three tenths:

butt 2. Convert tenth drib 3.002 to mixed number

3.002 is three whole and two thousandths. Let's write down three goals

And we write down two thousand:

butt 3. Convert tenth drib 4.50 to mixed number

4.50 tse several hundred and fifty hundred. We record four whole things

And I charge fifty hundred:

Before speaking, let's understand the remaining example from the previous one. We were told that tens of fractions are 4.50 and 4.5 rubles. They also told us that the zero can be thrown in. Let's try to figure out that tens are 4.50 and 4.5 equals. For this purpose, let us convert tens fractions to mixed numbers.

After translation into the mixture, the number of tenths of 4.50 is reduced to , and the number of tenths of 4.5 is reduced to

There are two mixed numbers and . Let's convert mixed numbers to improper fractions:

Now we can use two fractions. The time has come to guess the main power of the fraction, that is to say, when multiplying (and dividing) the number and the sign of the fraction by the same number, the value of the fraction does not change.

Let's divide the first drib by 10

They took it away, but this is another drib. This means that they are equal to each other and equal to the same meaning:

Try dividing 450 by 100 on a calculator, and then 45 by 10. It’s a funny thing.

Conversion of tenth fraction to primary fraction

No matter how many dozens of dollars you can transfer back from the first few. For this purpose, it is enough to remember to read tens fractions. For example, let's convert it to 0.3 for the average drib. 0.3 one point zero and three tenths. We write down zero wholes:

and about three tenths 0. As is tradition, it is not necessary to write zero, so the residual evidence will not be 0, but simply.

butt 2. Convert tenth drib 0.02 to zero drib.

0.02 point zero and two hundredths. Zero is not written down, so two hundredths are written down

butt 3. Retranslate 0.00005 at the extreme value

0.00005 one hundred thousandths. Zero is not written down, therefore five hundred thousandths is written down

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We dedicate this material to such an important topic as tens of fractions. Firstly, it is important to note the main meanings, let’s look at the rules of tenth notation, as well as what the ranks of tenth fractions are. Below you can see the main types: end and unend, periodic and non-periodic fractions. In the final part we will show how points that represent fractional numbers are scaled on the coordinate axis.

What is the tenth notation of shot numbers?

Thus, the tenth notation of fractional numbers can be used both for natural numbers and for fractional numbers. It looks like two or more numbers are dialed, with a coma between them.

A ten-piece coma is required in order to cream the whole part into shot glass. As a rule, the last digit of the tenth fraction is not zero, after the first zero, if the tenth digit is immediately after the first zero.

How can you apply shot numbers to the tenth notation? Tse can be 34, 21, 0, 35035044, 0, 0001, 11231552, 9 and in.

In some podruchniks, it is possible to develop vicoristic spots instead of coma (5, 67, 6789, 1011 and so on.) This option is considered equal to, but not more typical for, Anglophone jellies.

Values of tens fractions

Based on the meaning of tens notation, we can formulate the meaning of tens fractions:

Viznachennya 1

Tens' fractions are fractional numbers in tenth notation.

Do we really need to write fractions in this form? It gives us some advantages over the basic ones, for example, a compact entry, especially in these cases, if the sign contains 1000, 100, 10, or a mixed number. For example, instead of 6 10 we can enter 0, 6 after 25 10000 - 0 0023 after 512 3 100 - 512 03.

How to correctly identify the primary fractions with tens, hundreds, and thousands in a tenth grader will be discussed within the framework of the accompanying material.

How to read tens fractions correctly

Learn the rules for reading tens fractions. So, those tens fractions, which indicate their correct primary equivalents, can be read the same way, except with the addition of the words “zero tenths” on the cob. Thus, the entry 0, 14, which indicates 14100, is read as “zero point four hundred.”

Since the tenth fraction can be placed in the same place as a mixed number, then it is read the same way as the whole number. So, since we have the number 56, 002, which corresponds to 56 2 1000, we read the following entry as “fifty-six point two thousand”.

The meaning of the digits in the notation of the tenth fraction depends on the location of the calculations (the same as in the case of natural numbers). So, for the tenth fraction 0.7 semka - tens of tens, for 0.0007 - ten thousandths, and for the fraction 70,000.345 won, it means tens of thousands of whole units. Thus, in tens fractions the concept of number rank also applies.

The names of the discharges, expanded to a coma, are similar to those that appear in natural numbers. Name those that have been expanded after being explicitly presented in the table:

Let's remove the butt.

Butt 1

We have a tenth fraction 43, 098. She has a four in the tens category, a three in the ones category, a zero in the tens category, and a zero in the hundredths category, 9 thousandths - 8.

It is customary to separate the ranks of tens fractions according to seniority. As we follow the numbers from the left to the right, we can classify them from senior ranks to junior ones. It turns out that hundreds of people are older than tens, and millions of people are younger, less than hundreds. If we take that final tenth fraction, which we pointed as a butt more, then in the new senior or highest there will be a discharge of a hundred, and for the younger or lower - a discharge of 10-thousandths.

Any tenth of a drib can be spread out over the next few ranks, so that the taxes will appear in the sum. This action works out the same way as for natural numbers.

Butt 2

Let's try to distribute the 56.0455 dribble after the ranks.

We have:

56 , 0455 = 50 + 6 + 0 , 4 + 0 , 005 + 0 , 0005

If we understand the power added, then we can detect this difference in other forms, for example, as a sum of 56 + 0.0455, or 56.0055 + 0.4 and so on.

What are tens fractions?

All the fractions that we talked about most are end tens fractions. This means that a number of numbers are added after the end. Vivedemo vyznachennya:

Viznachennya 1

Terminal tens fractions are a type of tens fractions, which have a terminating number of signs after the sign.

The butts of such shot can be 0, 367, 3, 7, 55, 102567958, 231032, 49 and etc.

Any of these fractions can be converted either to a mixed number (where the value of their fractional part is subtracted from zero), or to a simple fraction (when the whole part is zero). To that end, we have dedicated this material. Here we will simply introduce a couple of applications: for example, the final tenth fraction 5, 63 can be reduced to the form 5 63 100, and 0, 2 suggests 2 10 (or any other equal fraction, for example, 4 20 or 1 5.)

Ale gate process, then. recording a subordinate fraction in a tenth form can be done every time. Thus, 5 13 cannot be replaced by the equal fraction from the sign 100, 10 and otherwise, and the final tenth fraction cannot be seen from it.

Main types of non-decimal fractions: periodic and non-periodic fractions

We were told that trailing fractions are called so because they have a trailing number of digits after them. However, they may be uncounted as a whole, and in this case the fractions themselves will also be called uncounted.

Vicennia 2

Uncounted tens fractions are those that have a number of digits after them.

Obviously, it is simply impossible to add such numbers to the records, so we indicate only some of them and then put a check mark. This is a sign to talk about the endless continuation of the sequence of signs after coma. The butts of multiple tens fractions can be 0, 143346732…, 3, 1415989032…, 153, 0245005…, 2, 666666666666…, 69, 74876815, etc.

At the “tail” of such a fraction there may be symbols, which would give rise to a sequence of numbers, or a constant repetition of the same sign or group of signs. Fractions from drawings after tens of coma are called periodic.

Vicenzennya 3

Periodic tens fractions are those uncounted tens fractions that have one group of many digits repeated after them. The part that is repeated is called the fraction period.

For example, for a shot 3 444444 .... The period will be the number 4, and for 76, 134134134134... - Group 134.

What is the minimum number of characters that can be removed from a periodic fraction? For periodic fractions, it is enough to write the entire period once in the round arms. So, drib 3, 444444…. It would be correct to write yak 3, (4), and 76, 134134134134... - yak 76, (134).

In general, the records with many periods in the arms have the same sense: for example, periodic drib 0, 677777 - the same as 0, 6 (7) and 0, 6 (77) etc. Also valid entries are 0.67777 (7), 0.67 (7777) and others.

In order to eliminate the consequences, we introduce the same value. It is possible to write down just one period (the shortest possible sequence of digits), which is worth closer to a tenth coma, and place it in round arms.

So, for the indicated fraction, the main entry in each case is 0, 6 (7), and, for example, in the case of the fraction 8, 9134343434 we will write 8, 91 (34).

If the signifier of a prime fraction contains simple multipliers that are not equal to 5 and 2, then when transferred to the tenth notation, uncounted fractions will appear from them.

In principle, we can write any kind of terminal difference in the form of a periodic one. For this we simply need to add a lot of zeros to the right-hander. What does the recording look like? Let us assume that we have a terminal 45, 32. The periodical appearance is 45, 32 (0). This action is possible because adding zeros to the right hand of any tenth fraction gives us, as a result, an equal fraction.

Closely follow on periodic fractions with period 9, for example, 4, 89 (9), 31, 6 (9). There is an alternative notation for similar fractions with a period of 0; therefore, they are often replaced when reading with fractions with a zero period. If this is the value of the offensive discharge, add one, and in the round arches indicate (0). It is easy to verify the equality of numbers by imagining them as equal fractions.

For example, the number 8, 31 (9) can be replaced with the similar number 8, 32 (0). Abo 4, (9) = 5, (0) = 5.

Numerous tens of periodic fractions are reduced to rational numbers. Otherwise, seemingly, any periodic differences can be detected in the appearance of the primary, and inadvertently.

There are fractions that have a sequence that repeats endlessly, every day. In this case, they are called non-periodic fractions.

Vicenchennya 4

Non-periodic tens fractions are preceded by those non-periodic tens fractions that have no period after them, then. repeated group of numbers.

Sometimes non-periodic fractions look even similar to periodic ones. For example, 9, 03003000300003 ... at first glance there appears to be a period, but the report’s analysis of the signs after the coma confirms that it is still a non-periodic period. With such numbers we need to be more respectful.

Non-periodic fractions are raised to irrational numbers. In simple cases, fractions cannot be translated.

Basic actions with tens fractions

With tens fractions you can work with the following actions: equalization, subdivision, addition, subdivision and multiplication. Let's ripen the skin around them.

The leveling of tens fractions can be reduced to the leveling of elementary fractions, which represent the final tenths. However, it is not possible to reduce numerous non-periodic fractions to this form, and converting tens fractions is often a difficult task. How can we quickly earn this task, as we need to earn it at the hour of the greatest task? It is easy to equalize tens fractions after digits in the same way as we equalize natural numbers. I will dedicate this article to this method.

To add tens fractions with other fractions, manually use the stacking method for natural numbers. To add periodic tens fractions, you must first replace them with basic ones and follow the standard scheme. Since our minds have been given the need to combine countless non-periodic fractions, then before doing so we need to round them to the tenth place, and then add them up. The smaller the digit to which we round, the greater the accuracy of the calculation. For the sake of clarity, multiplication and subdivision of unskewed fractions, forward rounding is also necessary.

Finding the difference in tens fractions back to the date added. In fact, with additional help we can find the number, the sum of which is a fraction that appears, give us a change. The report is consistent within the framework of the surrounding material.

The multiplication of tens fractions is carried out in the same way as for natural numbers. The method of calculating the cashier is also suitable for this. This equation with periodic fractions can again be reduced to a multiplication of primary fractions following the already established rules. Continuous fractions, as we remember, need to be rounded before adding fractions.

The process of sub-tens fractions is a reverse process of multiplication. At the peak of the hour, we are also forced into the stack.

You can set the exact relationship between the end tenth fraction and a point on the coordinate axis. It is clear how to identify a point on the axis that exactly corresponds to the required tenth fraction.

We have already learned how there are points that represent prime fractions, and tens fractions can be reduced to this form. For example, the initial value is 14 10 – the same as 1, 4, so the corresponding point will be removed from the cob in a positive direction exactly to the same distance:

You can do without replacing the tenth fraction with the prime fraction, but use the method of sorting into ranks as a basis. So, if we need to identify a point whose coordinate is similar to 15, 4008, then we first represent the number in the form of 15 + 0, 4 +, 0008. For the cob, 15 whole single cuts are added to the cob in the positive direction, then 4 tenths of one cut, and then 8 ten-thousandths of one cut. The result is a coordinate point, which is indicated by 15, 4008.

For an unfinished tenth fraction, it is better to use this method, the fragments of wine allow you to get as close as possible to the required point. In some cases, you can find the exact similarity of the unskewed fraction on the coordinate axis: for example, 2 = 1.41421. . . , and this fraction can be associated with a point on the coordinate exchange, removed from 0 to the half of the diagonal of the square, the side of which corresponds to one single section.

If we find not a point on the axis, but a tenth of a fraction, which is indicative of it, then this action is called a tenth of a fraction. I marvel at how you can earn money.

Let’s say we need to move from zero to a given point on the coordinate axis (or get as close as possible to the point with the unskewed fraction). For this step, we place one section at a time into the coordinate system until we reach the required point. After entire sections, if necessary, tens, hundredths and fractions of fractions are measured so that the correspondence is as accurate as possible. From the results, we took a tenth fraction, which indicates the specified point on the coordinate axis.

Most of all, we pointed the little ones at point M. Marvel at this once again: in order to touch this point, you need to subtract from zero one single section and at least ten parts from the new one, the rest of which this point is representative of tenth drib 1, 4.

Since we cannot put it into a speck in the process of a tenth of a millimeter, it means that it represents an endless tenth of a millimeter.

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