EntranceWhat does it mean to write as a decimal? Decimal
For example.$\frac(3)(10), 4 \frac(7)(100), \frac(11)(10000)$
Such fractions are usually written without a denominator, and the meaning of each digit depends on the place in which it stands. For such fractions whole part is separated by a comma, and after the comma there must be as many digits as there are zeros in the denominator of the common fraction. The fractional digits are called decimals.
For example.$\frac(21)(100)=0.21 ; 3 \frac(21)(100)=$3.21
The first decimal place after the decimal point corresponds to tenths, the second to hundredths, the third to thousandths, etc.
If the number of zeros in the denominator of a decimal fraction is greater than the number of digits in the numerator of the same fraction, then the required number of zeros is added after the decimal point before the numerator digits.
Since there are four zeros in the denominator, and two digits in the numerator, in the decimal notation of the fraction we add $4-2=2$ zeros before the numerator.
The main property of a decimal fraction
Property
If you add several zeros to the decimal fraction on the right, the value of the decimal fraction will not change.
For example.$12,034=12,0340=12,03400=12,034000=\ldots$
Comment
Thus, zeros at the end of the decimal are not taken into account, so when doing various actions these zeros can be crossed out/discarded.
Comparison of decimals
To compare two decimal fractions (to find out which of two decimal fractions is larger), you need to compare their whole parts, then tenths, hundredths, etc. If the whole part of one of the fractions is greater than the whole part of another fraction, then the first fraction is considered larger. In the case of equality of whole parts, the fraction with more tenths is greater, etc.
Example
Exercise. Compare fractions $2,432$ ; $2.41$ and $1,234$
Solution. The fraction $1.234$ is the smallest fraction because its integer part is 1, and $1
Let us now compare the size of the fractions $2.432$ and $1.234$. Their whole parts are equal to each other and equal to 2. Let's compare the tenths: $4=4$. Compare hundredths: $3>1$. Thus, $2.432>$2.41.
fractional number.
Decimal notation of a fractional number is a set of two or more digits from $0$ to $9$, between which there is a so-called \textit (decimal point).
Example 1
For example, $35.02$; $100.7$; $123\456.5$; $54.89$.
The leftmost digit in the decimal notation of a number cannot be zero, the only exception being when the decimal point is immediately after the first digit $0$.
Example 2
For example, $0.357$; $0.064$.
Often the decimal point is replaced with a decimal point. For example, $35.02$; $100.7$; $123\456.5$; $54.89$.
Decimal definition
Definition 1
Decimals-- these are fractional numbers that are represented in decimal notation.
For example, $121.05; $67.9$; $345.6700$.
Decimal fractions are used to write regular fractions more compactly. ordinary fractions, the denominators of which are the numbers $10$, $100$, $1\000$, etc. and mixed numbers, the denominators of the fractional part of which are the numbers $10$, $100$, $1\000$, etc.
For example, the common fraction $\frac(8)(10)$ can be written as a decimal $0.8$, and the mixed number $405\frac(8)(100)$ can be written as a decimal $405.08$.
Reading Decimals
Decimals, which correspond to regular fractions, are read in the same way as ordinary fractions, only the phrase “zero integers” is added in front. For example, the common fraction $\frac(25)(100)$ (read “twenty-five hundredths”) corresponds to the decimal fraction $0.25$ (read “zero point twenty-five hundredths”).
Decimal fractions that correspond to mixed numbers are read the same way as mixed numbers. For example, mixed number$43\frac(15)(1000)$ corresponds to the decimal fraction $43.015$ (read “forty-three point fifteen thousandths”).
Places in decimals
In writing a decimal fraction, the meaning of each digit depends on its position. Those. in decimal fractions the concept also applies category.
Places in decimal fractions up to the decimal point are called the same as places in natural numbers. The decimal places after the decimal point are listed in the table:
Picture 1.
Example 3
For example, in the decimal fraction $56.328$, the digit $5$ is in the tens place, $6$ is in the units place, $3$ is in the tenths place, $2$ is in the hundredths place, $8$ is in the thousandths place.
Places in decimal fractions are distinguished by precedence. When reading a decimal fraction, move from left to right - from senior rank to younger.
Example 4
For example, in the decimal fraction $56.328$, the most significant (highest) place is the tens place, and the low (lowest) place is the thousandths place.
A decimal fraction can be expanded into digits similar to the digit decomposition of a natural number.
Example 5
For example, let's break down the decimal fraction $37.851$ into digits:
$37,851=30+7+0,8+0,05+0,001$
Ending decimals
Definition 2
Ending decimals are called decimal fractions, the records of which contain a finite number of characters (digits).
For example, $0.138$; $5.34$; $56.123456$; $350,972.54.
Any finite decimal fraction can be converted to a fraction or a mixed number.
Example 6
For example, the final decimal fraction $7.39$ answers a fractional number$7\frac(39)(100)$, and the final decimal fraction $0.5$ corresponds to the proper common fraction $\frac(5)(10)$ (or any fraction that is equal to it, for example, $\frac(1) (2)$ or $\frac(10)(20)$.
Converting a fraction to a decimal
Converting fractions with denominators $10, 100, \dots$ to decimals
Before converting some proper fractions to decimals, they must first be “prepared.” The result of such preparation should be the same number of digits in the numerator and the same number of zeros in the denominator.
The essence of “preliminary preparation” of proper ordinary fractions for conversion to decimal fractions is adding such a number of zeros to the left in the numerator that the total number of digits becomes equal to the number of zeros in the denominator.
Example 7
For example, let's prepare the fraction $\frac(43)(1000)$ for conversion to a decimal and get $\frac(043)(1000)$. And the ordinary fraction $\frac(83)(100)$ does not need any preparation.
Let's formulate rule for converting a proper common fraction with a denominator of $10$, or $100$, or $1\000$, $\dots$ into a decimal fraction:
write $0$;
after it put a decimal point;
write down the number from the numerator (along with added zeros after preparation, if necessary).
Example 8
Convert the proper fraction $\frac(23)(100)$ to a decimal.
Solution.
The denominator contains the number $100$, which contains $2$ and two zeros. The numerator contains the number $23$, which is written with $2$.digits. This means that there is no need to prepare this fraction for conversion to a decimal.
Let's write $0$, put a decimal point and write down the number $23$ from the numerator. We get the decimal fraction $0.23$.
Answer: $0,23$.
Example 9
Write the proper fraction $\frac(351)(100000)$ as a decimal.
Solution.
The numerator of this fraction contains $3$ digits, and the number of zeros in the denominator is $5$, so this ordinary fraction must be prepared for conversion to a decimal. To do this, you need to add $5-3=2$ zeros to the left in the numerator: $\frac(00351)(100000)$.
Now we can form the desired decimal fraction. To do this, write down $0$, then add a comma and write down the number from the numerator. We get the decimal fraction $0.00351$.
Answer: $0,00351$.
Let's formulate rule for converting improper fractions with denominators $10$, $100$, $\dots$ into decimal fractions:
write down the number from the numerator;
Use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.
Example 10
Convert the improper fraction $\frac(12756)(100)$ to a decimal.
Solution.
Let's write down the number from the numerator $12756$, then separate the $2$ digits on the right with a decimal point, because the denominator of the original fraction $2$ is zero. We get the decimal fraction $127.56$.
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 a 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 the integer part is separated from the fractional part 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 is: 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
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 it. 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.
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 now get 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.
Fractions written in the form 0.8; 0.13; 2.856; 5.2; 0.04 is called decimal. In fact, decimals are a simplified notation for ordinary fractions. This notation is convenient to use for all fractions whose denominators are 10, 100, 1000, and so on.
Let's look at examples (0.5 is read as zero point five);
(0.15 read as, zero point fifteen);
(5.3 read as, five point three).
Please note that in the notation of a decimal fraction, a comma separates the integer part of a number from the fractional part, the integer part of a proper fraction is 0. The notation of the fractional part of a decimal fraction contains as many digits as there are zeros in the notation of the denominator of the corresponding ordinary fraction.
Let's look at an example, 
, 
, 
.
In some cases, it may be necessary to treat a natural number as a decimal whose fractional part is zero. It is customary to write that 5 = 5.0; 245 = 245.0 and so on. Note that in the decimal notation of a natural number, the unit of the least significant digit is 10 times less than the unit of the adjacent most significant digit. Writing decimal fractions has the same property. Therefore, immediately after the decimal point there is a place of tenths, then a place of hundredths, then a place of thousandths, and so on. Below are the names of the digits of the number 31.85431, the first two columns are the integer part, the remaining columns are the fractional part.
This fraction is read as thirty-one point eighty-five thousand four hundred and thirty-one hundred thousandths.
Adding and subtracting decimals
The first way is to convert decimal fractions into ordinary fractions and perform addition. 
As can be seen from the example, this method is very inconvenient and it is better to use the second method, which is more correct, without converting decimal fractions into ordinary ones. In order to add two decimal fractions, you need to:
- equalize the number of digits after the decimal point in the terms;
- write the terms one below the other so that each digit of the second term is under the corresponding digit of the first term;
- add the resulting numbers the same way you add natural numbers;
- Place a comma in the resulting sum under the commas in the terms.
Let's look at examples:
- equalize the number of digits after the decimal point in the minuend and subtrahend;
- write the subtrahend under the minuend so that each digit of the subtrahend is under the corresponding digit of the minuend;
- perform subtraction in the same way as natural numbers are subtracted;
- put a comma in the resulting difference under the commas in the minuend and subtrahend.
Let's look at examples:
In the examples discussed above, it can be seen that the addition and subtraction of decimal fractions was performed bit by bit, that is, in the same way as we performed similar operations with natural numbers. This is the main advantage of the decimal form of writing fractions.
Multiplying Decimals
In order to multiply a decimal fraction by 10, 100, 1000, and so on, you need to move the decimal point in this fraction to the right by 1, 2, 3, and so on, respectively. Therefore, if the comma is moved to the right by 1, 2, 3 and so on digits, then the fraction will increase accordingly by 10, 100, 1000 and so on times. In order to multiply two decimal fractions, you need to:
- multiply them as natural numbers, ignoring commas;
- in the resulting product, separate as many digits on the right with a comma as there are after the commas in both factors together.
There are cases when a product contains fewer digits than is required to be separated by a comma; the required number of zeros are added to the left before this product, and then the comma is moved to the left by the required number of digits.
Let's look at examples: 2 * 4 = 8, then 0.2 * 0.4 = 0.08; 23 * 35 = 805, then 0.023 * 0.35 = 0.00805.
There are cases when one of the multipliers is equal to 0.1; 0.01; 0.001 and so on, it is more convenient to use the following rule.
- To multiply a decimal by 0.1; 0.01; 0.001 and so on, in this decimal fraction you need to move the decimal point to the left by 1, 2, 3, and so on, respectively.
Let's look at examples: 2.65 * 0.1 = 0.265; 457.6 * 0.01 = 4.576.
Properties of Multiplication natural numbers are also performed for decimal fractions.
- ab = ba- commutative property of multiplication;
- (ab) c = a (bc)- the associative property of multiplication;
- a (b + c) = ab + ac is a distributive property of multiplication relative to addition.
Decimal division
It is known that if you divide a natural number a to a natural number b means to find such a natural number c, which when multiplied by b gives a number a. This rule remains true if at least one of the numbers a, b, c is a decimal fraction.
Let's look at an example: you need to divide 43.52 by 17 with a corner, ignoring the comma. In this case, the comma in the quotient should be placed immediately before the first digit after the decimal point in the dividend is used.
There are cases when the dividend is less than the divisor, then the integer part of the quotient is equal to zero. Let's look at an example:
Let's look at another interesting example.
The division process has stopped because the digits of the dividend have run out and the remainder does not have a zero. It is known that a decimal fraction will not change if any number of zeros are added to it on the right. Then it becomes clear that the numbers of the dividend cannot end.
In order to divide a decimal fraction by 10, 100, 1000, and so on, you need to move the decimal point in this fraction to the left by 1, 2, 3, and so on digits. Let's look at an example: 5.14: 10 = 0.514; 2: 100 = 0.02; 37.51: 1000 = 0.03751.
If the dividend and divisor are increased simultaneously by 10, 100, 1000, and so on times, then the quotient will not change.
Consider an example: 39.44: 1.6 = 24.65, increase the dividend and divisor by 10 times 394.4: 16 = 24.65 It is fair to note that dividing a decimal fraction by a natural number in the second example is easier.
In order to divide a decimal fraction by a decimal, you need to:
- move the commas in the dividend and divisor to the right by as many digits as there are after the decimal point in the divisor;
- divide by a natural number.
Let's consider an example: 23.6: 0.02, note that the divisor has two decimal places, therefore we multiply both numbers by 100 and get 2360: 2 = 1180, divide the result by 100 and get the answer 11.80 or 23.6: 0, 02 = 11.8.
Comparison of decimals
There are two ways to compare decimals. Method one, you need to compare two decimal fractions 4.321 and 4.32, equalize the number of decimal places and start comparing place by place, tenths with tenths, hundredths with hundredths, and so on, in the end we get 4.321 > 4.320.
The second way to compare decimal fractions is done using multiplication; multiply the above example by 1000 and compare 4321 > 4320. Which method is more convenient, everyone chooses for themselves.
A decimal fraction differs from an ordinary fraction in that its denominator is a place value.
For example:
Decimal fractions are separated from ordinary fractions into a separate form, which led to their own rules for comparing, adding, subtracting, multiplying and dividing these fractions. In principle, you can work with decimal fractions using the rules of ordinary fractions. Own rules for converting decimal fractions simplify calculations, and rules for converting ordinary fractions to decimals, and vice versa, serve as a link between these types of fractions.
Writing and reading decimal fractions allows you to write them down, compare them, and perform operations on them according to rules very similar to the rules for operations with natural numbers.
The system of decimal fractions and operations on them was first outlined in the 15th century. Samarkand mathematician and astronomer Dzhemshid ibn-Masudal-Kashi in the book “The Key to the Art of Counting”.
The whole part of the decimal fraction is separated from the fractional part by a comma; in some countries (the USA) they put a period. If a decimal fraction does not have an integer part, then the number 0 is placed before the decimal point.
You can add any number of zeros to the fractional part of a decimal on the right; this does not change the value of the fraction. The fractional part of a decimal is read at the last significant digit.
For example:
0.3 - three tenths
0.75 - seventy-five hundredths
0.000005 - five millionths.
Reading the whole part of a decimal is the same as reading natural numbers.
For example:
27.5 - twenty seven...;
1.57 - one...
After the whole part of the decimal fraction the word “whole” is pronounced.
For example:
10.7 - ten point seven
0.67 - zero point sixty-seven hundredths.
Decimal places are the digits of the fractional part. The fractional part is not read by digits (unlike natural numbers), but as a whole, therefore the fractional part of a decimal fraction is determined by the last significant digit on the right. The place system of the fractional part of the decimal is somewhat different than that of natural numbers.
- 1st digit after busy - tenths digit
- 2nd decimal place - hundredths place
- 3rd decimal place - thousandths place
- 4th decimal place - ten-thousandth place
- 5th decimal place - hundred thousandths place
- 6th decimal place - millionth place
- The 7th decimal place is the ten-millionth place
- The 8th decimal place is the hundred millionth place
The first three digits are most often used in calculations. The large digit capacity of the fractional part of decimals is used only in specific branches of knowledge where infinitesimal quantities are calculated.
Converting a decimal to a mixed fraction consists of the following: the number before the decimal point is written as an integer part of the mixed fraction; the number after the decimal point is the numerator of its fractional part, and in the denominator of the fractional part write a unit with as many zeros as there are digits after the decimal point.