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; $67.9$; $345.6700$.
Decimals are used to more compactly write proper 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 fraction $0.8$, and mixed number$405\frac(8)(100)$ -- as a decimal fraction of $405.08$.
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 as mixed numbers. For example, the 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.
The places in decimal fractions before the decimal point are called the same as the 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 whose records contain final number 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 correct common fraction$\frac(5)(10)$ (or any fraction that is equal to it, such as $\frac(1)(2)$ 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 $127.56$.
| Ending decimals |
| Multiplying and dividing decimals by 10, 100, 1000, 10000, etc. |
| Converting a trailing decimal to a fraction |
Decimals are divided into the following three classes: finite decimals, infinite periodic decimals, and infinite non-periodic decimals.
Ending decimals
Definition . Final decimal fraction (decimal fraction) called a fraction or mixed number having a denominator of 10, 100, 1000, 10000, etc.
For example,
Decimal fractions also include those fractions that can be reduced to fractions with a denominator of 10, 100, 1000, 10000, etc., using the basic property of fractions.
For example,
Statement . An irreducible simple fraction or an irreducible mixed non-integer number is a finite decimal fraction if and only if the factorization of their denominators into prime factors contains only the numbers 2 and 5 as factors, and in arbitrary powers.
For decimal fractions there is special recording method , using a comma. To the left of the decimal point the whole part of the fraction is written, and to the right is the numerator of the fractional part, before which such a number of zeros are added so that the number of digits after the decimal point is equal to the number of zeros in the denominator of the decimal fraction.
For example,
Note that the decimal fraction will not change if you add several zeros to the right or left of it.
For example,
3,14 = 3,140 =
=
3,1400 = 003,14 .
The numbers before the decimal point (to the left of the decimal point) in decimal notation of final decimal fraction, form a number called whole part decimal.
The numbers after the decimal point (to the right of the decimal point) in the decimal notation of the final decimal fraction are called decimals.
A final decimal has a finite number of decimal places. Decimals form fractional part of a decimal.
Multiplying and dividing decimals by 10, 100, 1000, etc.
In order to multiply a decimal by 10, 100, 1000, 10000, etc., enough move comma to the right by 1, 2, 3, 4, etc. decimal places respectively.
There is another representation of the rational number 1/2, different from representations of the form 2/4, 3/6, 4/8, etc. We mean representation in the form of a decimal fraction 0.5. Some fractions have finite decimal representations, e.g.
while the decimal representations of other fractions are infinite:
These infinite decimals can be obtained from the corresponding rational fractions by dividing the numerator by the denominator. For example, in the case of the fraction 5/11, dividing 5.000... by 11 gives 0.454545...
Which rational fractions have finite decimal representations? Before answering this question in general, let's look at a specific example. Let's take, say, the final decimal fraction 0.8625. We know that
and that any finite decimal fraction can be written as a rational decimal fraction with a denominator equal to 10, 100, 1000, or some other power of 10.
Reducing the fraction on the right to an irreducible fraction, we get
The denominator 80 is obtained by dividing 10,000 by 125 - the largest common divisor 10,000 and 8625. Therefore, in expansion into prime factors the number 80, like the number 10,000, includes only two prime factors: 2 and 5. If we started not with 0.8625, but with any other finite decimal fraction, then the resulting irreducible rational fraction would also have this property. In other words, the expansion of the denominator b into prime factors could only include prime numbers 2 and 5, since b is a divisor of some power of 10, and . This circumstance turns out to be decisive, namely, the following general statement holds:
An irreducible rational fraction has a finite decimal representation if and only if the number b has no prime factors of 2 and 5.
Note that b does not have to have both numbers 2 and 5 among its prime factors: it can be divisible by only one of them or not be divisible by them at all. For example,
here b is equal to 25, 16 and 1, respectively. What is significant is that b has no other divisors other than 2 and 5.
The above sentence contains the expression if and only if. So far we have proved only the part that relates to turnover only then. It was we who showed that the decomposition of a rational number into a decimal fraction will be finite only in the case when b has no prime factors other than 2 and 5.
(In other words, if b is divisible by a prime number other than 2 or 5, then the irreducible fraction has no finite decimal expression.)
The then part of the sentence states that if the integer b has no prime factors other than 2 and 5, then the irreducible rational fraction can be represented by a finite decimal fraction. To prove this, we must take an arbitrary irreducible rational fraction, for which b has no prime factors other than 2 and 5, and make sure that the corresponding decimal fraction is finite. Let's look at an example first. Let
To obtain the decimal expansion, we transform this fraction into a fraction whose denominator is an integer power of ten. This can be achieved by multiplying the numerator and denominator by:
The above reasoning can be extended to the general case as follows. Suppose b is of the form , where the type is non-negative integers (i.e., positive numbers or zero). Two cases are possible: either less than or equal (this condition is written), or greater (which is written). When we multiply the numerator and denominator of the fraction by
Since the integer is not negative (that is, positive or equal to zero), then , and therefore a is a positive integer. Let's put it. Then
Remember how in the very first lesson about decimals I said that there are numerical fractions that cannot be represented as decimals (see lesson “ Decimals”)? We also learned how to factor the denominators of fractions to see if there were any numbers other than 2 and 5.
So: I lied. And today we will learn how to convert absolutely any numerical fraction into a decimal. At the same time, we will get acquainted with a whole class of fractions with an infinite significant part.
A periodic decimal is any decimal that:
- The significant part consists of infinite number numbers;
- At certain intervals, the numbers in the significant part are repeated.
The set of repeating digits that make up the significant part is called the periodic part of a fraction, and the number of digits in this set is called the period of the fraction. The remaining segment of the significant part, which is not repeated, is called the non-periodic part.
Since there are many definitions, it is worth considering a few of these fractions in detail:
This fraction appears most often in problems. Non-periodic part: 0; periodic part: 3; period length: 1.
Non-periodic part: 0.58; periodic part: 3; period length: again 1.
Non-periodic part: 1; periodic part: 54; period length: 2.
Non-periodic part: 0; periodic part: 641025; period length: 6. For convenience, repeating parts are separated from each other by a space - this is not necessary in this solution.
Non-periodic part: 3066; periodic part: 6; period length: 1.
As you can see, the definition of a periodic fraction is based on the concept significant part of a number. Therefore, if you have forgotten what it is, I recommend repeating it - see the lesson “”.
Transition to periodic decimal fraction
Consider an ordinary fraction of the form a /b. Let's factorize its denominator into prime factors. There are two options:
- The expansion contains only factors 2 and 5. These fractions are easily converted to decimals - see the lesson “Decimals”. We are not interested in such people;
- There is something else in the expansion other than 2 and 5. In this case, the fraction cannot be represented as a decimal, but it can be converted into a periodic decimal.
To define a periodic decimal fraction, you need to find its periodic and non-periodic parts. How? Convert the fraction to an improper fraction, and then divide the numerator by the denominator using a corner.
The following will happen:
- Will split first whole part, if it exists;
- There may be several numbers after the decimal point;
- After a while the numbers will start repeat.
That's all! Repeating numbers after the decimal point are denoted by the periodic part, and those in front are denoted by the non-periodic part.
Task. Convert ordinary fractions to periodic decimals:
All fractions without an integer part, so we simply divide the numerator by the denominator with a “corner”:
As you can see, the remainders are repeated. Let's write the fraction in the “correct” form: 1.733 ... = 1.7(3).
The result is a fraction: 0.5833 ... = 0.58(3).
We write it in normal form: 4.0909 ... = 4,(09).
We get the fraction: 0.4141 ... = 0,(41).
Transition from periodic decimal fraction to ordinary fraction
Consider the periodic decimal fraction X = abc (a 1 b 1 c 1). It is required to convert it into a classic “two-story” one. To do this, follow four simple steps:
- Find the period of the fraction, i.e. count how many digits are in the periodic part. Let this be the number k;
- Find the value of the expression X · 10 k. This is equivalent to shifting the decimal point to the right a full period - see the lesson "Multiplying and dividing decimals";
- The original expression must be subtracted from the resulting number. In this case, the periodic part is “burned” and remains common fraction;
- Find X in the resulting equation. We convert all decimal fractions to ordinary fractions.
Task. Reduce to ordinary improper fraction numbers:
- 9,(6);
- 32,(39);
- 0,30(5);
- 0,(2475).
We work with the first fraction: X = 9,(6) = 9.666 ...
The parentheses contain only one digit, so the period is k = 1. Next, we multiply this fraction by 10 k = 10 1 = 10. We have:
10X = 10 9.6666... = 96.666...
Subtract the original fraction and solve the equation:
10X − X = 96.666 ... − 9.666 ... = 96 − 9 = 87;
9X = 87;
X = 87/9 = 29/3.
Now let's look at the second fraction. So X = 32,(39) = 32.393939...
Period k = 2, so multiply everything by 10 k = 10 2 = 100:
100X = 100 · 32.393939 ... = 3239.3939 ...
Subtract the original fraction again and solve the equation:
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 give the calculations:
Period k = 1 ⇒ multiply everything 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.
Finally, the last fraction: X = 0,(2475) = 0.2475 2475... Again, for convenience, the periodic parts are separated from each other by spaces. We have:
k = 4 ⇒ 10 k = 10 4 = 10,000;
10,000X = 10,000 0.2475 2475 = 2475.2475 ...
10,000X − X = 2475.2475 ... − 0.2475 2475 ... = 2475;
9999X = 2475;
X = 2475: 9999 = 25/101.
To rational number m/n is written as a decimal fraction; you need to divide the numerator by the denominator. In this case, the quotient is written as a finite or infinite decimal fraction.
Write down given number as a decimal fraction.
Solution. Divide the numerator of each fraction into a column by its denominator: A) divide 6 by 25; b) divide 2 by 3; V) divide 1 by 2, and then add the resulting fraction to one - the integer part of this mixed number.
Irreducible ordinary fractions whose denominators do not contain prime factors other than 2 And 5 , are written as a final decimal fraction.
IN example 1 when A) denominator 25=5·5; when V) the denominator is 2, so we get the final decimals 0.24 and 1.5. When b) the denominator is 3, so the result cannot be written as a finite decimal.
Is it possible, without long division, to convert into a decimal fraction such an ordinary fraction, the denominator of which does not contain other divisors other than 2 and 5? Let's figure it out! What fraction is called a decimal and is written without a fraction bar? Answer: fraction with denominator 10; 100; 1000, etc. And each of these numbers is a product equal number of twos and fives. In fact: 10=2 ·5 ; 100=2 ·5 ·2 ·5 ; 1000=2 ·5 ·2 ·5 ·2 ·5 etc.
Consequently, the denominator of an irreducible ordinary fraction will need to be represented as the product of “twos” and “fives”, and then multiplied by 2 and (or) 5 so that the “twos” and “fives” become equal. Then the denominator of the fraction will be equal to 10 or 100 or 1000, etc. To ensure that the value of the fraction does not change, we multiply the numerator of the fraction by the same number by which we multiplied the denominator.
Express the following common fractions as decimals:
Solution. Each of these fractions is irreducible. Let's factor the denominator of each fraction into prime factors.
20=2·2·5. Conclusion: one “A” is missing.
8=2·2·2. Conclusion: three “A”s are missing.
25=5·5. Conclusion: two “twos” are missing.
Comment. In practice, they often do not use factorization of the denominator, but simply ask the question: by how much should the denominator be multiplied so that the result is one with zeros (10 or 100 or 1000, etc.). And then the numerator is multiplied by the same number.
So, in case A)(example 2) from the number 20 you can get 100 by multiplying by 5, therefore, you need to multiply the numerator and denominator by 5.
When b)(example 2) from the number 8 the number 100 will not be obtained, but the number 1000 will be obtained by multiplying by 125. Both the numerator (3) and the denominator (8) of the fraction are multiplied by 125.
When V)(example 2) from 25 you get 100 if you multiply by 4. This means that the numerator 8 must be multiplied by 4.
An infinite decimal fraction in which one or more digits invariably repeat in the same sequence is called periodic as a decimal. The set of repeating digits is called the period of this fraction. For brevity, the period of a fraction is written once, enclosed in parentheses.
When b)(example 1) there is only one repeating digit and is equal to 6. Therefore, our result 0.66... will be written like this: 0,(6) . They read: zero point, six in period.
If there are one or more non-repeating digits between the decimal point and the first period, then such a periodic fraction is called a mixed periodic fraction.
An irreducible common fraction whose denominator is together with others multiplier contains multiplier 2 or 5 , becomes mixed periodic fraction.
Write numbers as decimals.
