The decimal fraction must contain a comma. That numerical part of the fraction, which is located to the left of the decimal point, is called the whole; to the right - fractional:
5.28 5 - integer part 28 - fractional part
The fractional part of a decimal is made up of decimal places(decimal places):
- tenths - 0.1 (one tenth);
- hundredths - 0.01 (one hundredth);
- thousandths - 0.001 (one thousandth);
- ten-thousandths - 0.0001 (one ten-thousandth);
- hundred thousandths - 0.00001 (one hundred thousandth);
- millionths - 0.000001 (one millionth);
- ten millionths - 0.0000001 (one ten millionth);
- one hundred millionth - 0.00000001 (one hundred millionth);
- billionths - 0.000000001 (one billionth), etc.
- read the number that is the integer part of the fraction and add the word " whole";
- read the number that makes up the fractional part of the fraction and add the name of the least significant digit.
For example:
- 0.25 - zero point twenty-five hundredths;
- 9.1 - nine point one tenth;
- 18.013 - eighteen point thirteen thousandths;
- 100.2834 is one hundred and two thousand eight hundred and thirty-four ten thousandths.
Writing decimals
To write a decimal fraction, you must:
- write down the integer part of the fraction and put a comma (the number meaning the integer part of the fraction always ends with the word " whole");
- write the fractional part of the fraction in such a way that the last digit falls into the desired digit (if there are no significant digits in certain decimal places, they are replaced by zeros).
For example:
- twenty point nine - 20.9 - in this example, everything is simple;
- five point one hundredth - 5.01 - the word "hundredth" means that there should be two digits after the decimal point, but since there is no tenth place in the number 1, it is replaced by zero;
- zero point eight hundred and eight thousandths - 0.808;
- three point fifteen - it is impossible to write such a decimal fraction, because a mistake was made in the pronunciation of the fractional part - the number 15 contains two digits, and the word "tenths" means only one. Correct will be three point fifteen hundredths (or thousandths, ten thousandths, etc.).
Decimal Comparison
Comparison of decimal fractions is carried out similarly to comparison of natural numbers.
- first, the integer parts of the fractions are compared - the decimal fraction with the larger integer part will be larger;
- if the integer parts of the fractions are equal, the fractional parts are compared bit by bit, from left to right, starting from the comma: tenths, hundredths, thousandths, etc. The comparison is carried out until the first discrepancy - that decimal fraction will be larger, which will have a larger unequal digit in the corresponding digit of the fractional part. For example: 1.2 8 3 > 1,27 9, because in hundredths the first fraction has 8, and the second has 7.
A decimal fraction differs from an ordinary fraction in that its denominator is a bit unit.
For example:
Decimal fractions have been separated from ordinary fractions into a separate form, which has led to its own rules for comparing, adding, subtracting, multiplying and dividing these fractions. In principle, you can work with decimal fractions according to 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, compare and operate on them according to rules very similar to the rules for operations with natural numbers.
For the first time, the system of decimal fractions and operations on them was described in the 15th century. Samarkand mathematician and astronomer Jamshid ibn-Masudal-Kashi in the book "The Key to the Art of Accounting".
The integer part of the decimal fraction is separated from the fractional part by a comma, in some countries (USA) they put a period. If there is no integer part in the decimal fraction, then put the number 0 before the decimal point.
Any number of zeros can be added to the fractional part of the decimal fraction on the right, this does not change the value of the fraction. The fractional part of the decimal fraction is read by the last significant digit.
For example:
0.3 - three tenths
0.75 - seventy-five hundredths
0.000005 - five millionths.
Reading the integer part of a decimal is the same as reading natural numbers.
For example:
27.5 - twenty-seven ...;
1.57 - one...
After the integer part of the decimal fraction, the word "whole" is pronounced.
For example:
10.7 - ten point seven
0.67 - zero point sixty-seven hundredths.
Decimals are fractional digits. The fractional part is read not by digits (unlike natural numbers), but as a whole, therefore the fractional part of a decimal fraction is determined by the last significant digit to the right. The bit system of the fractional part of a decimal fraction is somewhat different than that of natural numbers.
- 1st digit after busy - tenths digit
- 2nd place after the decimal point - hundredth place
- 3rd place after the decimal point - thousandth place
- 4th place after the decimal point - ten-thousandth place
- 5th place after the decimal point - hundred-thousandth place
- 6th place after the decimal point - millionth place
- 7th place after the decimal point - ten-millionth place
- The 8th place after the decimal point is the hundred-millionth place
In calculations, the first three digits are most often used. The large bit depth of the fractional part of decimal fractions is used only in specific branches of knowledge, where infinitesimal values are calculated.
Decimal to mixed fraction conversion consists of the following: write the number before the decimal point as the 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 one with as many zeros as there are digits after the decimal point.
3.4 Correct order
In the previous section, we compared numbers by their position on the number line. This is a good way to compare magnitudes of numbers in decimal notation. This method always works, but it is laborious and inconvenient to do it every time you need to compare two numbers. There is another good way to figure out which of two numbers is greater.
Example A
Consider the numbers from the previous section and compare 0.05 and 0.2.
To find out which number is greater, we first compare their integer parts. Both numbers in our example have an equal number of integers - 0. Then compare their tenths. The number 0.05 has 0 tenths and the number 0.2 has 2 tenths. That the number 0.05 has 5 hundredths doesn't matter, because the tenths determine that the number 0.2 is greater. We can thus write:
Both numbers have 0 integers and 6 tenths, and we can't yet determine which one is greater. However, the number 0.612 has only 1 hundredth part, and the number 0.62 has two. Then, we can determine that
0,62 > 0,612
The fact that the number 0.612 has 2 thousandths does not matter, it is still less than 0.62.
We can illustrate this with a picture:
|
0,612 | |||
 |
0,62 | |||
In order to determine which of the two numbers in decimal notation is greater, you need to do the following:
1. Compare whole parts. The number whose integer part is greater and will be greater.
2 . If the integer parts are equal, compare tenths. That number, which has more tenths, will be more.
3 . If tenths are equal, compare hundredths. That number, which has more hundredths, will be more.
4 . If hundredths are equal, compare thousandths. That number, which has more thousandths, will be more.
In this article, we will cover the topic decimal comparison". First, let's discuss the general principle of comparing decimal fractions. After that, we will figure out which decimal fractions are equal and which are unequal. Next, we will learn how to determine which decimal fraction is greater and which is less. To do this, we will study the rules for comparing finite, infinite periodic and infinite non-periodic fractions. We will supply the whole theory with examples with detailed solutions. In conclusion, let us dwell on the comparison of decimal fractions with natural numbers, ordinary fractions and mixed numbers.
Let's say right away that here we will only talk about comparing positive decimal fractions (see positive and negative numbers). The remaining cases are analyzed in the articles comparing rational numbers and comparison of real numbers.
Page navigation.
General principle for comparing decimal fractions
Based on this principle of comparison, the rules for comparing decimal fractions are derived, which make it possible to do without converting the compared decimal fractions into ordinary fractions. These rules, as well as examples of their application, we will analyze in the following paragraphs.
By a similar principle, finite decimal fractions or infinite periodic decimal fractions are compared with natural numbers, ordinary fractions and mixed numbers: the compared numbers are replaced by their corresponding ordinary fractions, after which ordinary fractions are compared.
Concerning comparisons of infinite non-recurring decimals, then it usually comes down to comparing final decimal fractions. To do this, consider such a number of signs of compared infinite non-periodic decimal fractions, which allows you to get the result of the comparison.
Equal and unequal decimals
First we introduce definitions of equal and unequal final decimals.
Definition.
The two trailing decimals are called equal if their corresponding common fractions are equal, otherwise these decimal fractions are called unequal.
Based on this definition, it is easy to justify the following statement: if at the end of a given decimal fraction we attribute or discard several digits 0, then we get a decimal fraction equal to it. For example, 0.3=0.30=0.300=… and 140.000=140.00=140.0=140 .
Indeed, adding or discarding zero at the end of a decimal fraction on the right corresponds to multiplying or dividing by 10 the numerator and denominator of the corresponding ordinary fraction. And we know the basic property of a fraction, which says that multiplying or dividing the numerator and denominator of a fraction by the same natural number gives a fraction equal to the original one. This proves that adding or discarding zeros to the right in the fractional part of a decimal fraction gives a fraction equal to the original one.
For example, a decimal fraction 0.5 corresponds to an ordinary fraction 5/10, after adding zero to the right, a decimal fraction 0.50 is obtained, which corresponds to an ordinary fraction 50/100, and. So 0.5=0.50 . Conversely, if in decimal fraction 0.50 discard 0 on the right, then we get a fraction 0.5, so from an ordinary fraction 50/100 we will come to a fraction 5/10, but 
. Therefore, 0.50=0.5 .
Let's move on to definition of equal and unequal infinite periodic decimal fractions.
Definition.
Two infinite periodic fractions equal, if the ordinary fractions corresponding to them are equal; if the ordinary fractions corresponding to them are not equal, then the compared periodic fractions are also not equal.
Three conclusions follow from this definition:
- If the records of periodic decimal fractions are exactly the same, then such infinite periodic decimal fractions are equal. For example, the periodic decimals 0.34(2987) and 0.34(2987) are equal.
- If the periods of the compared decimal periodic fractions start from the same position, the first fraction has a period of 0 , the second has a period of 9 , and the value of the digit preceding period 0 is one more than the value of the digit preceding period 9 , then such infinite periodic decimal fractions are equal. For example, the periodic fractions 8.3(0) and 8.2(9) are equal, and the fractions 141,(0) and 140,(9) are also equal.
- Any two other periodic fractions are not equal. Here are examples of unequal infinite periodic decimal fractions: 9.0(4) and 7,(21) , 0,(12) and 0,(121) , 10,(0) and 9.8(9) .
It remains to deal with equal and unequal infinite non-periodic decimal fractions. As you know, such decimal fractions cannot be converted into ordinary fractions (such decimal fractions represent irrational numbers), so the comparison of infinite non-periodic decimal fractions cannot be reduced to a comparison of ordinary fractions.
Definition.
Two infinite non-recurring decimals equal if their entries match exactly.
But there is one nuance: it is impossible to see the “finished” record of infinite non-periodic decimal fractions, therefore, it is impossible to be sure of the complete coincidence of their records. How to be?
When comparing infinite non-periodic decimal fractions, only a finite number of signs of the compared fractions are considered, which allows us to draw the necessary conclusions. Thus, the comparison of infinite non-periodic decimal fractions is reduced to the comparison of finite decimal fractions.
With this approach, we can talk about the equality of infinite non-periodic decimal fractions only up to the considered digit. Let's give examples. Infinite non-periodic decimal fractions 5.45839 ... and 5.45839 ... are equal to within hundred thousandths, since the final decimal fractions 5.45839 and 5.45839 are equal; non-recurring decimal fractions 19.54 ... and 19.54810375 ... are equal to the nearest hundredth, since the fractions 19.54 and 19.54 are equal.
The inequality of infinite non-periodic decimal fractions with this approach is established quite definitely. For example, the infinite non-periodic decimal fractions 5.6789… and 5.67732… are not equal, since the differences in their records are obvious (the final decimal fractions 5.6789 and 5.6773 are not equal). The infinite decimals 6.49354... and 7.53789... are also not equal.
Rules for comparing decimal fractions, examples, solutions
After establishing the fact that two decimal fractions are not equal, it is often necessary to find out which of these fractions is greater and which is less than the other. Now we will analyze the rules for comparing decimal fractions, allowing us to answer the question posed.
In many cases, it is sufficient to compare the integer parts of the compared decimals. The following is true decimal comparison rule: greater than the decimal fraction, the integer part of which is greater, and less than the decimal fraction, the integer part of which is less.
This rule applies to both finite decimals and infinite decimals. Let's consider examples.
Example.
Compare decimals 9.43 and 7.983023….
Decision.
Obviously, these decimal fractions are not equal. The integer part of the final decimal fraction 9.43 is equal to 9, and the integer part of the infinite non-periodic fraction 7.983023 ... is equal to 7. Since 9>7 (see comparison of natural numbers), then 9.43>7.983023.
Answer:
9,43>7,983023 .
Example.
Which of the decimals 49.43(14) and 1,045.45029... is less?
Decision.
The integer part of the periodic fraction 49.43(14) is less than the integer part of the infinite non-periodic decimal fraction 1 045.45029…, therefore, 49.43(14)<1 045,45029… .
Answer:
49,43(14) .
If the integer parts of the compared decimal fractions are equal, then to find out which of them is greater and which is less, one has to compare the fractional parts. Comparison of fractional parts of decimal fractions is carried out bit by bit- from the category of tenths to the younger ones.
First, let's look at an example of comparing two final decimal fractions.
Example.
Compare the end decimals 0.87 and 0.8521 .
Decision.
The integer parts of these decimal fractions are equal (0=0 ), so let's move on to comparing the fractional parts. The values of the tenths place are equal (8=8 ), and the value of the hundredths place of the fraction 0.87 is greater than the value of the hundredths place of the fraction 0.8521 (7>5 ). Therefore, 0.87>0.8521 .
Answer:
0,87>0,8521 .
Sometimes, in order to compare trailing decimals with different numbers of decimals, you have to append a number of zeros to the right of the fraction with fewer decimals. It is quite convenient to equalize the number of decimal places before starting to compare the final decimal fractions by adding a certain number of zeros to the right of one of them.
Example.
Compare the trailing decimals 18.00405 and 18.0040532.
Decision.
Obviously, these fractions are unequal, since their records are different, but at the same time they have equal integer parts (18=18).
Before bitwise comparison of the fractional parts of these fractions, we equalize the number of decimal places. To do this, we assign two digits 0 at the end of the fraction 18.00405, while we get the decimal fraction equal to it 18.0040500.
The decimal places of 18.0040500 and 18.0040532 are equal up to hundred thousandths, and the value of the millionth place of 18.0040500 is less than the value of the corresponding fraction place of 18.0040532 (0<3 ), поэтому, 18,0040500<18,0040532 , следовательно, 18,00405<18,0040532 .
Answer:
18,00405<18,0040532 .
When comparing a finite decimal fraction with an infinite one, the final fraction is replaced by an infinite periodic fraction equal to it with a period of 0, after which a comparison is made by digits.
Example.
Compare the ending decimal 5.27 with the infinite non-recurring decimal 5.270013….
Decision.
The integer parts of these decimals are equal. The values of the digits of the tenths and hundredths of these fractions are equal, and in order to perform further comparison, we replace the final decimal fraction with an infinite periodic fraction equal to it with a period of 0 of the form 5.270000 ... . Before the fifth decimal place, the values of the decimal places 5.270000... and 5.270013... are equal, and on the fifth decimal place we have 0<1 . Таким образом, 5,270000…<5,270013… , откуда следует, что 5,27<5,270013… .
Answer:
5,27<5,270013… .
Comparison of infinite decimal fractions is also carried out bit by bit, and ends as soon as the values of some bit are different.
Example.
Compare the infinite decimals 6.23(18) and 6.25181815….
Decision.
The integer parts of these fractions are equal, the values of the tenth place are also equal. And the value of the hundredths place of the periodic fraction 6.23(18) is less than the hundredths place of the infinite non-periodic decimal fraction 6.25181815…, therefore, 6.23(18)<6,25181815… .
Answer:
6,23(18)<6,25181815… .
Example.
Which of the infinite periodic decimals 3,(73) and 3,(737) is greater?
Decision.
It is clear that 3,(73)=3.73737373… and 3,(737)=3.737737737… . At the fourth decimal place, the bitwise comparison ends, since there we have 3<7 . Таким образом, 3,73737373…<3,737737737… , то есть, десятичная дробь 3,(737) больше, чем дробь 3,(73) .
Answer:
3,(737) .
Compare decimals with natural numbers, common fractions and mixed numbers.
To get the result of comparing a decimal fraction with a natural number, you can compare the integer part of this fraction with a given natural number. In this case, periodic fractions with periods of 0 or 9 must first be replaced with their equal final decimal fractions.
The following is true rule for comparing decimal fraction and natural number: if the integer part of a decimal fraction is less than a given natural number, then the whole fraction is less than this natural number; if the integer part of a fraction is greater than or equal to a given natural number, then the fraction is greater than the given natural number.
Consider examples of the application of this comparison rule.
Example.
Compare natural number 7 with decimal fraction 8.8329….
Decision.
Since the given natural number is less than the integer part of the given decimal fraction, then this number is less than the given decimal fraction.
Answer:
7<8,8329… .
Example.
Compare the natural number 7 and the decimal 7.1.
