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 point, 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's 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 will now get an improper fraction, which is very convenient for further calculations.
Task. Convert decimal fractions to ordinary fractions: 0.008; 3.107; 2.25; 7,2008.
Cross out the zeros on the left and the commas - we get the following numbers (these will be the numerators): 8; 3107; 225; 72008.
In the first and second fractions there are 3 decimal places, in the second - 2, and in the third - as many as 4 decimal places. We get the denominators: 1000; 1000; 100; 10000.
Finally, let's combine the numerators and denominators into ordinary fractions:
As can be seen from the examples, the resulting fraction can very often be reduced. Let me note once again that any decimal fraction can be represented as an ordinary fraction. The reverse conversion may not always be possible.
Topic: Decimal fractions. Adding and subtracting decimals
Lesson: Decimal notation of fractional numbers
The denominator of a fraction can be expressed by any natural number. Fractional numbers in which the denominator is expressed as 10; 100; 1000;…, where n, we agreed to write it without a denominator. Any fractional number whose denominator is 10; 100; 1000, etc. (that is, a one followed by several zeros) can be represented in decimal notation (as a decimal). First write the whole part, then the numerator of the fractional part, and the whole part is separated from the fraction by a comma.
For example,
If an entire part is missing, i.e. If the fraction is proper, then the whole part is written as 0.
To write a decimal correctly, the numerator of the fraction must have as many digits as there are zeros in the fraction.
1. Write as a decimal.
2. Represent a decimal as a fraction or mixed number.
3. Read the decimals.
12.4 - 12 point 4;
0.3 - 0 point 3;
1.14 - 1 point 14 hundredths;
2.07 - 2 point 7 hundredths;
0.06 - 0 point 6 hundredths;
0.25 - 0 point 25;
1.234 - 1 point 234 thousandths;
1.230 - 1 point 230 thousandths;
1.034 - 1 point 34 thousandths;
1.004 - 1 point 4 thousandths;
1.030 - 1 point 30 thousandths;
0.010101 - 0 point 10101 millionths.
4. Move the comma in each digit 1 place to the left and read the numbers.
34,1; 310,2; 11,01; 10,507; 2,7; 3,41; 31,02; 1,101; 1,0507; 0,27.
5. Move the comma in each number 1 place to the right and read the resulting number.
1,37; 0,1401; 3,017; 1,7; 350,4; 13,7; 1,401; 30,17; 17; 3504.
6. Express in meters and centimeters.
3.28 m = 3 m + .
7. Express in tons and kilograms.
24.030 t = 24 t.
8. Write the quotient as a decimal fraction.
1710: 100 = 
;
64: 10000 = 
803: 100 = 
407: 10 = 
9. Express in dm.
5 dm 6 cm = 5 dm + 
;
9 mm = 
We will devote this material to such an important topic as decimal fractions. First, let's define the basic definitions, give examples and dwell on the rules of decimal notation, as well as what the digits of decimal fractions are. Next, we highlight the main types: finite and infinite, periodic and non-periodic fractions. In the final part we will show how the points corresponding to fractional numbers are located on the coordinate axis.
What is decimal notation of fractional numbers
The so-called decimal notation of fractional numbers can be used for both natural and fractional numbers. It looks like a set of two or more numbers with a comma between them.
The decimal point is needed to separate the whole part from the fractional part. As a rule, the last digit of a decimal fraction is not a zero, unless the decimal point appears immediately after the first zero.
What are some examples of fractional numbers in decimal notation? This could be 34, 21, 0, 35035044, 0, 0001, 11,231,552, 9, etc.
In some textbooks you can find the use of a period instead of a comma (5. 67, 6789. 1011, etc.). This option is considered equivalent, but it is more typical for English-language sources.
Definition of decimals
Based on the above concept of decimal notation, we can formulate the following definition of decimal fractions:
Definition 1
Decimals represent fractional numbers in decimal notation.
Why do we need to write fractions in this form? It gives us some advantages over ordinary ones, for example, a more compact notation, especially in cases where the denominator contains 1000, 100, 10, etc., or a mixed number. For example, instead of 6 10 we can specify 0.6, instead of 25 10000 - 0.0023, instead of 512 3 100 - 512.03.
How to correctly represent ordinary fractions with tens, hundreds, thousands in the denominator in decimal form will be discussed in a separate material.
How to read decimals correctly
There are some rules for reading decimal notations. Thus, those decimal fractions that correspond to their regular ordinary equivalents are read almost the same way, but with the addition of the words “zero tenths” at the beginning. Thus, the entry 0, 14, which corresponds to 14,100, is read as “zero point fourteen hundredths.”
If a decimal fraction can be associated with a mixed number, then it is read in the same way as this number. So, if we have the fraction 56, 002, which corresponds to 56 2 1000, we read this entry as “fifty-six point two thousandths.”
The meaning of a digit in a decimal fraction depends on where it is located (the same as in the case of natural numbers). So, in the decimal fraction 0.7, seven is tenths, in 0.0007 it is ten thousandths, and in the fraction 70,000.345 it means seven tens of thousands of whole units. Thus, in decimal fractions there is also the concept of place value.
The names of the digits located before the decimal point are similar to those that exist in natural numbers. The names of those located after are clearly presented in the table:
Let's look at an example.
Example 1
We have the decimal fraction 43,098. She has a four in the tens place, a three in the units place, a zero in the tenths place, 9 in the hundredths place, and 8 in the thousandths place.
It is customary to distinguish the ranks of decimal fractions by precedence. If we move through the numbers from left to right, then we will go from the most significant to the least significant. It turns out that hundreds are older than tens, and parts per million are younger than hundredths. If we take that final decimal fraction that we cited as an example above, then the highest, or highest, place in it will be the hundreds place, and the lowest, or lowest, place will be the 10-thousandth place.
Any decimal fraction can be expanded into individual digits, that is, presented as a sum. This action is performed in the same way as for natural numbers.
Example 2
Let's try to expand the fraction 56, 0455 into digits.
We will get:
56 , 0455 = 50 + 6 + 0 , 4 + 0 , 005 + 0 , 0005
If we remember the properties of addition, we can represent this fraction in other forms, for example, as the sum 56 + 0, 0455, or 56, 0055 + 0, 4, etc.
What are trailing decimals?
All the fractions we talked about above are finite decimals. This means that the number of digits after the decimal point is finite. Let's derive the definition:
Definition 1
Trailing decimals are a type of decimal fraction that has a finite number of decimal places after the decimal sign.
Examples of such fractions can be 0, 367, 3, 7, 55, 102567958, 231 032, 49, etc.
Any of these fractions can be converted either to a mixed number (if the value of their fractional part is different from zero) or to an ordinary fraction (if the integer part is zero). We have devoted a separate article to how this is done. Here we’ll just point out a couple of examples: for example, we can reduce the final decimal fraction 5, 63 to the form 5 63 100, and 0, 2 corresponds to 2 10 (or any other fraction equal to it, for example, 4 20 or 1 5.)
But the reverse process, i.e. writing a common fraction in decimal form may not always be possible. So, 5 13 cannot be replaced by an equal fraction with the denominator 100, 10, etc., which means that a final decimal fraction cannot be obtained from it.
Main types of infinite decimal fractions: periodic and non-periodic fractions
We indicated above that finite fractions are so called because they have a finite number of digits after the decimal point. However, it may well be infinite, in which case the fractions themselves will also be called infinite.
Definition 2
Infinite decimal fractions are those that have an infinite number of digits after the decimal point.
Obviously, such numbers simply cannot be written down in full, so we indicate only part of them and then add an ellipsis. This sign indicates an infinite continuation of the sequence of decimal places. Examples of infinite decimal fractions include 0, 143346732…, 3, 1415989032…, 153, 0245005…, 2, 66666666666…, 69, 748768152…. etc.
The “tail” of such a fraction may contain not only seemingly random sequences of numbers, but also a constant repetition of the same character or group of characters. Fractions with alternating numbers after the decimal point are called periodic.
Definition 3
Periodic decimal fractions are those infinite decimal fractions in which one digit or a group of several digits is repeated after the decimal point. The repeating part is called the period of the fraction.
For example, for the fraction 3, 444444…. the period will be the number 4, and for 76, 134134134134... - the group 134.
What is the minimum number of characters that can be left in the notation of a periodic fraction? For periodic fractions, it will be enough to write the entire period once in parentheses. So, fraction 3, 444444…. It would be correct to write it as 3, (4), and 76, 134134134134... – as 76, (134).
In general, entries with several periods in brackets will have exactly the same meaning: for example, the periodic fraction 0.677777 is the same as 0.6 (7) and 0.6 (77), etc. Records of the form 0, 67777 (7), 0, 67 (7777), etc. are also acceptable.
To avoid mistakes, we introduce uniformity of notation. Let's agree to write down only one period (the shortest possible sequence of numbers), which is closest to the decimal point, and enclose it in parentheses.
That is, for the above fraction, we will consider the main entry to be 0, 6 (7), and, for example, in the case of the fraction 8, 9134343434, we will write 8, 91 (34).
If the denominator of an ordinary fraction contains prime factors that are not equal to 5 and 2, then when converted to decimal notation, they will result in infinite fractions.
In principle, we can write any finite fraction as a periodic one. To do this, we just need to add an infinite number of zeros to the right. What does it look like in recording? Let's say we have the final fraction 45, 32. In periodic form it will look like 45, 32 (0). This action is possible because adding zeros to the right of any decimal fraction results in a fraction equal to it.
Special attention should be paid to periodic fractions with a period of 9, for example, 4, 89 (9), 31, 6 (9). They are an alternative notation for similar fractions with a period of 0, so they are often replaced when writing with fractions with a zero period. In this case, one is added to the value of the next digit, and (0) is indicated in parentheses. The equality of the resulting numbers can be easily verified by representing them as ordinary fractions.
For example, the fraction 8, 31 (9) can be replaced with the corresponding fraction 8, 32 (0). Or 4, (9) = 5, (0) = 5.
Infinite decimal periodic fractions are classified as rational numbers. In other words, any periodic fraction can be represented as an ordinary fraction, and vice versa.
There are also fractions that do not have an endlessly repeating sequence after the decimal point. In this case, they are called non-periodic fractions.
Definition 4
Non-periodic decimal fractions include those infinite decimal fractions that do not contain a period after the decimal point, i.e. repeating group of numbers.
Sometimes non-periodic fractions look very similar to periodic ones. For example, 9, 03003000300003 ... at first glance seems to have a period, but a detailed analysis of the decimal places confirms that this is still a non-periodic fraction. You need to be very careful with such numbers.
Non-periodic fractions are classified as irrational numbers. They are not converted to ordinary fractions.
Basic operations with decimals
The following operations can be performed with decimal fractions: comparison, subtraction, addition, division and multiplication. Let's look at each of them separately.
Comparing decimals can be reduced to comparing fractions that correspond to the original decimals. But infinite non-periodic fractions cannot be reduced to this form, and converting decimal fractions into ordinary fractions is often a labor-intensive task. How can we quickly perform a comparison action if we need to do this while solving a problem? It is convenient to compare decimal fractions by digit in the same way as we compare natural numbers. We will devote a separate article to this method.
To add some decimal fractions with others, it is convenient to use the column addition method, as for natural numbers. To add periodic decimal fractions, you must first replace them with ordinary ones and count according to the standard scheme. If, according to the conditions of the problem, we need to add infinite non-periodic fractions, then we need to first round them to a certain digit, and then add them. The smaller the digit to which we round, the higher the accuracy of the calculation will be. For subtraction, multiplication and division of infinite fractions, pre-rounding is also necessary.
Finding the difference between decimal fractions is the inverse of addition. Essentially, using subtraction we can find a number whose sum with the fraction we are subtracting will give us the fraction we are minimizing. We will talk about this in more detail in a separate article.
Multiplying decimal fractions is done in the same way as for natural numbers. The column calculation method is also suitable for this. We again reduce this action with periodic fractions to the multiplication of ordinary fractions according to the rules already studied. Infinite fractions, as we remember, must be rounded before calculations.
The process of dividing decimals is the inverse of multiplying. When solving problems, we also use columnar calculations.
You can establish an exact correspondence between the final decimal fraction and a point on the coordinate axis. Let's figure out how to mark a point on the axis that will exactly correspond to the required decimal fraction.
We have already studied how to construct points corresponding to ordinary fractions, but decimal fractions can be reduced to this form. For example, the common fraction 14 10 is the same as 1, 4, so the corresponding point will be removed from the origin in the positive direction by exactly the same distance:
You can do without replacing the decimal fraction with an ordinary one, but use the method of expansion by digits as a basis. So, if we need to mark a point whose coordinate will be equal to 15, 4008, then we will first present this number as the sum 15 + 0, 4 +, 0008. To begin with, let’s set aside 15 whole unit segments in the positive direction from the beginning of the countdown, then 4 tenths of one segment, and then 8 ten-thousandths of one segment. As a result, we get a coordinate point that corresponds to the fraction 15, 4008.
For an infinite decimal fraction, it is better to use this method, since it allows you to get as close as you like to the desired point. In some cases, it is possible to construct an exact correspondence to an infinite fraction on the coordinate axis: for example, 2 = 1, 41421. . . , and this fraction can be associated with a point on the coordinate ray, distant from 0 by the length of the diagonal of the square, the side of which will be equal to one unit segment.
If we find not a point on the axis, but a decimal fraction corresponding to it, then this action is called the decimal measurement of a segment. Let's see how to do this correctly.
Let's say we need to get from zero to a given point on the coordinate axis (or get as close as possible in the case of an infinite fraction). To do this, we gradually postpone unit segments from the origin until we get to the desired point. After whole segments, if necessary, we measure tenths, hundredths and smaller fractions so that the match is as accurate as possible. As a result, we received a decimal fraction that corresponds to a given point on the coordinate axis.
Above we showed a drawing with point M. Look at it again: to get to this point, you need to measure one unit segment and four tenths of it from zero, since this point corresponds to the decimal fraction 1, 4.
If we cannot get to a point in the process of decimal measurement, then it means that it corresponds to an infinite decimal fraction.
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A common fraction (or mixed number) in which the denominator is one followed by one or more zeros (i.e. 10, 100, 1000, etc.):
can be written in a simpler form: without a denominator, separating the integer and fractional parts from each other with a comma (in this case, it is considered that the integer part of a proper fraction is equal to 0). First, the whole part is written, then a comma is placed, and after it the fractional part is written:
Common fractions (or mixed numbers) written in this form are called decimals.
Reading and writing decimals
Decimal fractions are written according to the same rules that are used to write natural numbers in the decimal number system. This means that in decimals, as in natural numbers, each digit expresses units that are ten times larger than the neighboring units to the right.
Consider the following entry:
The number 8 stands for prime units. The number 3 means units that are 10 times smaller than simple units, i.e. tenths. 4 means hundredths, 2 means thousandths, etc.
The numbers that appear to the right after the decimal point are called decimals.
Decimal fractions are read as follows: first the whole part is called, then the fractional part. When reading a whole part, it should always answer the question: how many whole units are there in the whole part? . The word whole (or integer) is added to the answer, depending on the number of whole units. For example, one integer, two integers, three integers, etc. When reading the fractional part, the number of shares is called and at the end they add the name of those shares with which the fractional part ends:
3.1 reads like this: three point one tenth.
2.017 reads like this: two point seventeen thousandths.
To better understand the rules for writing and reading decimal fractions, consider the table of digits and the examples of writing numbers given in it:
Please note that after the decimal point, there are as many digits after the decimal point as there are zeros in the denominator of the corresponding ordinary fraction:
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.