Let's look at examples of how to round up to tenths of a number using the rounding rules.

Rule for rounding numbers to tenths.

To round a decimal to tenths, you must leave only one digit after the decimal point, and discard all other digits following it.

If the first of the discarded digits is 0, 1, 2, 3 or 4, then the previous digit is not changed.

If the first of the discarded digits is 5, 6, 7, 8 or 9, then the previous digit is increased by one.

Examples.

Round to tenths:

To round a number to tenths, leave the first digit after the decimal point, and discard the rest. Since the first discarded digit is 5, we increase the previous digit by one. They read: "Twenty-three point seventy-five hundredths is approximately equal to twenty-three point eight."

To round this number to tenths, leave only the first digit after the decimal point, discard the rest. The first discarded digit is 1, so the previous digit is not changed. They read: "Three hundred and forty-eight point thirty-one hundredth is approximately equal to three hundred and forty-one point three."

Rounding to tenths, we leave one digit after the decimal point, and discard the rest. The first of the discarded digits is 6, which means that we increase the previous one by one. They read: "Forty-nine point, nine hundred and sixty-two thousandths is approximately equal to fifty point, zero tenths."

We round up to tenths, so after the comma we leave only the first of the digits, the rest are discarded. The first of the discarded digits is 4, which means we leave the previous digit unchanged. They read: "Seven point twenty-eight thousandths is approximately equal to seven point zero tenths."

To round to tenths, this number leaves one digit after the decimal point, and discard all following after it. Since the first discarded digit is 7, therefore, we add one to the previous one. They read: "Fifty-six point eight thousand seven hundred and six ten-thousandths is approximately equal to fifty-six point nine-tenths."

And a couple more examples for rounding to tenths:

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 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.

We have already said that fractions are ordinary and decimal. At the moment, we have studied ordinary fractions a little. We learned that there are regular fractions and improper fractions. We also learned that ordinary fractions can be reduced, added, subtracted, multiplied and divided. And we also learned that there are so-called mixed numbers, which consist of an integer part and a fractional part.

We have not yet fully studied ordinary fractions. There are many subtleties and details that should be discussed, but today we will begin to study decimal fractions, since ordinary and decimal fractions quite often have to be combined. That is, when solving problems, you have to work with both types of fractions.

This lesson may seem complicated and incomprehensible. It's quite normal. These kinds of lessons require that they be studied and not skimmed over.

Lesson content

Expressing quantities in fractional form

Sometimes it is convenient to show something in fractional form. For example, one tenth of a decimeter is written like this:

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

Consider the following example. Show 6 cm and another 3 mm in centimeters in fractional form.

So, you want to show 6 cm and 3 mm in centimeters, but in fractional form. We already have 6 whole centimeters:

But there are still 3 millimeters left. How to show these 3 millimeters, while in centimeters? Fractions come to the rescue. One centimeter is ten millimeters. Three millimeters is three parts out of ten. And three parts out of ten are written as cm

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

As a result, we have six whole centimeters and three tenths of a centimeter:

In this case, 6 shows the number of whole centimeters, and the fraction shows the number of fractional. This fraction is read as "six point and three tenths of a centimeter".

Fractions, in the denominator of which there are numbers 10, 100, 1000, can be written without a denominator. First write the integer part, and then the numerator of the fractional part. The integer part is separated from the numerator of the fractional part by a comma.

For example, let's write without a denominator. First write down the whole part. The whole part is 6

The whole part is recorded. Immediately after writing the whole part, put a comma:

And now we write down the numerator of the fractional part. In a mixed number, the numerator of the fractional part is the number 3. We write the three after the decimal point:

Any number that is represented in this form is called decimal.

Therefore, you can show 6 cm and another 3 mm in centimeters using a decimal fraction:

6.3 cm

It will look like this:

In fact, decimals are the same common fractions and mixed numbers. The peculiarity of such fractions is that the denominator of their fractional part contains the numbers 10, 100, 1000 or 10000.

Like a mixed number, a decimal has an integer part and a fractional part. For example, in a mixed number, the integer part is 6 and the fractional part is .

In the decimal fraction 6.3, the integer part is the number 6, and the fractional part is the numerator of the fraction, that is, the number 3.

It also happens that ordinary fractions in the denominator of which the numbers 10, 100, 1000 are given without an integer part. For example, a fraction is given without an integer part. To write such a fraction as a decimal, first write down 0, then put a comma and write down the numerator of the fractional part. A fraction without a denominator would be written like this:

Reads like "zero point five tenths".

Convert mixed numbers to decimals

When we write mixed numbers without a denominator, we are converting them to decimals. When converting ordinary fractions to decimal fractions, there are a few things you need to know, which we'll talk about now.

After the integer part is written, it is imperative to count the number of zeros in the denominator of the fractional part, since the number of zeros in the fractional part and the number of digits after the decimal point in the decimal fraction must be the same. What does it mean? Consider the following example:

At first

And you could immediately write down the numerator of the fractional part and the decimal fraction is ready, but you must definitely count the number of zeros in the denominator of the fractional part.

So, we count the number of zeros in the fractional part of the mixed number. The denominator of the fractional part has one zero. So in the decimal fraction after the decimal point there will be one digit and this figure will be the numerator of the fractional part of the mixed number, that is, the number 2

Thus, the mixed number, when translated into a decimal fraction, becomes 3.2.

This decimal is read like this:

"Three whole two tenths"

"Tenths" because the fractional part of the mixed number contains the number 10.

Example 2 Convert mixed number to decimal.

We write down the whole part and put a comma:

And you could immediately write down the numerator of the fractional part and get the decimal fraction 5.3, but the rule says that after the decimal point there should be as many digits as there are zeros in the denominator of the fractional part of the mixed number. And we see that there are two zeros in the denominator of the fractional part. So in our decimal fraction after the decimal point there should be two digits, not one.

In such cases, the numerator of the fractional part needs to be slightly modified: add a zero before the numerator, that is, before the number 3

Now you can convert this mixed number to a decimal. We write down the whole part and put a comma:

And write the numerator of the fractional part:

The decimal fraction 5.03 reads like this:

"Five point three hundredths"

"Hundredths" because the denominator of the fractional part of the mixed number is the number 100.

Example 3 Convert mixed number to decimal.

From the previous examples, we learned that in order to successfully convert a mixed number to a decimal, the number of digits in the numerator of the fractional part and the number of zeros in the denominator of the fractional part must be the same.

Before converting a mixed number into a decimal fraction, its fractional part needs to be slightly modified, namely, to make sure that the number of digits in the numerator of the fractional part and the number of zeros in the denominator of the fractional part are the same.

First of all, we look at the number of zeros in the denominator of the fractional part. We see that there are three zeros:

Our task is to organize three digits in the numerator of the fractional part. We already have one digit - this is the number 2. It remains to add two more digits. They will be two zeros. Add them before the number 2. As a result, the number of zeros in the denominator and the number of digits in the numerator will become the same:

Now we can turn this mixed number into a decimal. We write down the whole part first and put a comma:

and immediately write down the numerator of the fractional part

3,002

We see that the number of digits after the decimal point and the number of zeros in the denominator of the fractional part of the mixed number are the same.

The decimal 3.002 reads like this:

"Three whole, two thousandths"

"Thousandths" because the denominator of the fractional part of the mixed number is the number 1000.

Converting common fractions to decimals

Ordinary fractions, in which the denominator is 10, 100, 1000 or 10000, can also be converted to decimal fractions. Since an ordinary fraction does not have an integer part, first write down 0, then put a comma and write down the numerator of the fractional part.

Here, too, the number of zeros in the denominator and the number of digits in the numerator must be the same. Therefore, you should be careful.

Example 1

The integer part is missing, so first we write 0 and put a comma:

Now look at the number of zeros in the denominator. We see that there is one zero. And the numerator has one digit. So you can safely continue the decimal fraction by writing the number 5 after the decimal point

In the resulting decimal fraction 0.5, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is correct.

The decimal fraction 0.5 reads like this:

"Zero point, five tenths"

Example 2 Translate common fraction into a decimal.

The whole part is missing. We write 0 first and put a comma:

Now look at the number of zeros in the denominator. We see that there are two zeros. And the numerator has only one digit. To make the number of digits and the number of zeros the same, add one zero in the numerator before the number 2. Then the fraction will take the form . Now the number of zeros in the denominator and the number of digits in the numerator are the same. So you can continue the decimal:

In the resulting decimal fraction 0.02, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is correct.

The decimal fraction 0.02 reads like this:

"Zero point, two hundredths."

Example 3 Convert common fraction to decimal.

We write 0 and put a comma:

Now we count the number of zeros in the denominator of the fraction. We see that there are five zeros, and there is only one digit in the numerator. To make the number of zeros in the denominator and the number of digits in the numerator the same, you need to add four zeros in the numerator before the number 5:

Now the number of zeros in the denominator and the number of digits in the numerator are the same. So you can continue the decimal. We write down the numerator of the fraction after the decimal point

In the resulting decimal fraction 0.00005, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is correct.

The decimal fraction 0.00005 reads like this:

"Zero point, five hundred-thousandths."

Convert improper fractions to decimals

An improper fraction is a fraction whose numerator is greater than the denominator. There are improper fractions that have the numbers 10, 100, 1000 or 10000 in the denominator. Such fractions can be converted to decimal fractions. But before converting to a decimal fraction, such fractions must have an integer part.

Example 1

The fraction is an improper fraction. To convert such a fraction to a decimal fraction, you must first select its integer part. We recall how to select the whole part of improper fractions. If you forgot, we advise you to return to and study it.

So, let's select the integer part in the improper fraction. Recall that a fraction means division - in this case, dividing the number 112 by the number 10

Let's look at this picture and assemble a new mixed number, like a children's construction set. The number 11 will be the integer part, the number 2 will be the numerator of the fractional part, the number 10 will be the denominator of the fractional part.

We got a mixed number. Let's convert it to a decimal. And we already know how to translate such numbers into decimal fractions. First we write down the whole part and put a comma:

Now we count the number of zeros in the denominator of the fractional part. We see that there is one zero. And the numerator of the fractional part has one digit. This means that the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part are the same. This gives us the opportunity to immediately write the numerator of the fractional part after the decimal point:

In the resulting decimal fraction 11.2, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is correct.

Means improper fraction when converted to decimal, it becomes 11.2

Decimal 11.2 reads like this:

"Eleven whole, two tenths."

Example 2 Convert improper fraction to decimal.

This is an improper fraction because the numerator is greater than the denominator. But it can be converted to a decimal fraction, since the denominator is the number 100.

First of all, we select the integer part of this fraction. To do this, divide 450 by 100 by a corner:

Let's collect a new mixed number - we get . And we already know how to translate mixed numbers into decimal fractions.

We write down the whole part and put a comma:

Now we count the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part. We see that the number of zeros in the denominator and the number of digits in the numerator are the same. This gives us the opportunity to immediately write the numerator of the fractional part after the decimal point:

In the resulting decimal fraction 4.50, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is translated correctly.

So the improper fraction, when translated into a decimal fraction, turns into 4.50

When solving problems, if there are zeros at the end of the decimal fraction, they can be discarded. Let's drop the zero in our answer. Then we get 4.5

This is one of interesting features decimal fractions. It lies in the fact that the zeros that are at the end of the fraction do not give this fraction any weight. In other words, the decimals 4.50 and 4.5 are equal. Let's put an equal sign between them:

4,50 = 4,5

The question arises: why is this happening? After all, 4.50 and 4.5 look like different fractions. The whole secret lies in the basic property of the fraction, which we studied earlier. We will try to prove why the decimal fractions 4.50 and 4.5 are equal, but after studying the next topic, which is called "converting a decimal fraction to a mixed number."

Decimal to mixed number conversion

Any decimal fraction can be converted back to a mixed number. To do this, it is enough to be able to read decimal fractions. For example, let's convert 6.3 to a mixed number. 6.3 is six whole points and three tenths. We write down six integers first:

and next three tenths:

Example 2 Convert decimal 3.002 to mixed number

3.002 is three integers and two thousandths. Write down three integers first.

and next we write two thousandths:

Example 3 Convert decimal 4.50 to mixed number

4.50 is four point and fifty hundredths. Write down four integers

and next fifty hundredths:

By the way, let's remember the last example from the previous topic. We said that the decimals 4.50 and 4.5 are equal. We also said that zero can be discarded. Let's try to prove that decimal 4.50 and 4.5 are equal. To do this, we convert both decimal fractions to mixed numbers.

After converting to a mixed number, the decimal 4.50 becomes , and the decimal 4.5 becomes

We have two mixed numbers and . Convert these mixed numbers to improper fractions:

Now we have two fractions and . It is time to remember the basic property of a fraction, which says that when multiplying (or dividing) the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

Let's divide the first fraction by 10

Received, and this is the second fraction. So and are equal to each other and equal to the same value:

Try dividing 450 by 100 first on a calculator, and then 45 by 10. A funny thing will work out.

Convert decimal to common fraction

Any decimal fraction can be converted back to a common fraction. To do this, again, it is enough to be able to read decimal fractions. For example, let's convert 0.3 to an ordinary fraction. 0.3 is zero and three tenths. We write zero integers first:

and next to three tenths 0 . Zero is traditionally not written down, so the final answer will not be 0, but simply.

Example 2 Convert decimal 0.02 to common fraction.

0.02 is zero and two hundredths. We don’t write down zero, so we immediately write down two hundredths

Example 3 Convert 0.00005 to fraction

0.00005 is zero and five hundred thousandths. Zero is not written down, so we immediately write down five hundred thousandths

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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 down the fractional part of the fraction in such a way that the last digit falls into the desired category (in the absence significant figures 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.

1. first, the integer parts of the fractions are compared - the decimal fraction with the larger integer part will be larger;
2. 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.