Decimals Reading and writing decimals. Decimal concept

A decimal fraction must contain a comma. The numerical part of the fraction that is located to the left of the decimal point is called the whole part; to the right - fractional:

5.28 5 - integer part 28 - fractional part

The fractional part of a decimal consists 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 thousandths);
millionths - 0.000001 (one millionth);
ten millionths - 0.0000001 (one ten millionth);
hundred millionths - 0.00000001 (one hundred millionths);
billionths - 0.000000001 (one billionth), etc.

read the number that makes up the whole 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 - one hundred point two thousand eight hundred thirty-four ten thousandths.

Writing Decimals

To write a decimal fraction:

write down the whole part of the fraction and put a comma (the number meaning the whole 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 one hundredth - 5.01 - the word “hundredth” means that there should be two digits after the decimal point, but since the number 1 does not have a tenth place, it is replaced by zero;
zero point eight hundred eight thousandths - 0.808;
three point fifteen tenths - such a decimal fraction cannot be written down, because there was an error in the pronunciation of the fractional part - the number 15 contains two digits, and the word “tenths” implies only one. Correct would be three point fifteen hundredths (or thousandths, ten thousandths, etc.).

Comparison of decimals

Comparison of decimal fractions is carried out similarly to comparison of natural numbers.

first, the whole parts of fractions are compared - the decimal fraction whose whole part is larger will be greater;
if the whole parts of fractions are equal, compare the fractional parts bit by bit, from left to right, starting from the decimal point: tenths, hundredths, thousandths, etc. The comparison is carried out until the first discrepancy - the greater will be the decimal fraction which has a larger unequal digit in the corresponding digit of the fractional part. For example: 1,2 8 3 > 1,27 9, because in the hundredths place the first fraction has 8, and the second has 7.

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 =

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.

Subject:

Target: introduce students to new numbers - decimal fractions, build knowledge and

Lesson type:

Equipment:

tasks.

View document contents
"Lesson summary on the topic "The concept of a decimal fraction. Reading and writing decimal fractions.""

Subject: The concept of a decimal fraction. Reading and writing decimals.

Target: introduce students to new numbers - decimal fractions, build knowledge and

mastery of mathematics methods; cultivate a culture of mathematical thinking.

Lesson type: lesson of learning new material.

Equipment: teacher's computer, screen, multimedia projector; on the tables: sheets with

tasks.

Lesson structure:

Organizing time.

Guys, today in class you must discover new knowledge, but, as you know, every new knowledge is related to what we have already learned. So let's start with a review.

Preparing to study new material.

Solve the anagram: fraction, angle, numerator, denominator.

Read the numbers in the table of digits.

From the numbers given, choose: natural numbers, proper fractions, improper fractions, mixed numbers.

Familiarization with new material.

	Our lesson will be dedicated to One interesting person. Listen to me carefully Answer the questions That's it, guys, take note. The topic of the lesson is “The concept of a decimal fraction. Reading and writing decimals." Lesson motto: Have excellent knowledge on the topic “Decimal Fractions.”
	Let's remember how the decimal number system works. Let's look at the table of categories and answer the questions: Questions: Read the numbers written in the table. How does the position of the unit change in each subsequent line compared to the previous one? How does the value of the corresponding number change? What arithmetic operation corresponds to this change? Conclusion : moving the unit one digit to the right, each time we decreased the corresponding number by 10 times and did this until we reached the last digit - the units digit. Is it possible to reduce one by 10 times? Certainly, Problem: But there is no place for this number in our table of digits yet. Think about how you need to change the table of digits so that you can write the number in it.
	We reason that we need to move the number 1 to the right by one place. But there are no digits to the right of the units digit, which means we need to add another column. Come up with a name for this column: tenths. Reasoning similarly: (hundredths) and: 10t. = (thousandths), etc.
	Since we reasoned correctly, we get the following table:
	2 units 3 tenths. And in order to write numbers outside the table, we need to separate the whole part from the fractional part with some sign. We agreed to do this using a comma or period. In our country, as a rule, a comma is used, and in the USA and some other countries, a period is used. We read the numbers as follows: a) 2,3 or 2.3 (two point three or two, comma, three or two, point, three)
	You and I have made a discovery. And this discovery is the rule for reading and writing decimal fractions. It coincided with the rule proposed by the author of the textbook. Rule: If a comma (or period) is used in the decimal notation of a number, then the number is said to be written as a decimal fraction. For brevity, numbers are simply called decimals. Note that the decimal fraction is not a new type of number, but a new way recording numbers.
	In science and industry, in agriculture, decimal fractions are used much more often than ordinary fractions. This is due to the simplicity of the rules for calculations with decimal fractions and their similarity to the rules for operations with natural numbers. 1703 - In Russia, the doctrine of decimal fractions was presented by Leonty Filippovich Magnitsky in the textbook “Arithmetic, that is, the science of numerals.”
	We have every reason to complete tasks on the topic of the lesson. First task. Read the number
	Read decimals What can you say about these three numbers? (they are equal) What can you conclude about the zeros that end a decimal? (you don’t have to write them, they don’t change the number) You can add zeros at the end of a decimal fraction or discard zeros, but this will not change the decimal fraction. The same fraction is written.
	A comma is placed between the whole and fractional parts. If there is no fractional digit, we replace it with 0 when writing the number. The number of digits after the decimal point must be equal to the number of zeros in the denominator of the common fraction. Write in decimal fraction:
	Write decimal fractions from dictation. 7 point 8 2 point 25 hundredths 0 whole 92 hundredths 12 point 3 hundredths 5 point 187 thousandths 24 whole 24 thousandths 7 point 7 7 point 7 hundredths 7 point 7 thousandths 0 point 5 ten thousandths Now we are doing independent work, during which we will test our knowledge on the topic of the lesson.
	Independent work (5 minutes)
	Check yourself: Write as a decimal fraction (on a line); Check the answers in the table, putting the corresponding letter for each number (under each number without punctuation) What word did you get? WELL DONE
	Reflection
	Homework: No. 647 a), 648 av), 649 a), 650 c)

Numbers

Mixed numbers

Natural

Improper fractions

Proper fractions

NAME THE NATURAL NUMBERS

NAME mixed NUMBERS

NAME common fractions

What numbers are left?

FRACTIONAL NUMBERS

DECIMAL RECORDING.

DECIMALS.

TODAY'S LESSON TOPIC:

Decimal fractions. Reading and writing decimal fractions.

THE PURPOSE OF THE LESSON:

Introduce the concept of decimal fractions. Learn to read and write decimals Learn to translate common fractions with denominators 10, 100, 1000, etc. to decimal and vice versa Develop logical thinking in a new situation Foster independence and responsibility for one’s own activities.

Fractions

Ordinary

Decimals, fractions

Decimal fractions.

RECORDING

READING

Decimal

ACTIONS

WITH DECIMALS

COMPARE

If a comma is used in the decimal notation of a number, the number is said to be written as a decimal fraction.

Numbers with a denominator 10; 100; 1000, etc. agreed to write without a denominator

MATHEMATICAL DICCTATION

WRITE OUT THE NUMBERS

THREE POINT SEVEN
SIX POINT ONE HUNDREDTH
FIVE POINT FOUR THOUSANDTHS

MATHEMATICAL DICCTATION

WRITE OUT THE NUMBERS

First write the whole part, and then the numerator of the fractional part

The integer part is separated from the fractional part by a comma

Numbers with denominators 10, 100, 1000, etc.

agreed to write without a denominator

After the decimal point, the numerator of the fractional part must have as many digits as there are zeros in the denominator

ALGORITHM

1. WRITE THE WHOLE PART OF A NUMBER

2. PUT A COMMA

3. AFTER THE decimal place put as many dots as there are zeros in the denominator

4. FROM THE LAST POINT WE WRITE THE NUMERATOR

5. REPLACE THE REMAINING POINTS WITH ZEROS

Decimal fractions consist of an integer part and a fraction

Integer digits

Fractional digits

thousandths

ten thousandths

hundred thousandths

millionths

FIVE POINT THREE

TWENTY-ONE POINT SEVEN

THREE POINT SEVEN

TWO POINT HUNDRED FIFTY-SIX THOUSANDTHS

SEVEN POINT TWENTY NINE HUNDREDTHS

SIX POINT ONE HUNDREDTH

FIVE POINT FOUR THOUSANDTHS

NINE point eight

= 9,0008

FIND AND WRITE THE MISSING NUMBERS

The origin and development of decimal fractions

Uzbekistan, XV century

Europe, 16th century

Russia, XVIII century

Ancient China, 2nd century BC.

The origin and development of decimal fractions in China was closely related to metrology (the study of measures). Already in the 2nd century BC. there was a decimal system of length measures.

IN 1427 year, mathematician

and astronomer from Uzbekistan ,

Al-Kashi wrote a book

"The Key to Arithmetic"

in which he formulated

basic

rules of action

with decimals

Uzbekistan, XV century

EUROPE,

century

IN 1579 year, decimal fractions are used in the “Canon of Mathematics” by the French mathematician François Vieta (1540-1603), published in Paris.

Wide

decimal propagation

in Europe began only after the publication of the book “The Tenth” by the Flemish mathematician Simone Stevina (1548-1620 ). He is considered the inventor of decimal fractions.

Russia, XVIII century

IN Russia first

systematic information

about decimals

found in Arithmetic

L.F. Magnitsky (1703)

2,135436

2 | 135436

Uzbekistan

France

Russia

Europe

1 cun,

3 beats,

5 serial,

4 hairs,

3 thinnest,

6 cobwebs

2,135436

China

2 135436

2 0 1 1 3 2 5 3 4 4 3 5 6 6

Are you probably tired?

Well, then everyone stood up together.

We stretch our arms, shoulders,

To make it easier for us to sit.

And don’t get tired at all.

check

Write the following fractions as decimals:

Write the following fractions as fractions or mixed numbers:

Summarize:

What fraction can be used to replace an ordinary fraction, the denominator of the fractional part of which is expressed unit with one or several zeros?