**Decimal fraction** — a fraction, the denominator of which is 10^{n}, where n is a natural number.

Recorded **decimal** is read according to the scheme

1 | 2 | 3 | 4 | , | 5 | 6 | 7 | 8 |

thousands | hundreds | dozens | unit | tenth | hundredths | thousands | ten |

For example *the decimal fraction* *10,436 read *"ten four hundred thirty-six thousandths".

Among the fractions most frequently used in daily life are fractions with denominators 10, 100, 1000, etc.

For example, 10 g kg,

1mm cm

2cm 5mm cm, etc.

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

First write the integer part, then the numerator of the fractional part. The integer part is separated from fractional part by a comma.

For example, instead of writing (read: "2 whole and 5 tenths").

Any number that the denominator of the fractional part of which is expressed by a unit with one or more zeros can be represented in the form of **a decimal**.

If roll is correct, before the comma, write the number 0.

For example, instead of writing (read: "as much as 0 and 33 thousand").

Pay attention! *After the decimal point the numerator of the fractional parts must have the same number of digits as zeroes in the denominator.*

### Table of digits after the decimal point

Decimal, and any number that consists of digits (0,1,2,3,4,5,6,7,8,9).

Place each digit in the number is important: it defines the bit of the number.

**The decimal fraction** consists of *integer part* (all the numbers before the decimal point) and *fractional parts* (all the figures after the decimal point).

*The integer part of* a decimal can be divided into discharges as well as natural numbers: units, tens, hundreds, thousands, etc.

*The fractional part is* a decimal, divide on categories: ten (denominator fractions 10), hundredths (tenths (in the denominator of the fractions 100) thousandths (tenths (in the denominator of the fractions 1000), etc.

*The table digits can be supplemented by any desired number of columns.*

- 1-th digit after the decimal point — decimal,
- 2-th digit after the decimal point — discharge hundredths,
- 3-th digit after the decimal point — discharge thousandths,
- 4-th digit after the comma — category of ten,
- 5-th digit after the decimal point — the discharge of one hundred thousandth,
- 6-th digit after the comma — category of the vast
- 7-th digit after the comma — category of the ten million,
- 8-th digit after the decimal point — discharge Stalina.

## Addition and subtraction of decimal fractions

*To add or subtract decimals, you need to: *

- To equate these fractions the number of digits after the decimal point;
- Write them under each other so that the decimal point was written under the comma;
- To perform addition (subtraction), not paying attention to the decimal point;
- To put the answer in a comma under comma in these fractions.

*Example:*

**Properties of addition for decimals:**

a + b = b + a - adjustable property

(a + b) + c = a + ( b + c ) - binding property

## Multiplying decimals

To multiply two decimals, we must:

1. perform the multiplication, ignoring the commas

2. separate with a comma as many figures on the right, how many of them after the decimal point in both factors together.

Read more here

## Dividing decimals

To divide a decimal fraction by a natural number, it is necessary:

1. to divide a fraction by that number, ignoring the decimal point;

2. to put a comma in private, when delenna a part.

Read more here

## Comparing decimals

To compare two decimals, you must first call them at the number of decimal places, attributing to one of them right zeros, and then, dropping a comma, to compare the resulting positive integers.

**Example: **

Compare two decimals 0,642 and 0.65. Even the number of decimal places assigned to the number of 0,65 case zero. Get the fraction 0,564 and 0,650.

Write them as fractions:

The denominators of the fractions are the same.

From two fractions with the same denominators, the larger, the fraction that has a greater numerator.

Since , then , therefore,

*Decimals* can be compared **by bits**.

In decimal fractions and 26,63 6,553 enough to compare the whole parts. As , then ;.

## Finite and infinite decimal fractions

**Definition:**a Finite decimal fraction is called a fraction, which contains a finite number of digits after the decimal point.

Example: 222,35

**Definition:**** **an Infinite decimal fraction is called a fraction, which does not contain a finite number of digits after the decimal point.

Example: 222,35...

**Definition:**an Infinite periodic decimal (periodic roll) is called a recurring decimal at the end contains a group of digits that repeat.

Example: 222,489898989...

Period of an infinite periodic decimal fraction is called the group of digits that repeat. In the previous example, is 89.

Periodic decimal fraction is called a pure periodic fraction, if her period starts immediately after the comma, and period can contain any finite number of digits.

Example: 8,44444....

Periodic decimal fraction is called a mixed fraction, if periodic decimal fraction contains a number that is placed between a part and the period. The number of recurring decimal that stands between a part and the period is called prepared this fraction.

Example: 8,4578787878...