**Real numbers **

**Andhra Pradesh**

**class-9 Introduction to Real numbers (9th class ) chapter-1 important points**After Reading

**Mathematics**

**Real numbers Definition**

**Notes With important points**, please do share it with your friends. You can

**Learn Maths for All Classes here.**

**Natural Numbers:**

**The counting numbers 1,2,3,4,5,6.....are called Natural numbers. It is represented by 'N' .**

**N= (1, 2, 3, 4, 5, 6 ... .}**

**The smallest a natural number is 1 and the greatest natural number does not exist**

**Whole Numbers: ****Natural numbers
including zero are called whole numbers. It is Represented by W. W=
{0,1,2,3,4,5,6,... } The least whole number is 0 and the greatest whole number
does not exist. **

**Integers: ****The set containing
the positive numbers, the negative numbers together with zero is called the set
of integers.**

**Whole numbers
including negative numbers are called Integers. It is represented by Z. The set
of integers is represented by Z. **

**Z = {... - 4, -3,
-2, -1, 0,1,2,3,4.....}**

**Note:**

**1{ 1.2,3,............}
is called Set of positive integers.**

**2. { -1,-2,-3 ,...............}
is called Set of negative integers. ,**

**3. Zero is
considered neither positive nor negative.**

**4. In the number
line all negative integers are on the left side of zero, all positive integers
are on the right side of zero.**

**5. Set of positive
integers and zero considered as Non – negative integers. {0,1,2,3,.......} is
set of non – negative integers. **

**6. Set of negative
integers and zero considered as Non - positive integers. (0,-1,-2,....} is set of non - positive integers.**

**7. The differences
between any two consecutive integers is 1.**

**
**

**8. The absolute
value of an integer is the numerical value of the integer regardless of its
sign.**

***Numbers of the form P/q where p and q are integers
and q≠0 are called rational numbers, represented by 'Q'. **

***There are infinitely many rational numbers between
any two integers**

** E.g. 3 < 29
/6, 20/6, 21/6, 22/6, 23/6 ....... < 4 **

***There are infinitely many rational numbers between
any two rational numbers.**

** E.g. 3/4< 29 /8, 71/ 16, 81/ 14........****<****13/2 **

***To find the decimal representation of a rational number we divide the numerator of a rational number by its denominator.**

**E.g.
: The decimal representation of 5/6**

**7/6=1.1666....**

**Also Check**

**Introduction to Knowing Our Numbers Key Points**

**Introduction to integers (7th class)**

**Trigonometry Do This & Try this solution**

**SSC (10th class) Trigonometry Exercise - 11.1 Solution**

**SSC(10th class) Trigonometry Exercise - 11.1 Solutions**

**
**

***Every rational number can be expressed as a
terminating decimal or as non-terminating repeating decimal. Conversely, every
terminating decimal or non- terminating recurring decimal can be expressed as a
rational number 7-29**

** =361/495**

*** A rational number whose denominator consists of
only 2's or 5's or a combination of 2's and 5's can be expressed as a
terminating decimal. Can be expressed as a terminating decimal.**

**EG: 13/32 **** can be expressed as a terminating decimal ****(****32 = ****2 x 2 x ****2 x 2 x 2) **

**7/125 can be expressed as a
terminating decimal (125 =5 x 5 x 5) **

**24 /40 Can be expressed as a terminating decimal (40
= 2 x 2x2x5) **

***Numbers which can't be written in the form where p
and q are integers and q ≠0.are called irrational numbers. E.g: V2, V3, V5...etc.
**

*******The decimal form of an irrational number is neither
terminating nor recurring decimal.**

***If 'n' is a natural number which is not a perfect
square, then Vn is always an irrational number. E.g. : 2, 3, 5, 7, 8, . etc.,
are not perfect squares.**

** V2, V3, V5,
V7 and ****V****8 are irrational numbers.**

***The
collection of rational numbers together with irrational numbers is called set
of Real numbers. R = QUS**

*** If a and b are two positive rational numbers such
that ab is not a perfect square, then Vab is an irrational number between 'a'
and 'b'.**

** E.g: Consider any two rational numbers 7 and 4.**

** 7 x 4 = 28
is not a perfect square; then 28 lies between 4 and 7.**

** i.e.****4 <
V28 < 7**

** ***** If 'a' is a rational number
and 'b' is any irrational number then a + b, a - b, a. b or a/b is an
irrational number.**

**E.g.: Consider 8 and V 7 then 8+V7, 8 - V7, 8/V7 and are all irrational
numbers**

*** If the product of any two irrational numbers
is a rational number, then they are said to be the rationalising factor of each
other.**

** E.g. : Consider any two irrational number 7/**** V**** 5 and 6/**** V****5.**

** 7****V**** 5 x 5**** V**** 5 = 7 x 5 x5 = 175 a rational number. **

**Also 73 x 3 = 219 - a rational
number.**

** 5 V 4 x V 4 = 20 - a rational number.**

** So The rationalising factor of an irrational number is not unique.**

** *The general form of rationalizing factor (R.F.) of (a **_+ **vb) is (a +****v****b). They
are called conjugates to each other.**

***Laws of Exponents: **

**1. a ^{m }. a^{n }= a ^{m+n}**

**2. (a ^{m})^{n}= a^{mn}**

** a ^{m}
/b^{n =} a ^{m-n }if m > n**

** =1 if m=n**

** =1/ a ^{n}**

^{-}

^{m}

^{ }if m < n*****1/ a^{n}=a^{-n}

*****a^{m }.
b^{n}= (ab)^{n}

*****a^{0 }= 1

***(****Va+b)= a+2**

**Va**

**b**

**+**

**b**

^{2}*******Va**×**Vb =****Vab**

***(****Va+b)****(****Va-b)****=a-b**^{2}^{ }

**Where a, b are rationales and m, n are
integers.**

*** Let a, b be any two rational numbers such
that a = b ^{n} then b = ^{n}**

**V**

**a = a**

^{1/n}Here 'b' is called n^{th}root of a.*** Let
'a' be a positive number and n > 1 then n**

** i.e, n ^{th} root of a is called a surd.**

^{}**
**

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