Hi friends
and my dear students! In this post, I have covered **Andhra
Pradesh class -10 Introduction to Real numbers (chapter-1) important
points** After Reading **Mathematics Real numbers (10th class
) Notes With important points**, please do share it with your friends.
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Key concepts:

*The collection of counting numbers
are called natural numbers. Set of natural numbers are denoted by 'N'

N={1,2,3,..}

* The collection of natural numbers
includes zero are called whole numbers. Set of whole numbers is denoted by
W'

W=(0} U { 1,2,3,.. }

= {0,1,2,3,. }

* The collection of whole numbers
includes negative numbers are called integers.

Set of integers are denoted by
Z' or I

Z={...3,-2,-1,} U {0,1,2,3,.... }

= {.. 3,-2,-1,0,1,2,3,... }

Note: 1) The number O' is neither positive
nor negative

2) There is no integer between any
two consecutive integers.

Lemma: Lema is a proven statement used for
proving another statement Dividend = Divisor x quotient + remainder Euclid's
division lemma: for a pair of given positive integers 'a' and b' there exist
unique pair of integers q' and 'r' such that a=b q+ r where o ≤ r<b.

Eg: Given positive integers 17 and 6

Let a=17 and b=6

17 = 6x2+5 where 0≤5< 6

Steps to obtain HCF by Euclid's division lemma.

Step 1: Consider two positive integers 'a'
and b' such that a>b

Step 2: by Euclid's lemma. To find whole
numbers 'q' and 'r' such that a=b q+ r

Step 3: If 'r'=0 then b' is the HCF of a
and b Step 4 If r + 0 then consider the division of b with the remainder r,
continue this process till the remainder becomes zero

Eg:- Find HCF of 972 and 21 let
a=972, b=21

by using Euclid's division lemma,

972=21 x 46+6

Here remainder is 6

By Euclid's division lemma

21= 6x3+3

Here remainder is 3.

By Euclid's division lemma,

6 =3x2+0

Now remainder is 'O' .

HCF of 972 and 21 is 3

Note: Euclid's division lemma is stated for only positive integers but it can be extended for all integers except 'O'

**Prime
number:** A natural number N>1have exactly two distinct factors (1 and itself)
are called prime numbers.

Eg: 2,3,5,7,11,13,..

**Composite number:** Natural number
N(>1) have more than two factors are called composite numbers

Eg:- 4,6,8,9,10,....

**Note****:** 1) There are infinite primes and composite
numbers

2) 1 is neither prime nor composite

3) 2' is even prime

**Co-prime:** Two numbers are
said to be co-primes it their HCF in 1

Ex: (2,3), (4,5)....

** Factor:** If a
number a divides another number b exactly then 'a' is a factor of b'

Eg:- 2
divides 6 exactly so 2 is a factor of 6

**Also Check**

**Introduction to Knowing Our Numbers Key Points**

## IITJEE 7th class Introduction to Algebra Notes

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

## Introduction to Real number

**Trigonometry Do This & Try this solution**

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

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

**FUNDAMENTAL THEOREM OF
ARITHMETIC:** Every composite number can be expressed as a product of primes and this
factorisation is unique.

Eg:- 24= 2 x 2 x 2 x 3.

To obtain LCM and HCF of given number by prime factorization
method.

Step
1: Let us consider given numbers be 'a' and b'

Step
2: Express a and b as product of prime powers

Step
3: HCF=product of the smallest power of each common prime factors

LCM=product of the greatest power of each prime factors of the
numbers

**Rational
numbers: **

The collection of numbers which are in the form p/q where p and q are integers
and q≠ 0 are called rational numbers set of rational numbers is denoted by
Q.

Eg:- 3 /2 ,-5 ,1/2

Every rational number can be written in the form of terminating decimal
or non-terminating repeating decimals.

Between any two rationals there exist infinite rational numbers

A rational number between any two rational number a and b is
a+b/2

**Terminating decimals in
rational numbers:** Let x=p/q be a rational number such that the prime factorisation of is
of the form 2^{n} x 5^{m} where n and m are non-negative
integers. Then x has decimal expansion which terminates.

Eg:- x=3/20=p/q

Q=20=2^{2} x 5

Which is the form of 2^{n}
x 5^{m}

3/20 is** **terminating
decimals

**Non terminating,
recuring decimals in rational numbers: **Let x=p/q be a rational number,
such that the prime factorisation of q is not in the form of 2^{n} x 5^{m}
where n and m are non-negative integers. Then x has a decimal expansion, which
in non-terminating repeating.

Eg:-**1)** x=23 /35=p/q

q= 35 = 5' x7 which is not in the form of 2^{n} x 5m

23/35 is non-terminating
repeating decimal

**2)** x= 1/7 =p/q

Q=7 which not in the form of 2^{n} x 5^{m}

1/7= 0.142857.... is a non-terminating repeating decimal.

**Irrational number:** Number which
cannot be written in the form of p/q are
called irrational numbers. the set of irrational numbers are denoted by Q or S.

Eg:
0.101001000...,1.256789124569....

Every non-perfect square number is an irrational number.

Eg:- V2, 3, V6,.

Note:-1) 7 is an irrational
number

2) An irrational number between a and b is√ab

Ex: irrational number between 2 and 3 is 2x3 =
√ 6

**Properties:**- 1) Sum of a rational number and an irrational number is an
irrational number Eg:- 2+√3,5+√7.

2)Difference of a rational and an irrational
number is an irrational number

Eg;4 -V5, 4-V11

3) Product of a non zero rational and an
irrational numbers is an irrational number

Eg:- 5√3,6√7 ,.. ...

4) Quotient of a non zero rational and an irrational number is an irrational number

Eg:- 5/√3, √7/4

5) the sum of the two irrational numbers need not be irrational.

Eg:- a = 3- √2,b = 3+ V2

Then a+b=3-V2 +3+ V2

= 6 (rational)

6) The
product of two irrational numbers need not be irrational,

Eg:- a =
√5,b =2V125

Then a x
b=√ 5 x 2√125

= 2x5x 5

= 50 (rational)

**Real numbers: **The set of rational
and irrational numbers together are called real numbers. set of real numbers
are denoted by R=QUQ'

**Logarithm: **Logarithmic form and exponent form are two sides of the same
coin. ie) every logarithmic form can be written in the exponent form and vice
versa. The theory of logarithm is obtained from the theory of indices.

If a>1. Then a^{x} increases as x increases. and If
a<1, then a^{x} decreases as x decreases.

Eg:- 2^{x }increases if x
increases

(1/2)^{x} decreases if x decreases.

Hence log_{a }^{x} is an increasing function if
a>1 and log_{a }^{x} is a decreasing function if a<1

Logarithms are used for all sorts of

calculations in engineering, science, business and economics.

** Defiinition:-** If a^{x} =
N then x = log when (a # 1) > 0 and N>0 for some a,N € R.

Eg:- log_{2}^{5}, log_{10} ^{100 }There
are two systems of logarithms.

1) Common logarithms (or) briggs logarithms

2) 2) Natural logarithms (or) Naperian
logarithms.

**Common Logarithms:** Logarithms to the
base '10' are called common logarithms.

Eg:- log_{10}^{50},
log_{10} ^{100}

**Natural logarithms:** Logarithms to the
base 'e' are called as natural logarithms. Eg:-** ****log _{e}^{x}, log_{e} ^{5 }**

Here e is an irrational number also called exponential number the value
of e=2.718 (Approximately)

**Laws of logarithms:**

First
law: log_{a}^{ xy} = log_{a}^{ x}+ log_{a}^{y}
where x,y and 'a' are positive real numbers and a≠1 Proof:- Let log_{a}^{x}**, **= m

and **log _{a}^{y}, ** = n

By definition; x= a^{n.. (}1)

y = a^{m.... }(2)

By multiplying 1 & 2; xy = a^{n} x a^{m}

Xy = a^{m+n }

log_{a}^{ xy} = m+n

log_{a}^{
xy} = log_{a}^{ x}+ log_{a}^{y}** **

hence
proved

2) log_{a}^{
x/y} = log_{a}^{ x}-log_{a}^{y}

3) log_{a}^{
1}= 0^{}

4) a^{log}_{a}^{m
}= m

log_{a }x^{m }=mlog_{a}^{x}

**characteristic and
Mantissa: **Consider a number N>0. Then let the value of log_{10}^{N}
consist of two parts One a integral part, The other a proper fraction. The
integral part in called the characternstic and the fractional (or) decimal
part is called the mantissa. The mantissa is always lie between 0 and 1

Eg:- log_{10}^{15}
= 1.176 Here 1 is characteristic 0.176 is mantissa

Note:-1) Characteristic of 'n' digited number is n-1

2) If the characteristic is n then
the number of digits is n+1

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