Question

Please explain me the Ch REAL NUMBERS of maths class 10th, CBSE. With all the imp topics with examples.

Please ans it fast
POINTS : 25
( it should be a detailed one )

Answer 1

actually I am 12th student , so, I dunno your syllabus . but some important point I can give you for solving all questions belongs to this chapter .


First of all you understand what is Real numbers ???

Real numbers :- real numbers is a group of rational and irrational number .
example :- 1 , 1/2 , 3/4 , -3 , 0, √2 , π etc

Q :- is √(-1) is a real number ???
answer :- no , becoz in real number only rational and irrational value exist . but √(-1) is an imaginary number .

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Rational numbers :- any number which is in the form of P/Q , where P and Q are integer numbers , and Q ≠ 0 , then
P/Q is known as rational number.
example :- 1, 2/3 , 4/5 , 0.24 , 0.333... etc
are rational numbers .

Q:- is 0.333333...... is rational number ??
ans :- yes , becoz 0.3333.... = 1/3 and 1/3 is a rational number , so 0.333... is a rational number .

# all natural numbers , whole numbers , integers , must be rational number.
example :- 0 , 1 , -2 , -5 etc are rational number .

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irrational number :- any number which is not in the form of P/Q is known as irrational number .
example :- π, √2, √3 , √7 etc are irrational numbers .

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★ R + R = R
example :-
R = 2/3
R = 4/3
then, R + R = 2/3 + 4/3 = 6/3 =2 =R

★ R + I = I
example :-
R = 1
I = √2
R + I = 1+ √2 = I

★ R × R = R
example :-
R = -3
R = 0.2
R × R = (-3)× (0.2) = -0.6 = -3/5 = R

★ R ×I = I
example :-
R = -1
I = √2
R × I = (-1)×√2 = -√2 = I

★ I × I = R or I
example :-

case 1 :-
I = √2
I = √2
I × I = √2 × √2 = 2 = R

case 2 :-
I = √2
I = √3
I × I = √2 × √3 = √6 = I

★ R/R = R
example :-
R = 2/3
R = 1/2
R/R = (2/3)/(1/2) = 4/3 = R

★ I/I = R or I
example :-
case 1 :-
I = √2
I = √2
I/I = √2/√2 = 1 = R

case2 :-
I = √6
I =√2
I/I = √6/√2 = √3 = I

where R = rational number
I = irrational number

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★★ most important of this chapter ★★
///Euclid lemma ///

according to this theory , two positive integer numbers a and b are given two another positive integer numbers q and r by
a = b× q + r
where 0 ≤ r < b

this is also known as Euclid division algorithm .

e.g
Dividend = divisor × quotient + remainder

example :- divide , 25 by 4
25 = 4 × 6 + 1
here 25 is dividend
4 is divisor
6 is quotient
1 is remainder

use of Euclid theorem in finding of HCF of two positve integer numbers .
How to find HCF of two numbers ???
see this example /-

example :- find HCF of 125 and 35 .

use Euclid lemma ,
a = bq + r for 125 and 35

125 = 3×35 + 20

again apply , Euclid lemma , for divisor and remainder e.g 35 and 20

35 = 20 × 1 + 15

similarly , again apply . Euclid lemma , for 20 and 15

20 = 15 × 1 + 5

again , for 15 and 5

15 = 5 × 3 + 0

hence , when , remainder =0
then, divisor = HCF
so, HCF { 125, 35 } = 5