Real Numbers - Proofs based on irrational numbers for Class 10th maths.
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Transcript of Real Numbers - Proofs based on irrational numbers for Class 10th maths.
REAL NUMBERS
Decimal Expansion Of Rational numbersEuclid’s Division AlgorithmFundamental Theorem Of ArithmeticPROOFS BASED ON IRRATIONAL NUMBERS
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REAL NUMBERS
Proofs Based On Irrational Numbers
Q) Prove that √7 is irrational.
SOLUTION:
Let’s say that √7 is actually a rational number.
Since √7 is a rational number we can express √7 as
√7 =
where a & b belong to integers and have no common factor other than 1 and b 0. So , since √7 =
Squaring on both the sides we get,
7 =
Or, 7b2 = a2 ------------------------------ (1)
Proofs based on irrational numbers
Chapter : Real Numbers Website: www.letstute.com
a2 is divisible by 7. (7b2 is divisible by 7)
a is divisible by 7. (7 is prime & divides a2, 7 divides a)
Let, a = 7 c ( where c is some integer)
Substituting a = 7c in (1), we get
7b2 = (7c)2
7b2 = 49 c2
b2 = 7 c2
b2 is divisible by 7. (7 c2 is divisible by 7)
b is divisible by 7. (7 is prime & divides b2, 7 divides b)
Proofs based on irrational numbers
a and b are divisible by 7. Chapter : Real Numbers Website: www.letstute.com
7 is a common factor of a and b, but this contradicts the fact that a and b have no common factor other than 1.
This contradiction has arisen because of our incorrect assumption that √7 is rational.
Hence, √7 is an irrational number.
Proofs based on irrational numbers
Chapter : Real Numbers Website: www.letstute.com
Chapter : Real Numbers Website: www.letstute.com
IF ‘p’ (PRIME NUMBER) DIVIDES ‘’ THEN ‘p’ DIVIDES ‘a’ WHERE ‘a’ IS POSITIVE INTEGER.
RATIONAL NUMBER + RATIONAL NUMBER = RATIONAL NUMBER.
RATIONAL NUMBER RATIONAL NUMBER = RATIONAL NUMBER.
= MAY OR MAY NOT BE A RATIONAL NUMBER.
ALWAYS TRY TO PROVE THESE TYPES OF PROBLEMS BY THE CONTRADICTION METHOD.
APPLY THE DEFINATION OF RATIONAL NUMBERS.
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Chapter : Real Numbers Website: www.letstute.com