Real Numbers - Proofs based on irrational numbers class 10th maths.

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Decimal Expansion Of Rational numbers
Euclid’s Division Algorithm
Fundamental Theorem Of Arithmetic

Q)Prove that 4 √7 is irrational.
Let’s assume that 4 √7 is a rational number.
Then, there exist co-prime a and b (b ≠ 0) such that,
4√7 = a/b
Or, √7 = a/4b
(because 4, a and b are integers& 4b ≠ 0
𝒂
𝟒𝒃
is a rational
number)
This contradicts that √7 is rational.
Thus our assumption was incorrect.
Hence, 4√7 is irrational.
Chapter : Real Numbers Website: www.letstute.com

Chapter : Real Numbers Website: www.letstute.com
 IF ‘p’ (PRIME NUMBER) DIVIDES ‘𝐚 𝟐
’ THEN ‘p’ DIVIDES ‘a’ WHERE
‘a’ IS POSITIVE INTEGER.
 RATIONAL NUMBER + RATIONAL NUMBER = RATIONAL NUMBER.
 RATIONAL NUMBER − RATIONAL NUMBER = RATIONAL NUMBER.

𝐑𝐀𝐓𝐈𝐎𝐍𝐀𝐋 𝐍𝐔𝐌𝐁𝐄𝐑
𝐑𝐀𝐓𝐈𝐎𝐍𝐀𝐋 𝐍𝐔𝐌𝐁𝐄𝐑
= MAY OR MAY NOT BE A RATIONAL NUMBER.

Chapter : Real Numbers Website: www.letstute.com
 ALWAYS TRY TO PROVE THESE TYPES OF PROBLEMS BY
THE CONTRADICTION METHOD.
 APPLY THE DEFINATION OF RATIONAL NUMBERS.

Please visit www.letstute.com to
take a test.
Chapter : Real Numbers Website: www.letstute.com

Real Numbers - Proofs based on irrational numbers class 10th maths.

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Real Numbers - Proofs based on irrational numbers class 10th maths.

1. Decimal Expansion Of Rational numbers Euclid’s Division Algorithm Fundamental Theorem Of Arithmetic

2. 2

3. Q)Prove that 4 √7 is irrational. Let’s assume that 4 √7 is a rational number. Then, there exist co-prime a and b (b ≠ 0) such that, 4√7 = a/b Or, √7 = a/4b (because 4, a and b are integers& 4b ≠ 0 𝒂 𝟒𝒃 is a rational number) This contradicts that √7 is rational. Thus our assumption was incorrect. Hence, 4√7 is irrational. Chapter : Real Numbers Website: www.letstute.com

4. Chapter : Real Numbers Website: www.letstute.com  IF ‘p’ (PRIME NUMBER) DIVIDES ‘𝐚 𝟐 ’ THEN ‘p’ DIVIDES ‘a’ WHERE ‘a’ IS POSITIVE INTEGER.  RATIONAL NUMBER + RATIONAL NUMBER = RATIONAL NUMBER.  RATIONAL NUMBER − RATIONAL NUMBER = RATIONAL NUMBER.  𝐑𝐀𝐓𝐈𝐎𝐍𝐀𝐋 𝐍𝐔𝐌𝐁𝐄𝐑 𝐑𝐀𝐓𝐈𝐎𝐍𝐀𝐋 𝐍𝐔𝐌𝐁𝐄𝐑 = MAY OR MAY NOT BE A RATIONAL NUMBER.

5. Chapter : Real Numbers Website: www.letstute.com  ALWAYS TRY TO PROVE THESE TYPES OF PROBLEMS BY THE CONTRADICTION METHOD.  APPLY THE DEFINATION OF RATIONAL NUMBERS.

6. Please visit www.letstute.com to take a test. Chapter : Real Numbers Website: www.letstute.com

Real Numbers - Proofs based on irrational numbers class 10th maths.

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