Are there any rational square roots of numbers?
Yes, if a number is a perfect square, its square root is rational (e.g., √9=3).
What happens when you add a rational number to an irrational number?
The result is always irrational, as shown in examples like 5 - √3.
Why is the square root of a prime number irrational?
The square root of a prime number cannot be expressed as a fraction, so it’s irrational. We use proof by contradiction and Theorem 1 to prove this.
How does the theorem help in finding HCF and LCM?
Prime factorization allows us to identify the common factors for HCF and all factors for LCM.
Why is prime factorization unique?
The uniqueness comes from the fact that no two different sets of prime numbers can be multiplied to produce the same composite number.
What does the Fundamental Theorem of Arithmetic state?
The theorem states that every composite number can be uniquely factorized as a product of prime numbers, apart from the order of factors.
What’s the difference between rational and irrational numbers?
Rational numbers can be expressed as a fraction of two integers and have either terminating or repeating decimal expansions. Irrational numbers cannot be expressed as fractions and have non-terminating, non-recurring decimals.
How does prime factorization help in finding the LCM of numbers?
By expressing each number in terms of its prime factors, we can identify the highest power of each prime factor present in the numbers. Multiplying these factors gives the LCM.
Why are certain square roots like 
irrational?
Certain square roots cannot be expressed as a fraction because their decimal expansions are non-terminating and non-repeating. The proof often involves assuming the number is rational and reaching a contradiction.
irrational?admin2024-11-25T14:03:00+05:30How is Euclid’s Division Lemma useful in finding the HCF?
Euclid’s Division Lemma allows us to systematically divide two numbers, using remainders to progressively reduce the numbers until we reach the HCF. This method is efficient and widely used in number theory.
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