Real Numbers online mcq quiz
Real Numbers Boostup points
Euclid’s Division Lemma
➤ An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
➤ A lemma is a proven statement used for proving another statement.
➤ Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers.
➤ To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:
Step 1: Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 ≤ r < d.
Step 2: If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.
Step 3: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.
The Fundamental Theorem of Arithmetic
➤ Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
Rational and Irrational Numbers
➤ A number ‘s’ is called rational if it can be written in the form p, qWhere p and q are integers and q ≠ 0.
➤ A number ‘s’ is called irrational if it cannot be written in the form p, q
Where p and q are integers and q ≠ 0.
Irrationality of Square Roots of 2, 3 and 5
➤ Let p be a prime number. If p divides a², then p divides a, wherea is a positive integer.
➤ 2 , 3 , 5 are irrational
Decimal Expansions of Rational Numbers
➤ Let x be a rational number whose decimal expansion terminates. Then we can express x in the form pq, where p and q are coprime, and the prime factorization of q is of the form 2n 5m, where n, m are non-negative integers.
➤ Let x = pq be a rational number, such that the prime factorization of q is of the form 2n 5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.
➤ Let x = pq be a rational number, such that the prime factorization of q is not of the form 2n 5m where n, m are non-negative integers. Then x has a decimal expansion which is nonterminating repeating (recurring)
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