NCERT Chapter Summary: Real Numbers

NCERT Chapter Summary: Real Numbers

Euclid’s division algorithm, as the name suggests, has to do with divisibility of integers. Stated simply, it says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b.

Euclid’s division lemma: Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r, 0 ≤ r < b.

Euclid’s division algorithm: This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers a and b, with a > b, is obtained as follows:

  • Step 1: Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b.
  • Step 2: If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r.
  • Step 3: Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). Also, HCF(a, b) = HCF(b, r).

The Fundamental Theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

If p is a prime and p divides a2, then p divides a, where a is a positive integer.

Let x be a rational number whose decimal expansion terminates. Then we can express x in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers.

Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).