Real Numbers

  • For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying the relation a bq r r b = + , 0 . ≤ < .
  • Euclid’s division algorithms: HCF of any two positive integers a and b. With a > b is obtained as follows:
    • Step 1: Apply Euclid’s division lemma to a and b to find q and r such that a bq r r b = + , 0 . ≤ <
      • a= Dividend
      • b=Divisor
      • q=quotient
      • r=remainder
    • Step II: If r HCF a b b if r = 0, ( , ) = ≠ 0, apply Euclid’s lemma to b and r.
    • Step III: 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. Ex : 24 = 2×2×2×3 = 3×2×2×2
  • Let p x , q ‘ 0 q = ≠ to be a rational number, such that the prime factorization of ‘q’ is of the form 2m 5n, where m, n are non-negative integers. Then x has a decimal expansion which is terminating. 
  • p is irrational, which p is a prime. A number is called irrational if it cannot be written in the form P q where p and q are integers and q ≠ 0.

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