LearnPick Navigation
Close

Real Numbers

Published in: Mathematics
813 views
  • Jay B

    • Mumbai
    • 3 Years of Experience
    • Qualification: B.Tech/B.E.
    • Teaches: Mathematics, Algebra, Physics, Chemistry
  • Contact this tutor

Real Numbers -Class X CBSE

  • 1
    REAL NUMBERS
  • 2
    CONTENTS ' Euclid's Division Lemma ' The Fundamental Theorem of Arithmetic ' Revisiting Irrational Numbers ' Revisiting Rational Numbers and Their Decimal Expansions ' Summary
  • 3
    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. THEORM 1 (Euclid's Division Lemma) : Given positive integers a and b, there exist unique integers q and r satisfying WHEREOsr
  • 4
    What exactly does the theorem mean? 'Euclid's division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.
  • 5
    EXAMPLE HCF of the integers 455 and 42. We start with the larger integer, that is, 455. Then we use Euclid's lemma to get 455 = 42 x 10 + 35 Now consider the divisor 42 and the remainder 35, and apply the division lemma to get 42=35x1+7 Now consider the divisor 35 and the remainder 7, and apply the division lemma to get
  • 7
    NOTE: 'The flow of the divisor, dividend and remainder! 'The second last remainder, before the remainder becomes zero is the HCF required!
  • 8
    To findinq properties of numbers Show that every positive even integer is of the form 24, and that every positive odd integer is of the form 2q + 1, where q is some integer. Solution : Let a be any positive integer and b = 2. Then, by Euclid's algorithm, a = 2q + r, for some integer q >=0, and r = 0 or r = 1, because r < 2. so, a = 2q or 2q + 1. If a is of the form 2q, then a is an even integer. Also, a positive integer can be either even or odd. Therefore, any positive odd integer is of the form 24 + 1.

Discussion

Copyright Infringement: All the contents displayed here are being uploaded by our members. If an user uploaded your copyrighted material to LearnPick without your permission, please submit a Takedown Request for removal.

Need a Tutor or Coaching Class?

Post an enquiry and get instant responses from qualified and experienced tutors.

Post Requirement

Related Notes

Query submitted.

Thank you!

Drop Us a Query:

Drop Us a Query