10th CBSE MATHS FORMULAE CHAPTER NO. : 1) REAL NUMBERS

CHAPTER NO :  1.  REAL NUMBERS

IMPORTANT FORMULAE:-

1. Euclid's Division Lemma :

      Let a and b be any two positive integers, then there exists unique integers
    q and r,
       Such that :   
                       a = bq + r ,   where 0 ≤ r < b

   

2. Euclid's Division Algorithm :

      Let a and b be any two positive integers,
      Such that : a > b and q = quotient , r = remainder
       Take a as dividend and b as divisor
         a = bq + r ,     where 0 ≤  r  < b

  Following steps are use to find HCF  of two positive integers using Euclid's Division Algorithm:

    Step 1:  Apply Euclid's lemma to a and b:    a = bq₁ +  r₁
                  If  r₁ = 0,    then HCF = b
                  If  r₁ ≠ 0,     then step 2
    Step 2: Apply Euclid`s lemma to b and r₁:    b = r₁q₁ +  r₂
                 If  r₂ = 0,    then HCF = r₁
                 If  r₂ ≠ 0,    then go to step 3

    Step 3: Apply Euclid's lemma to r₁ and r₂:     r₁ = r₂q₂ +  r₃
                 If  r₃ = 0,    then HCF = r₃
                 If  r₃ ≠ 0 ,    then go to step 4
    Step 4: Continue the steps untill  we get remainder zero

3. HCF and LCM :

     
     1. HCF(a, b) -  Highest Common Factor 
                       -  Product of  the smallest power of each common prime factor in the numbers.
         
          Example:  HCF of 6 and 20 - prime factor of 6   =  2¹ x 3
                                                  - prime factor of 20 = 2² x 5
                           HCF of 6 and 20 = 2
   
     2. LCM(a, b) - Least Common Multiple 
                        - Product of the greatest power of each prime factor involved in the numbers 
           
             Example:  LCM of 6 and 20 - 2² ⨯ 3 ⨯ 5  =60    (above example)

    3. HCF(a, b) ⨯ LCM(a, b)  =  a ⨯ b  for any positive integers a and b .           
              Example: HCF(6, 20) ⨯ LCM(6, 20) =  6 ⨯ 20                                                                                                                           2 ⨯ 60        =  120

4. Important Theorems:

Theorem 1: Euclid's Division Lemma ( explanation is given in above )

Theorem 2: 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. ( NCERT)

      Explanation : In very easy words composite number X are the product of                               prime numbers .

                               X = P₁ ⨯ P₂ ⨯ P₃ ⨯ …..⨯ Pₙ

                              where P₁,P₂,....,Pₙ are prime numbers.

                          prime numbers (2, 3, 7, 11.....)

Theorem 3: Let p be prime number ,  If p divides a², then p divides a ,where a positive integer.

    

Theorem 4: √2 is irrational number.

                    

     * Following steps are use to solve prove type questions of theorem 4:

      Step 1: Let us assume given √m (or any other √2 , √3 , 2√3+3.... etc)

                  is rational.

      Step 2:  consider two positive numbers a and b such that:

                      √m = a/b    ,where a and b are co prime i.e. their HCF is 1.

                   now,

                       √m = a/b

                    squaring on both sides

                         m = a²/b²

                        mb² = a²

                        m|a²    

                        m|a                …..  {by theorem 2} ….(ⅰ)

                       a=mc for some integer c

                    squaring on both sides

                       a²=m²c²

                       b²=mc²

                        m|b²

                        m|b                       …..  {by theorem 2}….(ⅱ)

 From equation (ⅰ) and (ⅱ), we observe that a and b have at least m as a common factor. But this contradicts the fact that a and b are co-prime. This means that our assumption is incorrect.Hence,  √m is a irrational number.

Theorem 5: 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 co-prime and the prime factorization of q is of the form 2ⁿ5ᵐ,where n, m are non-negative integers.

 

Theorem 6: Let x=p/q be a rational number, such that the prime factorization of q is of the form 2ⁿ5ᵐ, where n, m are non- negative integers. Then, x has a decimal expansion which terminates.

Theorem 7: Let x=p/q be a rational numbers, such that the prime factorization of q is not of the form 2ⁿ5ᵐ, where n, m are non- negative integers.                 Then, x has decimal expansion which is non-terminating repeating (recurring).