Real numbers

Real numbers

1. Theorems

   • Statement
     • 1. Let p be a prime no. If p divides a² . Then p divides a,where a is a positive integer.
       • 2. √2 is irrational
         • 3. Let x be a rational no. Whose decimal expansion terminates . Then x can be expressed in the form of p/q, where p &q are co prime, the prime factorisation of q is of the form 2ⁿ, 5 raise to power m , where n,m are non- negative integers.
           • 4. Let x = p/q be a rational number such that the prime factorisation of q is of the form 2ⁿ, 5 raise to power m . Where n, m are non - negative integers. Then x has a decimal expansion which terminates.
             • 5. Let x =p/q be a rational number, such that the prime factorisation of q is not of the form of 2ⁿ5 raise to power m . Where n, m are non - negative integers. Then x has a decimal expansion which is non - terminating repeating.

2. Euclid

   • Given positive integers are a, b. There exist unique integers q and r. Satisfying a= bq +r ; 0 5 r < b
   • Division algorithm.
     • Steps to obtain the HCF of two positive integers. Say c and d, with c > d.
       • Step 1- apply euclid division lemma, to c and d, c=dp +r.
         • Step 2- if r = zero, d is the HCF of c and d. If r is not equal to zero, apply euclid division to d and r.
           • Step 3- continue the process till the remainder is zero.
   • Division lemma.

3. Fundamental theorem of airthematic

   • Every composite no. Can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur
     • Composite no. X=p₁*p₂*p₃..*p₄ where p₁p₂..px are prime no.

4. prime factorisation method

   • for any two positive inteɡers , a and b
     • HCF (a,b)*LCM(a,b)=a*b
       • for example
         • f(x)=3x²y
           • ɡ(x)=6xy²
             • HCF=3xy
               • LCM=6x²y²