Bracket Function : Number Theory #Bracket #Function #: #Number #Theory

 

 Bracket Function : 

               The function I : R → Z defined by I(x) = n where n ≤ x < n+1 is called the bracket function or the step function or the integral part function .

 Notation : 

                  Integral part of x in R is denoted by I(x) or [x].

Definition :  

                 If x 𝞊 R , x - [x] is called the Fractional part of x. 

 Note : 

1.       [x] ≤ x < [x] + 1  or x-1 < [x] ≤ x.
2.       For every x 𝞊 R, x ≥ [x] ,  hence the fractional part of x is always non-negative
3.       x 𝞊 Z ⇔ x=[x].

 For example : 

       * For 14/3 = 4.6666... , [ 14/3] = 4.

       * For 3/4, [3/4] = 0   because 0 < 3/4 < 1.

       * For π = 22/7 , we have 3 <  π < 4 ⟹     [ π ] = 3.

Properties of Bracket Function : 

       

       * For x 𝞊  R , [x] ≤ x ≤ [x] + 1 .

       * If a 𝞊 Z and b 𝞊 Z then [a/b] = the quotient when a is divided by b.

       * If m 𝞊 Z and x 𝞊 R, then [x+m] = [x] + m

       * If x 𝞊 R then [x] + [-x] = 0    when x 𝞊 Z

                    and     [x] + [-x] = -1   when x 𝞊 R-Z

       * For x,y 𝞊 R, we have [x+y] ≥  [x] + [y] 

        * The product of n consecutive integers is divisible by n!.

 * Perfect Number

 *Historical background of Number Theory

 * Dirichlet Multiplication

 * Euclidean Algorithm

 * Fundamental Theorem of Arithmetic

 *  Properties of Numbers

 * Historical Introduction to Number Theory 

 * GCD of morethan 2 numbers

 *   The Mobius Function 𝝻 ( n ) .

 *  The Euler Totient Function 

 * Formal Power Series 

 * Liouville’s function λ(n) 

  * Congruences

 * Fermat's Theorem

 * Wilson's Theorem 

#Bracket #Function #: #Number #Theory