Laws of Arithmetic

Laws of Arithmetic

1) Commutative Law :

     *) For Addition : (Passes this law)

          a+b = b+a ; e.g. 3 + 4 = 4 + 3 = 7 
          The sum of two numbers is the same if                we change their order.

     *) For Subtraction : (Doesn't Pass this law)

          a-b ≠ b-a ; e.g. 3 - 4 ≠ 4 - 3 

     *) For Multiplication : (Passes this law)

          a.b= b.a ; e.g. 3 * 4 = 4 * 3 = 12
          The product of two numbers is the same            if we change their order.

     *) For Division : (Doesn't Pass this law)

2) Associative Law :

     *) For Addition : (Passes)

          a + (b +c) = (a + b) + c = (a + c) + b

          e.g. 2+(3+4) = (2+3)+4 = (2+4)+3 = 9
          The sum of three numbers remain the                same whichever way we group any two              of the three numbers.

     *) For Subtraction : (Doesn't Pass this law)

          a - (b - c) ≠ (a - b) - c ≠ (a - c) - b

          e.g. 2-(3-4) ≠ (2-3)-4 ≠ (2-4)-3

     *) For Multiplication : ( Passes)

          a * (b * c) = (a * b) * c = (a* c) * b

          e.g. 2*(3*4) = (2*3)*4 = (2*4)*3 = 24
          The product of the three numbers                        remain same whichever way we group                any two of the three numbers.

     *) For Division : (Doesn't Pass this law)

          a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c ≠ (a ÷ c) ÷ b

          e.g. 2÷(3÷4) ≠ (2÷3)÷4 ≠ (2÷4)÷3

   

3) Properties of Zero :
      
    *) For Addition (Identity element) :                           (Passes)
         e.g. 4+0 = 4 = 0 + 4  
         When we add zero to a number or add a             number to zero, the sum is the number               itself.

    *) For Subtraction : (doesn't pass this                       property)
         e.g. 4 - 0 = 4 but 0 - 4 = - 4 ; (4 ≠ - 4)

     *) For Multiplication :  (Passes)
         e.g. 4 * 0 = 0 = 0 * 4 
         Any number multiplied by zero is always           zero.

     *) For Division : (doesn't Pass this property)
         e.g. 0 ÷ 4 = 0 but 4 ÷ 0 = ∞ ; (0 ≠ ∞)
         Zero divided by any number is always                 zero . but division by zero is not                             permissible.

4) Properties of 1:

     *) For Addition :    ( X )

     *) For Subtraction :  ( X )

     *) For Multiplication : (Identity element)
         e.g. 4 * 1 = 4 = 1 * 4 ( Passes this property)
         When 1 multiplies a number , the                         product is the number itself.

      *) For Division : (doesn't pass this                               property)
           e.g. 4 ÷ 1 =4 but 1 ÷ 4 = .25 ;  (4 ≠ .25)
           Note:
           A number divide by itself is always = 1
           e.g. 4 ÷ 4 = 1

5) Distributive Law : (Meaning Multiplication                                             distributes over)
           *) Addition : (Satisfies)
               e.g. a (b + c) = ab + ac
               2(3 + 4) = 2(7) = 14 ; 2*3 + 2*4 = 6 + 8 = 14
               L.H.S. = R.H.S.

           *) Subtraction : (Satisfies)
                e.g. a (b - c) = ab - ac
                2(3 - 4) = 2 (-1) = -2 ; 2*3 - 2*4 = 6 - 8 = -2
                L.H.S. = R.H.S.

          *) Multiplication : (Satisfies)
               e.g. a (b * c) = ab * ac
                       2(3 * 4) = 2 ( 12) = 24 ;
                       (2*3) * (2*4) = 6 * 8 = 24
                       L.H.S. = R.H.S.
          *) Division : ( doesn't Satisfy)
              e.g. a ( b ÷ c) = ab ÷ ac
                      2 ( 8 ÷ 4) = 2(2) = 4
                      (2*8) ÷ (2*4) = 16 ÷ 8 = 2
                      L.H.S. ≠ R.H.S.