# CLASS-6IMPORTANT RULES OF INTEGERS

IMPORTANT THINGS TO REMEMBER ABOUT INTEGERS

a) ‘0’ (Zero) is less than every ‘+’ positive Integers.

b) ‘0’ (Zero) is greater than every ‘-’ negative Integers.

c) Every (+) positive Integers is greater than (-) negative Integers.

d) The greater is the Integers, the lesser is it’s negative.

Example; 10 < 15  but  -10 > -15

And when 50 > 25  but  -50  < -25

SOME RULES OF ADDITION OF INTEGERS –

1) The sum of two integers is always is an Integers. If, x & y are an Integers then the sum of both Integers (x + y) is always is an Integers.

2) For all Integers x & y, (x + y) = (y + x)    [Communicative Law]

3) For all Integers x,y & z, we consider (x + y) + z = x + (y + z)         [Associative Law]

4) The Integers ‘0’ zero is the additive identity in Integers, as ; x + 0 = 0 + x = x

5) For every Integers  (+ x) + (- x) = (- x) + (+x ) = 0   [ Existence of Additive Inverse ]

SOME  RULES OF SUBTRACTION OF INTEGERS –

1) For any two Integers x & y we define;  x – y = x + (-y) = x + ( Additive Inverse of y )

2) The difference between two Integers is always an Integer. If, x and y are any two integers, then (x – y) always would be an Integer.

3) For any Different Integers x and y ;  x – y = y - x  (Not Equal).

4) If any Integers x , y , z are not all ‘0’ zero,

( x - y ) – z =  x – ( y – z )    (NOT EQUAL)

5) If x is an Integers then, x – 0 = x  &  0 – x =  - x

6)  - (- x) =  x , which means that the additive inverse of  (- x)  is  (+ x)

Example. – (- 34) = (+ 34) ,   - (- 10) = (+ 10)

7) Sum of Two Integers = Given Integers + Other Integers ; ( x + y )

Or, Given Integers = Sum of two Integers – Other Integers ;

x = ( x + y ) - y

Or, Other Integers = Sum of two Integers – Given Integers ;

y = ( x + y ) – x

RULES OF MULTIPLICATION OF INTEGERS –

1) The product of two Integers is always is an Integers. If, x & y are an Integers then xy is always is an integers [Closure Property].

2) For all Integers x & y, we have (x . y) = (y . x)  [Communicative law]

3) If x , y & z are an Integers then we have (x . y) . z = x . (y . z)   [Associative Law]

4) The Integers 1 is the multiplicative identity, when X x 1 = X  for every Integers X or  Y x 1 = Y for every Integers Y [Existence of Multiplicative Identity]

5) If Y is an Integers, then Y x 0 = 0 or 0 x Y = 0 for every Integers Y  [Multiplication property of ‘0’ Zero]

6) If x, y and z are an Integers, then x. (y + z) = (x . y) + ( x.z ) [Distribution Law of Multiplication over Addition]

SOME RULES OF DIVISION OF INTEGERS –

1) The quotient of two Integers need not be an Integers.

If  (+ x) , (+ y) are Integers , but (+ x) ÷ (+ y) is not an Integers.

2) For every non-zero Integer x, we have x ÷ x = 1

- 7                        + 10

Example.2)  -------- =  1      or    ---------  =  1

- 7                        + 10

3) For every non-zero Integers x,  we have  0 ÷ x = 0

0 ÷ (+ 15) = 0,  0 ÷ (- 30) = 0

4) For unequal non-zero Integers x and y, there are  x ÷ y = y ÷ x (Not Equal)

5) If unequal non-zero Integers x , y and z, we have  (x ÷ y) ÷ z = x ÷ (y ÷ z)