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)