Integers


Integers :-
 
. . . . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, . . . . Integers
1,2,3,4, .....  Positive Integers (always > 0)
-1, -2, -3, -4, ...... Negative Integers (always < 0)
Number 0 Interger (neither positive nor negative)

Absolute Value of an Integer :-
Its a numerical value regardless of its sign. The absolute value of an integer is always non-negative.
The absolute value of an integer p is written as [ p ].

If p is an integer, then
[ p ] = p if p>=0
       = -p if p<0

Additive inverse of an integer :-
For every integer a, there exists integer -a such that
a + (-a) = 0 = (-a) + a
Thus, additive inverse of a = -a    and  additive inverse of -a = a.


Addition  of Integers :- 
1) Addition of two positive integers - Add them as natural numbers.
2) Addition of two negative integers - Find the sum & give negative sign to the sum.
3) Additio of a positive integer & a negative integer - Subtract the smaller number from the larger number & give the sign of the integer which has the larger absolute value to the result.

Properties of Addition of Integers :- 
 
Closure Property If p & q are two integers, then p+q is also an integer.
eg. 11 + 2 = 13 (13 is also an integer)
Commulative Property If p & q are two integers, then p+q = q+p
e.g. 6+(-19) = -13    &   (-19)+6 = -13.
Hence 6+(-19) = (-19)+6
Associative Property If p, q and r are any three integers, then (p+q)+r = p+(q+r)
e.g. ((-7)+3) + (-18) = -22   &   (-7)+ (3+(-18)) = -22
Hence  ((-7)+3) + (-18) = (-7)+ (3+(-18))
 


Subtraction of Integers :-
Subtracting one integer from another integer is same as adding the additive inverse of the integer that is being subtarcted to the other integer.

Properties of Subtraction of Integers :-
 
Closure Property If p and q are any two integers, then p-q is also an integer.
e.g.-11 - 3 = -8 is also an integer.
Not commutative If p and q are any two integers, then p-q ≠ q-p, p ≠ q
e.g. 8-(-5) = 13       &     (-5)-8 = -13
Hence 8-(-5)  ≠ (-5)-8
Not associative If p,q and r are any three integers, then (p-q)-r ≠ p-(q-r), c ≠ 0
e.g. (2-(-3)) - 12 = -7    &    2-((-3-(-12) = 7
Hence  (2-(-3)) - 12  ≠  2-((-3-(-12)



Multiplication of Integers :-
 
Case Examples Product
Multiplication of two positive integers Multiply them as natural numbers.
e.g 3x7 = 21, product (21) is positive integer. 
Positive integer
Multiplication of positive integer & a negative integer Multiply them as natural numbers & put the minus sign before the product.
eg. 3x(-7) = -21, product (-21) is an negative integer.
Negative integer
Multiplication of two negative integers Multiply them as natural numbers & put the positive sign before the product.
e.g (-3)x(-7) = 21, product (21) is a positive integer.
Positive integer
Multiplication by 0 If p is any integer, then px0 = 0 = 0xp Zero

 
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