Integers
Class 8
Integers :-
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 :-
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 :-
Multiplication of 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 |