Integers are whole numbers and their negatives. On a number line, adding a positive integer moves right and adding a negative integer moves left. Integers are closed under addition and subtraction. For any integers a and b, a + b and a - b are also integers. Addition is commutative but subtraction is not. Both operations are associative. For integers a and b, a * (-b) = (-a) * b and a * (-b) = - (a * b). The product of two negative integers is positive. If the number of negative factors in a product is even, the product is positive, and if odd, the product is negative. Integers are closed

1. INTEGERS
INTEGERS FROM A BIGGER COLLECTION OF
NUMBERS WHICH CONTAINS WHOLE NUMBER AND
NEGATIVE NUMBERS
A NUMBER LINE REPRESENTING INTEGERS
ON A NUMBER LINE WHEN WE
1. Add a positive integer, we move to the right.
2. Add a negative integer, we move to the left.
3. Subtract a positive integer, we, move to the left.
4. Subtract a negative unteger, we move to theright.

2. PROPERTIES OFINTEGERS
FOR ANY TWO INTEGERS A AND B
a-b=a + additive inverse of b=a + (-b)
a-b(-b) = a + additive inverse of (-b) = a+b
Closure under addition
Sum of two whole numbers is again a whole number. For
example, 17 + 24 = 41 which is again a whole number.
His property is known as the closure property for addition of
the whole numbers. Integers are closed under addition in
general. For any two integers a and b, a + b is an integer.
Integers are closed under subtraction. Thus, if a and b are
two integers then a-b is also an intger.

3. Commutative property
Addition is commutative for integers.
In general, for any two integers a and b, we can say
a + b = b+a
Subtraction is not commutative for integers.
Because 5 –(-3)=5+3=8, and (-3)-5=-3-5=-8.
Associative property
Addition is associative for integers. In
general for any intgers a,b and c, we can say
a+(b+c)=(a+b)+c

4. Additive identity
In general, for any integer a
a+0=a=0+a
1. (-8)+0= - 8
2. 0+(-8)= - 8
MULTIPLICATION OF A POSITIVE AND A
NEGATIVE INTEGER
FOR ANY TWO POSITIVE INTEGERS A AND B WE
CAN SAY
a X (-b) = (-a) X b = - (a X b )
(-33) X 5 = 33 X (-5) = -165

5. MULTILICATION OF TWO NEGATIVE INTEGERS
WE MULTIPLY THE TWO NEGATIVE INTEGERS AS
WHOLE NUMBERS AND PUT THE POSITIVE SIGN
BEFORE THE PRODUCT.
THUS, WE HAVE ( -10 ) x ( -12 ) = 120
SIMILARLY ( -15) x ( - 6 ) = 90
IN GENERAL, FOR ANY TWO POSITIVE INTEGERS
A AND B,
( -a ) x ( -b ) = a x b

6. Product of three or more Negative Integers
If the number of negative integers in a
product is even, then the product is a
positive integer; if the number of negative
integers in a product is odd. Then the
product is a negative integer.
a. ( -4 ) x ( -3 ) = 12
b. (-4 ) x ( -3 ) x ( -2 )= [ ( -4 ) x ( -3 ) ] x ( -2 ) = 12 x ( -2 )= -24

7. CLOSURE UNDER MULTIPLICATION
Integers are closed under multiplication.
In general,
a x b is an integer, for all integers a and b.
Statements Inferences
(-20 ) x ( -5 ) = 100 Product is an integer
MULTIPLICATION BY ZERO
IN GENERAL, FOR ANY INTEGER A.
a x 0 = 0 x a =0
( -3 ) x 0 = 0
0 x ( -4 ) = 0

8. MULTIPLICATIVE IDENTITY
IN GENERAL, FOR ANY INTEGER A WE HAVE.
ax1=1xa=a
( -3 ) x 1 = -3 1x5=5
ASSOCIATIVITY FOR MULTIPLICATION
Product of three integers does not depend upon the
grouping of integers and this is called the associative
property for multiplication of integers.
In general, for any three integers a, b and c
( a x b) x c = a x (b x c )
[ (-3) x (-2)] x 5 = 6 x 5 = 30
(-3) x [(-2) x 5] = (-3) x (-10) = 30

9. DISTRIBUTIVE PROPERTY
IN GENERAL, FOR ANY INTEGERS A, B AND C,
a x ( b + c ) =a x b + a x c
A ( -2 ) x (3+5) = - 2x8= -16
and [(-2) x 3] +[ (-2) x5]= (-6)+ (-10) =-16
IN GENERAL, FOR ANY INTEGERS A, B AND C,
ax(b-c)=a x b–a x c
4x(3-8)=4x(-5)= -20
4x3 - 4x8=12 -32= -20
4x(3-8)=4x3 - 4x8
(1) (-18) x (-10) x 9
(-18) x (-10) x 9 = [(-18)x(-10)]x9 = 180x9=1620
(2) (-20) x (-2) x (-5) x7
(-20) x (-2) x (-5) x 7 = -20x (-2 x -5) x7 = [-20x10] x7
= -1400

10. MADE BY –
ROHAN GOEL
CLASS- VII-B