1) Rules for adding and subtracting integers include keeping the sign the same when adding like signs, and using the sign of the larger number when subtracting or adding opposite signs. 2) When multiplying integers, the sign of the product is determined by the number of negative factors. If even, the product is positive, and if odd, the product is negative. 3) Integers are closed under addition, subtraction, and multiplication, and follow properties like commutativity and associativity for these operations.

1. INTEGERS

3. RULES FOR
INTEGER

4. ADDITION RULE
1)When the signs are same, ADD and
keep the sign.
(-2) + (-4) = (-6)
2)When the signs are different
, SUBTRACT and use the sign of the
larger number.
(-2) + 4 = 2
2 + (-4) = (-2)

5. SUBTRACTION RULE
When subtracting , change the subtraction to
adding the opposite and then follow your
addition rule.
Example 1) -4 – (-7)
- 4 +(+7)
Different sign---subtract and use larger sign
3
Example 2) -3 -7
-3 + (-7)
Same sign --- Add and keep the sign.
(-10)

6. The additive inverses (or
opposites) of two numbers add
to equal zero.
Example: The additive inverse
of 3 is -3
Proof: 3 + (-3) = 0
We will use the additive
inverses for subtraction
problems.

7. • Multiplying a positive integer and
a negative integer , we multiply
them as whole numbers and put a
minus sign(-) before the product.
We thus get a negative integer.
3 x (-4) = (-12)

8. Product of two negative
integers is a positive
integer. We multiply the
two negative integers as a
whole numbers and put the
positive sign(+) before the
product.
(-31) x (-10) = +310

9. 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.
(-9) x (-5) x (-6) x (-3) = 45 x 18 = 810
(-9) x (-5) x 6 x (-3) = 45 x (-18) = (-810)

10. 1 is the MULTIPLICATIVE IDENTITY for
integers.
For an integer ’ a’ we have, ax1 = a
MULTIPLICATION BY ZERO
Product of an integer and a zero is zero.
For an integer a , a x 0 = 0

11. DIVISION RULE
• When we divide a negative integer
by a positive integer or vice versa, we
divide them as whole number and
then put a minus sign(-) before the
quotient. We, thus get a negative
integer.
72 (-8) = (-9)

12. •Division of a negative
integer by another
negative integer gives a
positive integer as
quotient.
(-45) (-9) = 5

13. PROPERTIES UNDER ADDITION AND
SUBTRACTION
1)Integers are closed for addition and
subtraction both. That is, a+b and a-b are
again integers, where a and b are any integers.
7+3 =10 and 7-3 =4
2) Addition is commutative for integers, i.e., a+b
= b+a for all integers a and b.
4+5 = 5+4 = 9

14. 3) Addition is associative for
integers, i.e.,(a+b)+c=a+(b+c) for all
integers a,b and c.
(2+3)+4 = 2+(3+4) = 9
4) Integer 0 is the identity under addition.
That is , a+0=0+a = a for every integer a.
3+0 = 0+3 = 3

15. PROPERTIES UNDER MULTIPLICATION
a)Integers are closed under multiplication
i.e., a x b is an integer for any two
integers a and b.
3x4=12, which is an integer.
b)Multiplication is commutative for integers
i.e. axb=bxa for any integers a and b.
2x4 = 4x2 = 8
2x(-4) = (-4)x2 = (-8)

16. c)The integer 1 is the identity under
multiplication, i.e. 1xa = ax1 = a for any
integer a .
d)Multiplication is associative for
integers, i.e.(axb)xc = ax(bxc) for any three
integers a,b and c.
1x(3x5) = (1x3)x5 = 15