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Integers

Nature Numbers
All the counting numbers like 1, 2, 3, . . .  . and so on.

Whole Numbers
Natural numbers along with O are whole numbers.
Every narural number is a whole number. 

Integers
All the positive numbers, O and negative numbers make up integers.
Eg: . . . . . . . -4, -3, -2, -1, 0, 1, 2, 3, 4,  . . . . . are all integers.
O is neither positive nor negative.
The intergers can be represented on the number line as 

 

Numbers that lie to the right side of 'O' area called positive integers.
Numbers that lie to the left side of 'O' are called negative integers.

Absolute Value
The distance of the integers from O irrespective of its sign is the absolute value of that integers.

  • It is always positive or zero.
  • It is denoted by | integer |.

Addition of Integers

  • Rule 1:To add integers of same sign, we add their values and the sum carries their common sign.
    Eg:
    3 + 4 = 7
    -3 + (-4) = -7
  • Rule 2:
    To add integers of different sign, we find the difference between their numerical values and the difference carries the sign of the greater number.
    Eg:
    4 + (-11) = -7
    11 + (-4) = 7


Subtraction of Integers
To subtract one integer from the other, convert the integer to be subtracted to its negative and then add them.

Multiplication of Integers

  • Rule 1:
    Product of two integers with same signs is equal to the product of their numerical values with positive sign.
    Eg:
    (-2) \(\times\) (-5) = 10
    2 \(\times\) 5 = 10
  • Rule 2:
    Product of two integers with opposite sign is equal to the product of their numberical values with negative sign.
    Eg:
    -2 \(\times\) 5 = -10
    2 \(\times\) (-5) = -10

Properties of Multiplication
S.N Property Statement Examples
1 Closure If a and b are integers then a \(\times\) b is also an integer. 5 \(\times\) (-2) = -10
2 Commutative If a \(\times\) b area two integers then
a \(\times\) b = b \(\times\) a

2 \(\times\) (-3) = (-3) \(\times\) 2
or, -6 = -6

  Associative If a,b and c are three integers then
a \(\times\) (b \(\times\) c) = (a \(\times\) b) \(\times\) c
2 \(\times\) (-3 \(\times\) 4) = (2 \(\times\) (-3)) \(\times\) 4
or, 2 \(\times\) (-12) = (-6) \(\times\) 4
or, -24 = -24
4 Multiplication by zero

For any integer 'a'
a \(\times\) 0 = O \(\times\) a = 0

(-4) \(\times\) 0 = 0
or, 0 \(\times\) (-4) = 0
5 Multiplicative identity For any integer 'a'
a \(\times\) 1 = 1 \(\times\) a = a
-2 \(\times\) 1 = -2
5 \(\times\) 1 = 5
6 Distributive property For any integer a,b and c
a \(\times\) (b + c) = a \(\times\) b + a \(\times\) c and a \(\times\) (b - c) = a \(\times\) b - a \(\times\) c

-3 \(\times\) (2 + 5) = -3 \(\times\) 2 + (-3) \(\times\) 5
or, (-6) + (-15) = -21
 or, -2 \(\times\) (5 - 6) = (-2) \(\times\) 5 - (-2) \(\times\) 6
= -10 + 12
= 2


Division of Integer

  • Rule 1:
    If two integers of same sign are divided then the quotient is positive.
    Eg:
    36 \(\div\) 3 = 12
    -48 \(\div\) (-16) = 3
  • Rule 2:
    If two integers of different sign are divided, then the quotient is negative.
    Eg:
    36 \(\div\) (-3) = -12
    -48 \(\div\) 16 = - 2
    Both commutative and associative properties do not hold good in division.
    Eg: 15 \(\div\) (-5) \(\neq\) (-5) \(\div\) 15
    a \(\div\) 1 = a
    0 \(\div\) a = 0
    a \(\div\) 0 is not defined.


Properties at Glance

  Closure Commutative Associative
Addition a + b is an integer a + b = b + a a + (b + c) = (a + c
Subtraction a - b is a integer a - b \(\neq\) b - a a - (b - c) \(\neq\) (a - b) - c
Multiplication a \(\times\) b is an integer a \(\times\) b = b \(\times\) a a \(\times\) (b \(\times\) c) = (a \(\times\) b) \(\times\) c
Division a \(\div\) b may or may not be an integer a \(\div\) \(\neq\) b \(\div\) a a \(\div\) (b \(\div\) c) \(\neq\) (a \(\div\) b) \(\div\) c

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