Let’s take a moment to overview integers:
➢ Just Imagine you are playing a game where you can win or lose
points.
➢ If you win, you get +5 or +10 points. But if you lose a game, you
lose your points likes -5 or -10.
➢ These numbers, both positive and negative are called Integers.
➢ They help us to keep track of gains and losses in real life, just like
in the game!
➢ It is a key idea in mathematics, integers are the collection of entire
numbers that can be zero, positive, or negative.
➢ They are represented by the letter Z, which is derived from the
German word “Zahlen,” which means “number”.

Example for Integer: lets consider a set of integers which is represented as Z
Z= {…,-3,-3,-1,0,1,2,3,…}

Zero: Zero (0) is neutral; it is neither positive nor negative.

Positive Integers: Z= {1, 2, 3, …}

Negative Integers: Z= {-1, -2, -3 ,…}

1)Closure under Addition:

• For any two integers a and b, a + b is an integers.
  Let us see whether this property is true for integers or not.
  Statement Observation
  (i) 17 + 23 = 40 ___
  (ii) (-10) + 3 = -7 ___
  (iii) (-20) + 0 = _ ____
  (iv) 3 + (-5) = -2 ___

2)Closure under Subtraction:

• For any two integers a and b, a – b is an integers.
  What happens when we subtract an integer from another integer? Can we
  say that their difference is also an integer?
  Statement Observation
  (i) 7 – 9 = -2 ____
  (ii) (-10) – (-3) = 7 ____
  (iii) 17 – (-21) = _ _____
  (iv) 3 – (-5) = -2 _____

3)Commutative Property:

• For ant two integers a and b, we can say a + b = b + a
  We know that 3 + 5 = 5 + 3 = 8, addition is commutative for whole numbers.
  We have 5 + (-6) = -1 and (-6) + 5 = -1
  So, 5 + (-6) = (-6) + 5
  Are the following equal?
  (i) (-8) + (-9) and (-9) + (-8)
  (ii) (-45) + 0 and 0 + (-45)
  And also we know that subtraction is not commutative for whole numbers.
  Consider an integers 5 and (-3).
  Is 5 – (-3) the same as (-3) – 5? No, Because 5 – (-3) = 8 and (-3) – 5 = -8.
  

4)Associative Property:

• In general, for ant two integers a and b, we can say a + (b + c) = (a + b) +c
  Consider the integers 3, 2 and 5.
  Look at 3 + (2 + 5) and (3 + 2) + 5 are equal?
  Answer:
  Yes, In both cases we get 10.
  3 + (2 + 5) = 10 and (3 + 2) + 5 = 10.
  Similarly consider -3, 1 and -7.
  (-3) + [1 +(-7)] = -3 + _ =
  [(-3) + 1] + (-7) = -2 + =_
  Is (-3) + [1 +(-7)] same as [(-3) + 1] + (-7) ?

5)Additive Identity:

• 0 is the additive identity, (a + 0 = a).
  Observe the following and fill in the blanks:
  (i) (-8) + 0 = -8
  (ii) (-23) + 0 = _ (iii) 0 + (-59) = _
  (iv) -61 + _ = -61 (v) 0 + _ = -37
  (vi) _ + 0 = _