Chapter 1
Integers
• Integers are the collection of whole numbers and their negatives. Positive Integers are
1, 2, 3 ... . Negative Integers are 1, 2, 3 ... .
• Every positive integers is greater than every negative integers.
• Zero is less than every positive integers and greater than every negative integers.
• Number line : On a number line, when we
(a) add a positive integer, we move to the right.
(b) add a negative integer, we move to the left.
(c) subtract a positive integer, we move to the left.
(d) subtract a negative integer, we move to the right.
Whole Numbers
• Whole Numbers are all natural numbers along with zero (0) are called whole numbers.
• Zero is the only whole number that is not a natural number.
Addition
• When two positive integers are added we get a positive integer. Example: 44+ 71 =
116.
• When two negative integers are added we get a negative integer. Example: (-44) +
(-71)=-116.
• When one positive and one negative integers is added, we take their difference and
place the sign of the bigger integer. Example: (-44) + (71) = 27
• The additive inverse of any integer a is a and additive inverse of (-a) is a.
• Closure Property: For any two integers a and b, a+b is an integer. Example: 20+10 = 30
is an integer and -8 + 5 =-3 is an integer.
• Commutative Property: For any two integers a and b, a + b = b + a. Example: 7+(-6)=-1
and (-7) + 6 =-1 So, 6+(-7)= (-7) +6.
• Associative Property: For any three integers a, b, and c, we have a +(b + c) = (a +b) +C
Example: (-7) + [(-2) + (-1)] = [(-7) + (-1)] +(-2) = -10.
• Zero is an additive identity for integers. For any integer a, a + 0 = a = 0 + a.
Subtraction
• Closure Property: For any two integers a and b, a-b is an integer. Example: 20-10 = 10
is an integer.
• Commutative Property: The subtraction is not commutative for whole numbers. For
example, 20 – 30 = -10 and 30 – 20 = 10. So, 20 – 30 ≠ 30 – 20.
• Subtraction is not associative for integers.
Multiplication
• Product of a positive integer and a negative integer is a negative integer. a × (–b) = –
ab, where a and b are integers.
• Product of two negative integers is a positive integer. (–a) × (–b) = ab, where a and b
are integers.
• Product of even number of negative integers is positive, where as the product of odd
number of negative integers is negative.
• Closure Property: For all integers a and b, a×b is an integer. For example: (-3) (-5) =
150 is an integer.
• Commutative Property: For any two integers a and b, a×b = b×a. For example: (-3) ×
(-5) = (-5) × (-5) = -15.
• Associative Property: For any three integers a,b and c, (a×b) × c = a× (b × c). For
example, (-7) × [(-2) × (-1)] = [(-7) × (-1)] ×(-2) = -14.
• Distributivity Property: For any three integers a,b and c, a×(b + c)= a × b + a×c Example:
(-2) (3 + 5) = [(-2)×3] + [(-2)× 5] =-16.
• The product of a integer and zero is again zero.
• 1 is the multiplicative identity for negative integers.
Division
• When a positive integer is divided by a negative integer or vice-versa and the quotient
obtained is an integer, then it is a negative integer. a ÷ (–b) = (–a) ÷ b = – a/b , where a
and b are positive integers and a/b is an integer.
• When a negative integer is divided by another negative integer to give an integer, then it
gives a positive integer. (–a) ÷ (–b) = a/b, where a and b are positive integers and a/b is
also an integer.
• For any integer a, a ÷ 1 = a and a ÷ 0 is not defined.
