Introduction to Numbers
Natural Numbers : The collection of all the counting numbers is called set of natural numbers. It is denoted by N = {1,2,3,4….}
Whole Numbers: The collection of natural numbers along with zero is called a set of whole numbers. It is denoted by W = { 0, 1, 2, 3, 4, 5, … }
Properties of Addition and Subtraction of Integers
Closure under Addition and subtraction
For every integer a and b, a+b and a–b are integers.
Commutativity Property for addition
for every integer a and b, a+b=b+a
Associativity Property for addition
for every integer a,b and c, (a+b)+c=a+(b+c)
Additive Identity & Additive Inverse
Additive Identity
For every integer a, a+0=0+a=a here 0 is Additive Identity, since adding 0 to a number leaves it unchanged.
Example : For an integer 2, 2+0 = 0+2 = 2.
Additive inverse
For every integer a, a+(−a)=0 Here, −a is additive inverse of a and a is the additive inverse of-a.
Example : For an integer 2, (– 2) is additive inverse and for (– 2), additive inverse is 2. [Since + 2 – 2 = 0]
Properties of Multiplication of Integers
Properties of Multiplication of Integers
Closure under Multiplication
For every integer a and b, a×b=Integer
Commutative Property of Multiplication
For every integer a and b, a×b=b×a
Multiplication by Zero
For every integer a, a×0=0×a=0
Multiplicative Identity
For every integer a, a×1=1×a=a. Here 1 is the multiplicative identity for integers.
Associative property of Multiplication
For every integer a, b and c, (a×b)×c=a×(b×c)
Distributive Property of Integers
Under addition and multiplication, integers show the distributive property.
i.e., For every integer a, b and c, a×(b+c)=a×b+a×c
These properties make calculations easier.
Division of Integers
When a positive integer is divided by a positive integer, the quotient obtained is a positive integer.
Example: (+6) ÷ (+3) = +2
When a negative integer is divided by a negative integer, the quotient obtained is a positive integer.
Example: (-6) ÷ (-3) = +2
When a positive integer is divided by a negative integer or negative integer is divided by a positive integer, the quotient obtained is a negative integer.
Example: (-6) ÷ (+3) =−2 and Example: (+6) ÷ (-3) = −2
Number Line
Representation of integers on a number line
On a number line when we add a positive integer for a given integer, we move to the right, add a negative integer for a given integer, we move to the left.
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