properties of integers and their definition
Answers
Answer:
An integer is a number with no decimal or fractional part, from the set of negative and positive numbers, including zero. Examples of integers are: -5, 0, 1, 5, 8, 97, and 3,043.
A set of integers, which is represented as Z, includes:
Positive Integers: An integer is positive if it is greater than zero. Example: 1, 2, 3 . . .
Negative Integers: An integer is negative if it is less than zero. Example: -1, -2, -3 . . .
Zero is defined as neither negative nor positive integer. It is a whole number.
Z = {... -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, ...}
Step-by-step explanation:
Properties of Integers
The major Properties of Integers are:
Closure Property
Associative Property
Commutative Property
Distributive Property
Additive Inverse Property
Multiplicative Inverse Property
Identity Property
Closure Property:
The closure property states that the set is closed for any particular mathematical operation. Z is closed under addition, subtraction, multiplication, and division of integers. For any two integers, a and b:
a + b ∈ Z
a - b ∈ Z
a × b ∈ Z
a/b ∈ Z
Associative Property:
According to the associative property, changing the grouping of two integers does not alter the result of the operation. The associative property applies to the addition and multiplication of two integers.
For any two integers, a and b:
a + (b + c) = (a + b) + c
a ×(b × c) = (a × b) × c
Commutative Property:
According to the commutative property, swapping the positions of operands in an operation does not affect the result. The addition and multiplication of integers follow the commutative property.
For any two integers, a and b:
a + b = b + a
a × b = b × a
Distributive Property:
Distributive property states that for any expression of the form a (b + c), which means a × (b + c), operand a can be distributed among operands b and c as: (a × b + a × c) i.e.,
a × (b + c) = a × b + a × c
Additive Inverse Property:
The additive inverse property states that the addition operation between any integer and its negative value will give the result as zero.
For any integer, a:
a + (-a) = 0
Multiplicative Inverse Property:
The multiplicative inverse property states that the multiplication operation between any integer and it's reciprocal will give the result as one.
For any integer, a: a × 1/a = 1
Identity Property:
Integers follow the Identity property for addition and multiplication operations.
Additive identity property states that: a × 0 = a
Similarly, multiplicative identity states that: a × 1/a = 1