Integers are a fundamental concept in mathematics that includes both positive and negative whole numbers, as well as zero. They are an extension of the natural numbers and play a crucial role in various mathematical operations, equations, and real-life scenarios.

Definition of Integers:
Integers are the set of whole numbers, their negatives, and zero. They are represented by the symbol "Z" and can be written as {...,-3, -2, -1, 0, 1, 2, 3,...}.

Positive Integers:
Positive integers are numbers greater than zero. They are denoted by the symbol "+". Examples of positive integers include 1, 2, 3, 4, and so on.

Negative Integers:
Negative integers are numbers less than zero. They are denoted by the symbol "-". Examples of negative integers include -1, -2, -3, -4, and so on.

Zero:
Zero is a unique integer that represents the absence of quantity or value. It is neither positive nor negative and is denoted by the symbol "0".

Properties of Integers:
1. Closure Property: The sum or difference of any two integers is always an integer.
2. Commutative Property: The order of addition or subtraction does not affect the result.
3. Associative Property: The grouping of three or more integers when adding or subtracting does not affect the result.
4. Zero Property: Adding or subtracting zero does not change the value of an integer.
5. Identity Property: The sum of an integer and its additive inverse (negative) is always zero.

Operations with Integers:
1. Addition: Adding two integers with the same sign results in a positive or negative integer, depending on the sign. Adding integers with different signs requires subtracting the smaller absolute value from the larger, with the sign of the integer with the larger absolute value being preserved.
2. Subtraction: Subtracting an integer is equivalent to adding its additive inverse. The rules for addition apply to subtraction as well.
3. Multiplication: Multiplying two integers with the same sign results in a positive integer, while multiplying integers with different signs yields a negative integer.
4. Division: Dividing integers can result in a positive or negative integer, depending on the signs involved.

Applications of Integers:
1. Temperature: Integers are used to represent positive and negative temperatures.
2. Money: Integers are used to represent gains and losses in financial transactions.
3. Coordinates: Integers are used to represent positions in a coordinate plane.
4. Sports: Integers are used to represent scores, rankings, and points in sports.

Integers are a foundational concept in mathematics and have numerous applications in various fields. Understanding and working with integers is essential for solving problems, analyzing data, and making accurate calculations.