Integers are the set of whole numbers and their negative counterparts. They are denoted by the symbol “Z” and are used to represent quantities that include negative numbers. Integers have many properties that are similar to whole numbers, but they also have their own unique properties.

## Properties of Integers:

- Closure Property: The sum or product of any two integers is always an integer.
- Commutative Property: The order of the integers in addition and multiplication does not matter. For example, 2 + 3 = 3 + 2, and 2 x 3 = 3 x 2.
- Associative Property: The grouping of integers in addition and multiplication does not matter. For example, (2 + 3) + 4 = 2 + (3 + 4), and (2 x 3) x 4 = 2 x (3 x 4).
- Distributive Property: Multiplication is distributive over addition. For example, a x (b + c) = (a x b) + (a x c).
- Identity Property: The sum of any integer and zero is the same integer, and the product of any integer and one is the same integer.
- Even and Odd Integers: Even integers are integers that are divisible by 2, while odd integers are not divisible by 2.
- Positive and Negative Integers: Positive integers are greater than zero, while negative integers are less than zero.
- Absolute Value: The absolute value of an integer is the distance of the integer from zero on the number line. For example, the absolute value of -5 is 5.
- Opposite Integers: The opposite of an integer is the integer with the same absolute value but opposite sign. For example, the opposite of -5 is 5.
- Divisibility: An integer is said to be divisible by another integer if it can be divided exactly by that integer without leaving a remainder. For example, 6 is divisible by 3 because 6 divided by 3 is 2 with no remainder.

## Applications of Integers:

Integers and their properties are used in a wide range of applications in mathematics, science, and engineering. They are used to represent quantities that include negative numbers, such as temperature, elevation, or financial gain/loss. They are also used in calculating distances, areas, and volumes in geometry, in analyzing data in statistics, and in designing electrical circuits.

## Summary

Integers and their properties are fundamental concepts in mathematics. They are used to represent quantities that include negative numbers, and have many properties that are similar to whole numbers. Understanding these properties can help us perform calculations and solve problems in a variety of fields, and by mastering the properties of integers, we can develop a strong foundation for further mathematical study and problem-solving in a range of contexts.