Integers

Integers are a set of numbers that includes all natural numbers, their negatives, and the number zero. They are used to represent quantities that can be greater than, less than, or equal to zero.

Definition

The set of integers, commonly denoted as ℤ, can be defined as follows:

ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}

As you can see, integers extend indefinitely in both the positive and negative directions.

Properties of Integers

Integers have several important properties:

1. Closure Property: If you add or multiply two integers, or subtract one integer from another, the result is always an integer.

2. Associative Property: For all integers a, b, and c, (a + b) + c = a + (b + c), (a * b) * c = a * (b * c), and (a - b) - c = a - (b + c).

3. Commutative Property: For all integers a and b, a + b = b + a and a * b = b * a.

4. Distributive Property: For all integers a, b, and c, a * (b + c) = a * b + a * c.

5. Identity Property: The number 0 is the additive identity in the set of integers, and the number 1 is the multiplicative identity.

6. Additive Inverses: Every integer a has an additive inverse -a, such that a + (-a) = 0.

Relationship to Whole Numbers

Integers are an extension of whole numbers, which include all non-negative integers starting from 0. The main difference between integers and whole numbers is the inclusion of negative numbers in the set of integers. This allows for a representation of quantities that can decrease as well as increase.

Applications

Integers are used in a variety of mathematical contexts, such as representing gains and losses, measuring temperatures above and below zero, and solving equations. They are also the foundation for more complex number systems, such as rational numbers, real numbers, and complex numbers.