Whole Numbers

Whole numbers are a set of numbers that includes all natural numbers and the number zero. They are non-negative integers and are often used for counting, ordering, and basic arithmetic operations.

Definition

The set of whole numbers, commonly denoted as ℤ₊ or W, can be defined as follows:

ℤ₊ = {0, 1, 2, 3, 4, 5, ...}

As you can see, whole numbers start at 0 and continue indefinitely in the positive direction.

Properties of Whole Numbers

Whole numbers have several important properties:

  1. Closure Property: If you add or multiply two whole numbers, the result is always a whole number.

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

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

  4. Distributive Property: For all whole numbers 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 whole numbers, and the number 1 is the multiplicative identity.

  6. No Negatives: There is no whole number that, when added to another whole number, results in a negative number.

Relationship to Natural Numbers

Whole numbers are an extension of natural numbers, which include all positive integers starting from 1. The main difference between whole numbers and natural numbers is the inclusion of 0 in the set of whole numbers. This addition allows for a more complete representation of counting and basic arithmetic operations.

Applications

Whole numbers are used in a variety of mathematical contexts, such as counting objects, expressing quantities, and performing basic arithmetic. They are also the foundation for more complex number systems, such as integers, rational numbers, real numbers, and complex numbers.