Problem 141 - College Algebra

Question

How do the whole numbers differ from the natural numbers?

Step-by-Step Solution

Answer

The main difference between whole numbers and natural numbers is that whole numbers include all non-negative integers including 0, while natural numbers include all positive integers starting from 1. Thus, whole numbers encompass natural numbers and the number 0.

Step 1: Definition of Whole Numbers

Whole numbers are a set of numbers that includes all positive integers starting from 0. This set is often symbolized as W = {0, 1, 2, 3, 4, 5, ...}. The dots (...) denote that the numbers continue indefinitely in the positive direction.

Step 2: Definition of Natural Numbers

The natural numbers, on the other hand, are a set of numbers that includes all positive integers starting from 1. This set is often symbolized as N = {1, 2, 3, 4, 5, ...}. Like whole numbers, the series continues indefinitely, but it starts from 1 instead of 0.

Step 3: Difference between Whole Numbers and Natural Numbers

The primary difference between whole numbers and natural numbers is the inclusion of 0. In the set of whole numbers, 0 is included, thereby reflecting all non-negative integers. The set of natural numbers, however, starts from 1, so it includes only positive integers. Therefore, the whole numbers include the natural numbers and the number 0.