If you start with the natural numbers and introduce operations like addition and subtraction, you’ll find that you are missing something: the negative numbers. So you look at the integers, and try again. Now, trying to introduce multiplication and division, you’ll find you are missing something again: multiplicative inverses. So you look at the rational numbers, and you’ll find you are missing something yet again. That brings you to the real numbers. Our goal will be to discuss what’s missing in \(\mathbb {Q}\) , and to show you why \(\mathbb {R}\) has what’s missing. This is known as completeness of \(\mathbb {R}\) . In this and the next chapter, we’ll show you some wonderful applications of completeness. To start, we will explore the relation “less than or equal to” on \(\mathbb {R}\) . We will define bounded set, infimum, supremum, minimum, and maximum of a set. This will allow us to introduce some of our major tools: The completeness axiom of \(\mathbb {R}\) and the well-ordering principle of \(\mathbb {N}\) .