Basics

A mathematically rigorous definition of the number systems requires the use of axiomatic set theory. As with most of mathematics, however, the intuitive understanding of the natural numbers ℕ, the integers ℤ, the rationals ℚ, the reals ℝ, and the complex numbers ℂ gained in elementary mathematics is sufficient for the beginning student of algebra. The occasional intrusion of set theory and foundational problems can be dealt with later. In this section, we discuss some properties of ℕ and ℤ that are somewhat less than elementary. Throughout this book, we will use the convention that 0 ∈ N. The set ℕ \ 0 of all positive natural numbers will be denoted by ℕ+.