The natural numbers emerge here from Peano’s axioms and basic set theory. From these the ring of integers is formed. Then the increasingly larger fields of rational, real, and complex numbers are constructed. Real numbers are characterized within ordered fields as having the least upper bound property. They are also studied from the point of view of decimal expansions and infinite continued fractions with application to Pell’s equation. The complex numbers are not only Cauchy complete but enjoy the important property of algebraic completeness. The p-adic numbers are included as another example of a field with an absolute value. In the final section the cardinality of these sets of numbers is discussed on a general footing.