Numbers are at the heart of mathematics and have been a source of fascination and curiosity for mathematicians, astronomers, scientists, and even theologians, magicians, and astrologers. Some mathematicians pursue this fascination in a field of mathematics called number theory, in which number sets are used.

The emergence of the whole numbers 0, 1, 2, 3, 4, … naturally stems from the fundamental idea of counting. If a group of bananas and pineapples can be paired off, one banana for every pineapple, then the two groups, or sets, have the same "number" of objects. The members of the two sets do not resemble each other in appearance—one is the set of bananas, whereas the other is a set of pineapples—but both sets share a common property.

The common property that results from comparing the "size" of sets leads to the counting numbers. Any two sets whose members can be paired off...