3.1 Real Number System

Real Number System is the complete, nested hierarchy of number sets—natural numbers, whole numbers, integers, rational numbers, and irrational numbers—that together compose every value capable of being located at a single position on the number line, along with the notational conventions used to state which of these sets a given number belongs to.

Natural Numbers

Natural numbers are the counting numbers 1, 2, 3, 4, and so on, used to represent quantities of discrete, countable objects. Natural numbers form the most basic number set within the real number system, from which every larger set is built by successive extension.

Natural Number Convention for Zero

The natural number convention for zero addresses whether 0 is included among the natural numbers, a point on which mathematical convention varies: some treatments include 0 as the smallest natural number, while others begin the natural numbers at 1 and treat 0 as belonging only to the whole numbers. Whichever convention is adopted should be stated explicitly and applied consistently, since the boundary between the natural numbers and the whole numbers depends entirely on this choice.

Whole Numbers

Whole numbers extend the natural numbers by including 0, forming the set 0, 1, 2, 3, 4, and so on. Every natural number is a whole number under either convention for zero, but the whole numbers guarantee the inclusion of 0 regardless of which convention is used for the natural numbers themselves.

Integers

Integers extend the whole numbers by including their negative counterparts, forming the set consisting of all whole numbers together with −1, −2, −3, and so on. The integers allow the real number system to represent quantities below a reference point, such as debts or positions to the left of an origin.

Rational Numbers

Rational numbers extend the integers by including every value expressible as a ratio of two integers, a fraction of the form a divided by b where b is not zero. This includes all integers themselves, as well as every terminating and repeating decimal.

Irrational Numbers

Irrational numbers are values that cannot be expressed as a ratio of two integers, with decimal expansions that neither terminate nor repeat, such as the square root of 2 and pi. Irrational numbers occupy the remaining positions on the number line not filled by any rational number.

Real Numbers

Real numbers form the union of the rational and irrational numbers, encompassing every number that corresponds to a single position on an unbroken number line. The real numbers are the outermost and most inclusive set within this hierarchy, containing every set described above as a subset.

Nested Number Sets

Nested number sets describes the structural relationship among the sets composing the real number system, in which each set is entirely contained within the next larger set: the natural numbers within the whole numbers, the whole numbers within the integers, the integers within the rational numbers, and the rational numbers within the real numbers.

Number Set Membership Notation

Number set membership notation uses the symbol ∈ to indicate that a specific number belongs to a specific set, such as writing 3 ∈ ℤ to state that 3 is a member of the integers. This notation allows a number's set membership to be stated compactly without describing the set in full each time.

Number Set Nonmembership Notation

Number set nonmembership notation uses the symbol ∉ to indicate that a specific number does not belong to a specific set, such as writing √2 ∉ ℚ to state that the square root of 2 is not a member of the rational numbers. This notation is the direct negation of membership notation, asserting exclusion rather than inclusion.

Number Set Inclusion

Number set inclusion uses the symbol ⊂ to indicate that every member of one set is also a member of another, larger set, such as writing ℤ ⊂ ℚ to state that every integer is also a rational number. Number set inclusion is the notation underlying the nested structure of the real number system, formally expressing the containment relationships that link the natural numbers, whole numbers, integers, rational numbers, and real numbers into a single ordered hierarchy.

Together, these sets and notational conventions establish the complete real number system as a nested hierarchy—natural, whole, integer, rational, and irrational numbers combining into the real numbers—with membership, nonmembership, and inclusion notation providing the precise symbolic vocabulary needed to state exactly where any given number, or any given set, fits within that hierarchy.