Real Numbers Explained: Definition, Axioms & Properties

Real numbers are the complete set of values on the continuous number line, denoted ℝ, and they include both rational numbers (fractions and repeating or terminating decimals) and irrational numbers (non-repeating, non-terminating decimals such as √2, π, and e). Real numbers can be positive, negative, or zero, and they satisfy a completeness property that makes them suitable for representing continuous quantities such as length, time, and measurement in mathematics and science.

Key Takeaways

• Real numbers, denoted ℝ, include all rational numbers (such as 1/2 and 5) and all irrational numbers (such as √2, π, and e).
• Between any two real numbers there exist infinitely many other real numbers, a property called density.
• Two standard formal constructions of ℝ are Cauchy sequences of rational numbers and Dedekind cuts.
• The least-upper-bound property states that every non-empty subset of ℝ with an upper bound has a supremum in ℝ.
• The ordering axioms establish ℝ as a totally ordered field, meaning any two real numbers a and b satisfy either a ≤ b or b ≤ a.

What are real numbers?

Real numbers are the set of all values that can appear on the continuous number line. They include rational numbers, which are fractions and terminating or repeating decimals, and irrational numbers, which are non-repeating, non-terminating decimals. The set of real numbers is denoted by the symbol ℝ. Real numbers can be positive, negative, or zero, and they may be represented as finite decimals, repeating decimals, or infinite non-repeating decimals.

What are the key characteristics of real numbers?

Positive real numbers are values greater than zero, such as 1, 2, and 3.5. Negative real numbers are values less than zero, such as −1, −2.7, and −0.5. Zero is itself a real number and serves as the additive identity, meaning any number added to zero equals itself.

Rational numbers can be expressed as a fraction of two integers with a nonzero denominator, such as 1/2, −3/4, or 5. Irrational numbers cannot be written as a simple fraction and have non-repeating, non-terminating decimal expansions, such as √2, π, and e.

Real numbers are dense on the number line, meaning that between any two real numbers there are infinitely many other real numbers. This density property makes real numbers well suited to representing continuous quantities in mathematics and science.

Why do real numbers matter?

Real numbers are a fundamental concept in mathematics and are used widely to describe quantities, measurements, and relationships across many disciplines, from physics and engineering to economics and statistics.

Where did the concept of real numbers come from?

The idea of real numbers grew from the geometric notion of length. Ancient mathematicians observed that rational numbers alone could not measure every length, such as the diagonal of a unit square, and this observation led to methods such as Eudoxus’s “method of exhaustion.”

How are real numbers formally constructed?

Modern mathematics offers several equivalent constructions of the real numbers. Two standard constructions are Cauchy sequences and Dedekind cuts.

Cauchy sequences

Real numbers can be obtained as equivalence classes of Cauchy sequences of rational numbers, which are sequences whose terms get arbitrarily close to each other as the sequence progresses.

Dedekind cuts

Real numbers can also be defined as Dedekind cuts in the rational numbers, meaning partitions of the rationals into two nonempty sets with specific order properties. Both the Cauchy-sequence and Dedekind-cut constructions require care to formalize (the Cauchy-sequence approach relies on equivalence classes), but they produce isomorphic complete ordered fields.

What are the basic building blocks of the real number system?

The real number system is presented as a set with distinguished elements and operations. Its basic components are:

• The set ℝ, whose elements are called real numbers.
• The distinguished constants 0 (additive identity) and 1 (multiplicative identity).
• Binary operations: addition (+) and multiplication (·).
• A unary operation: negation (−).
• A partial unary operation: reciprocal, or multiplicative inverse, written x⁻¹ for x ≠ 0.
• An order relation: ≤.

What are the field axioms of real numbers?

The field axioms describe the algebraic properties of real numbers under addition and multiplication:

• Closure: a + b and a · b are in ℝ for all a, b in ℝ.
• Negation: −a is in ℝ for all a in ℝ.
• Inverses: if d ≠ 0, then d⁻¹ is in ℝ.
• Commutativity: a + b = b + a, and a · b = b · a.
• Associativity: a + (b + c) = (a + b) + c, and a · (b · c) = (a · b) · c.
• Distributivity: a · (b + c) = (a · b) + (a · c).
• Identities: a + 0 = a, and a · 1 = a.
• Inverses: a + (−a) = 0, and for d ≠ 0, d · d⁻¹ = 1.

What are the ordering axioms of real numbers?

The ordering axioms establish real numbers as a totally ordered field:

• Antisymmetry: if a ≤ b and b ≤ a, then a = b.
• Transitivity: if a ≤ b and b ≤ c, then a ≤ c.
• Comparability: for any a, b in ℝ, either a ≤ b or b ≤ a holds (total order).
• Addition compatibility: if a ≤ b, then a + c ≤ b + c for any c.
• Multiplication compatibility: if 0 ≤ c and a ≤ b, then a · c ≤ b · c.

What is the completeness axiom of real numbers?

The defining analytic property of the real numbers is completeness. One standard form is the least-upper-bound property: every non-empty subset of ℝ that has an upper bound has a least upper bound, or supremum, within ℝ.

An equivalent sequential form is the monotone convergence theorem, which states that every bounded monotone sequence of real numbers converges to a real limit. For example, if (fₙ) is an increasing sequence with an upper bound, the limit limₙ→∞ fₙ exists in ℝ.

How are subtraction and division defined in terms of the basic operations?

Subtraction is defined in terms of addition and negation: a − b is defined as a + (−b). Division is defined in terms of multiplication and the reciprocal: if d ≠ 0, then a / d is defined as a · d⁻¹.

Why are they called “real” numbers?

The term “real” in “real numbers” is simply a historical name distinguishing this number system from others, such as integers, rationals, or finite fields. It does not imply that real numbers are “more real” or more ordinary in the everyday sense than other number systems.

Frequently Asked Questions

What is the difference between rational and irrational numbers?

Rational numbers can be expressed as a fraction of two integers with a nonzero denominator, such as 1/2 or 5. Irrational numbers cannot be written as a simple fraction and have decimal expansions that never repeat or terminate, such as √2, π, and e.

What does it mean for real numbers to be dense?

Density means that between any two real numbers, no matter how close together, there are infinitely many other real numbers. This property distinguishes the real number line from sets like the integers, where numbers have clear gaps between them.

What is a Dedekind cut?

A Dedekind cut is a partition of the rational numbers into two nonempty sets with specific order properties, used to formally construct the real numbers. This construction, along with equivalence classes of Cauchy sequences, produces an isomorphic complete ordered field equivalent to ℝ.

What is the least-upper-bound property?

The least-upper-bound property states that every non-empty subset of ℝ that has an upper bound also has a least upper bound, or supremum, that itself belongs to ℝ. This property defines the completeness of the real number system.

Is zero a real number?

Yes, zero is a real number. It serves as the additive identity in the real number system, meaning that adding zero to any real number leaves that number unchanged.

Why can’t rational numbers alone measure every length?

Ancient mathematicians discovered that some lengths, such as the diagonal of a unit square, cannot be expressed as a ratio of two integers. This gap between rational numbers and geometric lengths motivated the development of irrational numbers and, eventually, the real number system.