Real numbers

The real numbers are a set of numbers that include both the rational numbers (such as 1, -3/4, 0.333...) and the irrational numbers (such as π, √2, and e). They can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. The real numbers are used to measure continuous quantities and are a fundamental component of mathematics.

Definition

Real numbers can be defined in various ways, such as through Dedekind cuts, Cauchy sequences, or infinite decimals. Each definition highlights different aspects of real numbers and their properties.

Dedekind Cuts

A Dedekind cut in the set of rational numbers is a partition of it into two non-empty subsets A and B, such that every element of A is less than every element of B, and A contains no greatest element. The real numbers can be constructed by considering each cut to represent a real number.

Cauchy Sequences

A Cauchy sequence of rational numbers is a sequence in which the numbers become arbitrarily close to each other as the sequence progresses. The real numbers can be defined as the set of limits of Cauchy sequences of rational numbers, where two sequences are considered the same real number if their difference converges to zero.

Infinite Decimals

A real number can also be represented by an infinite decimal expansion. Unlike finite decimals or repeating decimals, which represent rational numbers, non-repeating infinite decimals represent irrational numbers.

Properties

Real numbers have several important properties, including the following:

• Completeness: Every non-empty subset of the real numbers that is bounded above has a least upper bound. This property is known as the completeness or supremum property and is crucial for many areas of analysis.
• Order: The real numbers can be ordered in a line where each number has a unique position.
• Field Properties: The real numbers form a field, meaning they support addition, subtraction, multiplication, and division (except by zero).
• Archimedean Property: For any two real numbers, there is an integer which is greater than the ratio of the two numbers. This implies that the integers are unbounded in the real numbers.

Applications

Real numbers are used throughout mathematics and its applications, including in physics, engineering, economics, and beyond. They are essential for describing quantities that vary continuously, such as time, distance, and probability.

See also

• Complex numbers
• Number theory
• Calculus
• Metric space