Real number

Real numbers are the values used to measure continuous, one-dimensional quantities. This includes physical properties like length, duration, or temperature. In these measurements, values are continuous, meaning any two values can have an arbitrarily small difference. You can visualize real numbers as every single point on a continuous line. This is often called the real line or the number line.

In mathematics, the set of real numbers is often written using a bold or blackboard bold symbol, such as $\mathbb{R}$.

Real numbers are organized into several distinct groups based on their properties. The first group is the rational numbers. This group includes integers and fractions, which are ratios of two integers.

The second group consists of irrational numbers. These are real numbers that are not rational. Some irrational numbers are algebraic numbers, meaning they are roots of polynomials with rational coefficients. Others, like the number $\pi$, are transcendental numbers because they are not roots of such polynomials. All these different types exist together on the same number line.

Historically, the term "real" was used in the 17th century by René Descartes. He used the word to distinguish these numbers from imaginary numbers. Imaginary numbers include things like the square roots of negative numbers. For many centuries, mathematicians and physicists worked with real numbers without having a formal, rigorous definition. They simply used the rules of arithmetic that seemed to work. It was not until the second half of the 19th century that mathematicians developed a formal mathematical foundation. This achievement led to the creation of real analysis, which is the study of real functions and sequences.

Mathematically, real numbers form what is known as an ordered field. This means they follow specific rules for arithmetic and order. There are two primary operations: addition and multiplication. Adding two numbers produces a sum, while multiplying them produces a product. Both operations are commutative, meaning the order does not change the result. They are also associative, meaning the grouping of numbers does not change the result. Furthermore, multiplication is distributive over addition.

Specific identity elements exist within this ordered field. The number zero is the additive identity, because adding it to any number does not change that number. The number one is the multiplicative identity, because multiplying any number by it leaves the number unchanged. Every real number also has an additive inverse, which is a number that, when added, results in zero. Every nonzero real number has a multiplicative inverse, which results in one when multiplied. These properties allow for the operations of subtraction and division.

A defining characteristic of real numbers is Dedekind completeness. This property separates real numbers from rational numbers. It states that every non-empty set of real numbers with an upper bound must have a least upper bound. This ensures there are no gaps or "holes" in the number line. This completeness leads to the Archimedean property. This property states that for every real number, there is an integer that is larger than it. It also implies that every positive real number has a positive square root.

Another way to represent real numbers is through decimal expansion. Every real number can be almost uniquely represented by an infinite decimal expansion. A decimal representation consists of an integer and an infinite sequence of digits. These digits represent products of powers of ten. For example, the constant $\pi$ is represented by a sequence of digits that never ends and never repeats. This decimal system is a way to specify a number as the least upper bound of decimal fractions. This method works for any numeral base, not just base ten.

Real numbers are also central to the concept of topological completeness. This is often described using Cauchy sequences. A Cauchy sequence is a sequence of numbers where the terms eventually become and remain arbitrarily close to each other. This concept is vital for understanding limits in calculus. Because real numbers are complete, many mathematical sequences are guaranteed to converge to a limit. This makes the real number system a fundamental tool for higher mathematics and physics.