Introduction to Real Numbers

A real number is a number that can be used to measure a continuous one-dimensional quantity such as a length, duration, or temperature. The set of real numbers is usually denoted as **R** or ℝ and is fundamental in calculus and many other branches of mathematics. Real numbers can be almost uniquely represented by an infinite decimal expansion, and they include rational and irrational numbers. The adjective "real" distinguishes real numbers from imaginary numbers, such as the square roots of negative numbers.

The real numbers can be thought of as the points on a line, called the number line or real line, on which the points corresponding to integers are equally spaced. The informal descriptions of real numbers are not sufficient for rigorous reasoning, and the development of a suitable formal definition was a major achievement of 19th-century mathematics. One modern axiomatic definition is that real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field.

Real numbers are completely characterized by their fundamental properties, which can be summarized by saying that they form an ordered field that is Dedekind complete. This implies that one can manipulate real numbers and compute with them without knowing how they can be defined. The real numbers include the rational numbers, such as the integer −5 and the fraction 4/3, and real numbers that are not rational are irrational.

Those real numbers that are roots of polynomials with rational coefficients are algebraic numbers, which include all the rational numbers and also irrational numbers such as √2 = 1.414.... Other real numbers, such as π = 3.1415..., are not roots of polynomials; these are the transcendental numbers. The real numbers can be thought of as the points on a line, called the number line or real line, on which the points corresponding to integers are equally spaced.

Characterizing Properties of Real Numbers

Real numbers are completely characterized by their fundamental properties, which can be summarized by saying that they form an ordered field that is Dedekind complete. This means that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. The characterizing properties of real numbers include the following:

Commutativity of addition and multiplication: a + b = b + a and ab = ba for all real numbers a and b.
Associativity of addition and multiplication: (a + b) + c = a + (b + c) and (ab)c = a(bc) for all real numbers a, b, and c.
Distributivity of multiplication over addition: a(b + c) = ab + ac and (a + b)c = ac + bc for all real numbers a, b, and c.
Existence of additive and multiplicative identities: 0 and 1 for all real numbers a.
Existence of additive and multiplicative inverses: -a and a^(-1) for all real numbers a.

Arithmetic Operations on Real Numbers

The real numbers form an ordered field, which means that methods and rules of elementary arithmetic apply to them. The arithmetic operations on real numbers include addition, subtraction, multiplication, and division. The addition of two real numbers a and b produces a real number denoted a + b, which is the sum of a and b. The multiplication of two real numbers a and b produces a real number denoted ab, which is the product of a and b.

The total order on the real numbers is denoted a < b, and it has the following two properties: for any two real numbers a and b, exactly one of a < b, a = b, or b < a is true, and if a < b and b < c, then a < c. The order is compatible with addition and multiplication, which means that a < b implies a + c < b + c for every real number c, and 0 < ab implies a < b for every real number a.

Applications of Real Numbers

Real numbers have numerous applications in various fields, including mathematics, physics, engineering, and economics. They are used to model and analyze real-world phenomena, such as population growth, financial markets, and physical systems. Real numbers are also used in computer science, particularly in programming languages and algorithms. The study of real numbers is essential for understanding many mathematical concepts, including calculus, algebra, and geometry.

In conclusion, real numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. Their characterizing properties, including commutativity, associativity, and distributivity, make them a powerful tool for modeling and analyzing real-world phenomena. The study of real numbers is essential for understanding many mathematical concepts, and it has led to numerous breakthroughs in science, technology, and engineering.