What are real numbers in maths

What are real numbers in mathematics ?

 In mathematics, real numbers are a fundamental concept that plays a central role in many areas of mathematics, science, and engineering. They are a type of number that can be represented on the number line and include all the familiar numbers such as integers, fractions, decimals, and irrational numbers. In this article, we will discuss real numbers in detail, including their properties, applications, and history.

Definition of Real Numbers

Real numbers are a set of numbers that includes all the rational and irrational numbers. The real numbers can be represented on a number line, with the integers located at equally spaced points and the other numbers located between the integers. The real numbers are denoted by the symbol ℝ.

The real numbers are defined using the concept of Dedekind cuts, which was first introduced by Richard Dedekind in the 1870s. A Dedekind cut is a partition of the rational numbers into two non-empty sets A and B, such that every element of A is less than every element of B, and there is no rational number that is both greater than all the elements of A and less than all the elements of B.

For example, consider the real number √2. We can define a Dedekind cut for √2 as follows:

A = {x ∈ ℚ : x < 0 or x² < 2}

B = {x ∈ ℚ : x > 0 and x² > 2}

Here, A is the set of all negative rational numbers and all rational numbers whose square is less than 2. B is the set of all positive rational numbers whose square is greater than 2. It can be shown that the set of all Dedekind cuts can be identified with the set of real numbers, and that the usual arithmetic operations on real numbers can be defined using operations on Dedekind cuts.

Properties of Real Numbers

The real numbers have many important properties that make them useful in a wide range of mathematical and scientific applications. Here are some of the most important properties of real numbers:

1. The real numbers are closed under addition, subtraction, multiplication, and division. This means that if we add, subtract, multiply, or divide any two real numbers, the result will always be a real number.

2. The real numbers are ordered. This means that any two real numbers can be compared using the less than (<) or greater than (>) symbols. Moreover, the order of the real numbers is compatible with the arithmetic operations, meaning that if a < b and c < d, then a + c < b + d and ac < bd.

3. The real numbers satisfy the Archimedean property. This property states that for any two positive real numbers a and b, there exists a positive integer n such that na > b. In other words, the real numbers do not have any infinitesimal or infinitely large elements.

4. The real numbers are dense. This means that between any two real numbers, there exists another real number. In other words, there are no "gaps" in the real number line.

5. The real numbers have the least upper bound property. This means that any non-empty set of real numbers that is bounded above has a least upper bound, which is itself a real number.

Applications of Real Numbers

Real numbers have a wide range of applications in mathematics, science, and engineering. Here are some of the most important applications of real numbers:

1. Calculus: Real numbers are used extensively in calculus, which is the branch of mathematics that deals with the study of functions and their properties. Calculus relies heavily on the concepts of limits, derivatives, and integrals, which all involve real numbers.

2. Geometry: Real numbers are also used in geometry, which is the branch of mathematics that deals