Real number

Real number line

Number-systems

Real numbers are the set of all rational numbers and irrational numbers combined, encompassing all the numbers that can be found on the number line. This includes both positive and negative numbers, as well as zero. The concept of real numbers is fundamental in mathematics, particularly in fields such as algebra, calculus, and analysis. Real numbers are denoted by the symbol \(\mathbb{R}\).

Definition

A real number can be defined as any number that can be expressed as a finite or infinite decimal expansion. This includes:

• Rational numbers, which can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). Rational numbers have either terminating or repeating decimal expansions.
• Irrational numbers, which cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions. Examples include \(\sqrt{2}\), \(\pi\), and \(e\).

Properties

Real numbers have several important properties, including the following:

• Ordering: Real numbers can be compared using the greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤) relations.
• Completeness: Every non-empty set of real numbers that is bounded above has a least upper bound (or supremum) in the set of real numbers. This property is known as the completeness property of real numbers.
• Density: Between any two real numbers, there exists another real number. This property indicates that real numbers are densely packed, with no "gaps" between them.

Operations

The basic arithmetic operations of addition, subtraction, multiplication, and division can be performed with real numbers, except division by zero, which is undefined. Real numbers also support more complex operations, such as exponentiation and taking roots.

The Real Number Line

The real number line is a visual representation of real numbers as points on an infinitely long line. The point labeled "0" represents the number zero, points to the right of zero represent positive numbers, and points to the left represent negative numbers. The distance between any two points on the line corresponds to the absolute difference of their values.

Applications

Real numbers are used extensively in everyday life and in all areas of mathematics and science. They are used to measure quantities such as distance, time, mass, and temperature, and to perform calculations in engineering, physics, economics, and many other fields.

History

The development of the real numbers can be traced back to the ancient Greeks, who made distinctions between rational and irrational numbers. However, the formal definition of real numbers as it is known today was developed in the 19th century by mathematicians such as Richard Dedekind and Georg Cantor, who introduced more rigorous foundations for mathematics.