Irrational Numbers

Definition

An irrational number is a real number that cannot be expressed as a ratio of integers, i.e., as a fraction. Therefore, irrational numbers, when written in decimal form, do not terminate or repeat[^1^]. They are non-terminating, non-repeating decimals[^2^].

History

The concept of irrational numbers was first introduced by the ancient Greeks. The Pythagorean mathematicians initially believed that all numbers could be expressed as ratios of integers. However, the discovery of irrational numbers shattered this belief. The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus), who probably discovered them while trying to represent the square root of 2 as a fraction[^3^].

Properties

Irrational numbers possess several unique properties that distinguish them from rational numbers. Some of these properties are:

Non-terminating, non-repeating: As mentioned earlier, irrational numbers are non-terminating, non-repeating decimals[^4^]. This means that their decimal expansion goes on forever without repeating a pattern.
Dense in real numbers: Between any two irrational numbers, there exists another irrational number[^5^]. This property is known as density.
Uncountable: The set of all irrational numbers is uncountable, which means that they cannot be put into a one-to-one correspondence with the set of natural numbers[^6^].

Examples

Some common examples of irrational numbers include:

The square root of any prime number[^7^].
The number pi (π), which represents the ratio of the circumference of a circle to its diameter[^8^].
The number e, which is the base of the natural logarithm[^9^].
The golden ratio (φ), which is often used in art and architecture[^10^].

Representation

Irrational numbers can be represented in several ways:

Decimal expansion: As non-terminating, non-repeating decimals[^11^].
Continued fractions: Every irrational number can be represented as a unique infinite continued fraction[^12^].
Algebraic numbers: Some irrational numbers are algebraic, which means they are the roots of polynomial equations with integer coefficients[^13^].
Transcendental numbers: Irrational numbers that are not algebraic are called transcendental[^14^]. Examples include pi and e.