Irrational Number
An irrational number is a real number that cannot be expressed as a ratio of two integers—that is, it cannot be written as a fraction p/q where p and q are integers and q ≠ 0. This distinguishes irrational numbers from rational numbers, which can always be expressed in fractional form. Irrational numbers have decimal representations that neither terminate nor repeat in a predictable pattern; instead, their decimal expansions continue infinitely without entering a cycle.
Common Examples
The most well-known irrational numbers include π (pi), the ratio of a circle’s circumference to its diameter, and e (Euler’s number), the base of natural logarithms. Other important examples are √2, the square root of 2, which was the first number proven to be irrational in ancient mathematics, and φ (the golden ratio). Many square roots of non-perfect squares are irrational, as are most trigonometric and logarithmic values.
Properties and Significance
Irrational numbers are uncountably infinite in quantity, meaning there are more irrational numbers than rational numbers, even though both sets are infinite. The set of all real numbers consists of both rational and irrational numbers together. Irrational numbers appear naturally in geometry, physics, and many areas of mathematics, making them essential for accurate calculations involving continuous quantities and transcendental functions.