Irrational Number
An irrational number is a real number that cannot be expressed as a ratio of two integers (a fraction p/q where p and q are integers and q ≠ 0). This distinguishes them from rational numbers, which can always be written in fractional form. Irrational numbers have decimal representations that neither terminate nor repeat in a predictable pattern, continuing infinitely without settling into a cycle.
Common Examples
The most famous irrational numbers include π (pi), the ratio of a circle’s circumference to its diameter, and e, the base of natural logarithms. The square root of 2 (√2) was historically significant as one of the first numbers proven to be irrational by ancient Greek mathematicians. The golden ratio φ (phi) and various other roots of non-perfect numbers are also irrational.
Properties and Significance
Irrational numbers are uncountably infinite in quantity—there are actually more irrational numbers than rational numbers, despite both sets being infinite. When irrational numbers are combined with rational numbers, they form the complete set of real numbers. Irrational numbers are essential to mathematics and science, appearing naturally in geometry, calculus, probability, and physics. Their discovery challenged ancient beliefs about number systems and demonstrated the limitations of expressing all quantities as simple fractions.