An irrational number is a real number that cannot be expressed as a ratio of two integers. In other words, it cannot be written as a simple fraction. Irrational numbers have decimal expansions that neither terminate nor become periodic. They are a subset of the real numbers that includes numbers such as the ratio π of a circles circumference to its diameter, Eulers number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Some key properties of irrational numbers include:
- They cannot be expressed as a ratio of two integers.
- Their decimal expansions neither terminate nor become periodic.
- They are a subset of the real numbers.
Examples of irrational numbers include π, √2, √3, √5, Euler’s number (e = 2.718281…), and the golden ratio (φ = 1.618033…), among others. To determine if a number is irrational, one can try to write it as a simple fraction. If it cannot be expressed as a ratio of two integers, then it is irrational.
