Irrational numbers

Irrational Numbers

An irrational number is a real number that cannot be expressed as a ratio of two integers, i.e., it cannot be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. Irrational numbers have non-terminating and non-repeating decimal expansions. The union of rational and irrational numbers forms the set of real numbers.

Examples of Irrational Numbers

Example 1: $\sqrt{2}$ $\sqrt{2} = 1.4142135\ldots$ (non-terminating and non-repeating) It cannot be written as a fraction of two integers. Thus, $\sqrt{2}$ is irrational.

Example 2: $\sqrt{3}$ $\sqrt{3} = 1.7320508\ldots$ The decimal expansion is non-terminating and non-repeating. Hence, $\sqrt{3}$ is irrational.

Example 3: $\pi$ $\pi = 3.1415926535\ldots$ Pi is a transcendental irrational number. It cannot be written as a ratio and does not repeat or terminate.

Example 4: $e$ $e = 2.718281828\ldots$ The base of the natural logarithm. Its decimal expansion is infinite and non-repeating. Hence, irrational.

Example 5: $\sqrt{5}$ $\sqrt{5} = 2.236067977\ldots$ It cannot be simplified to a rational number. So, it is irrational.

Example 6: $\sqrt{7}$ $\sqrt{7} = 2.645751311\ldots$ Like all square roots of non-perfect squares, it is irrational.

Important Note

The decimal expansion of every irrational number:

• Never terminates (does not end)
• Never repeats (no pattern)

Irrational numbers cannot be written as fractions, and their decimal expansions go on forever without repeating. Famous irrational numbers include $\pi$, $e$, and square roots of non-perfect squares like $\sqrt{2}$, $\sqrt{3}$, etc.