Irrational Numbers
2026-02-28 17:36 Diff
https://brightchamps.com/en-us/math/numbers/irrational-numbers

Irrational numbers are real numbers that cannot be written as a simple fraction in the form of (p/q), where p and q are whole numbers, and q ≠ 0. Their decimal representations are non-terminating and non-repeating, meaning they go on forever without creating a repeating pattern. Irrational numbers are applied in fields like mathematics, science, and engineering. 
 

Examples of Irrational Numbers


Irrational numbers are commonly encountered in both theoretical and real-life applications. Some of the most famous irrational numbers are given below.
 

Is π (Pi) an irrational number?

Yes, the number π is irrational, because its decimal expansion is infinite and shows no repeating pattern. The decimal expansion of π = 3.14159…. It is widely used in geometry for calculating the circumference and area of circles.
 

Is e (Euler’s Number) an irrational number?

Euler’s number e approximately equals 2.71828… It is also an irrational number. It appears frequently in advanced mathematics, especially in exponential growth, calculus, and logarithmic functions.
 

Is the Golden Ratio (ϕ) an irrational number?

Yes. The golden ratio has the approximate value of 1.61803…and it is irrational. It can be found in art, architecture, design and even nature. It is derived from the expression \(ϕ = \frac{1 +\sqrt{5}}{2}\).

How Do You Know a Number is Irrational? 

To identify whether a number is irrational, you can check the following characteristics:

  • It cannot be written as a fraction: Irrational numbers cannot be expressed in the form p/q, where p and q are integers and q not equal to 0. 
     
  • Its decimal form is non-terminating and non-repeating: If a number’s decimal expansion is continuing without showing any repeating pattern, forever, then it is an irrational number. For instance, π = 3.1415926535… is continuous without pattern. 
     
  • Square roots of non-perfect squares are irrational: \(\sqrt{2}, \sqrt{3},\sqrt{5}\) and \(\sqrt{11}\) are irrational. But \(\sqrt{4} = 2\), and it is rational. 
     
  • Certain special mathematical constants are irrational: Numbers like π, e (Euler’s number), and the Golden ratio (ϕ) have been proven to be irrational.

Irrational Number Symbol

The common symbols used for the types of numbers are: N (Natural numbers), I (Imaginary numbers), R (Real numbers), and Q (Rational numbers). 
 

  • (R\Q) is the most formal way to express irrational numbers, which means all real numbers except rational numbers. 
     
  • R-Q defines that irrational numbers can be obtained by subtracting rational numbers from real numbers. 
     
  • ∉ Q conveys that the number is not rational.