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.

Irrational numbers are a subset of real numbers, which means they follow all the fundamental properties of the real number system. In addition, they have specific properties of their own. 
 

• Cannot be expressed as fractions: Irrational numbers cannot be expressed as simple fractions in the form p/q, where p and q are integers, and q is not equal to 0. 
   
• Non-terminating and non-repeating decimals: Their decimal representations continue forever without repeating. For example, the value of π ≈ 3.1415926…
   
• The sum of an irrational and a rational number always results in an irrational number. 
  For example, \(\sqrt{5} + 2\) is irrational. 
   
• Multiplying an irrational number by any non-zero rational number gives an irrational number. That is, if we assume xy = z is rational, then x = z/y would also be rational, contradicting the assumption that x is irrational. Thus, the product xy must be irrational. 
   
• Sometimes, adding two irrational numbers gives a rational number. For example, \(\sqrt{2} + (-\sqrt{2}) = 0\), which is rational. 
   
• The product of two irrational numbers can be rational in some cases. For example, \(\sqrt{2} × \sqrt{2} = 2\), which is rational. 
   
• Because the operation may sometimes result in a rational number, the set of irrational numbers is not closed under addition or multiplication.
   
• LCM of two irrational numbers may or may not exist, as the least common multiple for two irrational numbers is not always defined.

Rational and irrational numbers have many differences between them. Some of them are given below.
 

Understanding irrational numbers takes more than just memorizing definitions. Below are some helpful tips for students, parents, and teachers to support learning. 
 

• Always check decimal expansions for termination or repetition: If a decimal expansion goes on forever without repeating a pattern, like 3.1415926535…, consider it as an irrational number. Students should be careful not to round off decimal approximations simply. 
   
• Recognize common roots and constants: Students should keep in mind that the square roots of non-perfect squares, such as √2, √3, √5, etc., are classical examples of irrational numbers. Also remember that the constants such as π, e also are irrational. 
   
• Practice with different examples: Practice constantly by combining irrationals through various operations like addition, subtraction, and multiplication, and sometimes they result in rational numbers. 
   
• Work out roots if possible: Students should try to simplify radicals rather than simply leave them. For example, √50 = 5√2, which helps recognize whether the simplified form involves any irrational numbers. 
   
• Be careful with approximations and rounding: Always keep in mind that decimal approximations, like 3.14 for π or 2.718 for e, are just estimates. They do not change the fact that the actual value is non-terminating and non-repeating. 
   
• Apply real-life examples: Parents and teachers can use everyday examples like, measuring lengths, drawing shapes, or cooking measurements to show students how irrational numbers occur. This creates understanding beyond textbooks. 
   
• Encourage simplification: Teach children to simplify radicals, such as √50 = 5√2, rather than relying solely on approximate decimal values. This helps them to keep track of irrationality exactly. 
   
• Use simple, familiar irrational numbers first: While teaching or practicing, start with standard irrational numbers like π, √2, or e, and then gradually explore more complex topics like sums or product of irrational numbers, roots of non-perfect powers, etc. 
   
• Highlight common mistakes and misconceptions: Parents and teachers can help students by clarifying that terminating or repeating decimals are rational, or that not all sums or products of irrationals remain irrational, or that approximated decimals are not exact. 
   
• Provide regular practice: Students use exercises or irrational numbers worksheets. Consistent practice helps students to master the concept over time.

Irrational numbers are important mathematical concepts with wide applications in many real-life situations. Some of them are given below.

Engineering and Construction: Irrational numbers are used to achieve accurate and precise calculations in construction and engineering. For example, we use the golden ratio to get aesthetically pleasing structures. Similarly, we use 2 in diagonal measurements in square-based designs.

Geometry: In geometry, irrational numbers are used in calculating area, circumference of circular objects is an irrational number, which is a non-terminating number. An approximation of 3.14159 or 3.142 for is used for calculations.

Physics and Scientific Research: There are uses of irrational numbers in physics equations. For example, the number e (Euler’s number) is fundamental in modeling exponential growth and decay, like radioactive decay, population growth, and compound interest calculations. Irrational numbers are also used in wave mechanics and quantum physics.

Computer Graphics and Animation: In computer graphics, the irrational numbers are used in rendering curves, rotations, and smooth animations, especially when the modeling circles or spirals.

Astronomy: Scientists use the π and other irrational constants to calculate the planetary orbits, circular paths, and distances in space with high accuracy.