Irrational Number: Definition, Properties, and Examples

Irrational numbers are real numbers that cannot be expressed as a ratio of two integers. They include decimals that go on forever without repeating themselves. A well-known example of an irrational number is pi, which is used in mathematics to calculate the circumference and area of circles. Irrational numbers have many interesting properties and applications in mathematics, science, and engineering.

In this article, we'll provide an overview of irrational numbers, including its definition and properties. We will also explore how to recognize irrational numbers.

What is an Irrational Number?

An irrational number is a type of real number that cannot be represented as a simple fraction or ratio of any two integers. In simpler terms, it cannot be written in the form of p/q, where both p and q are integers, and q is not equal to zero.

Additionally, when we look at the decimal representation of an irrational number, we will find that it neither ends nor repeats in a regular pattern. The digits after the decimal point go on infinitely without any recurring sequence.

Some Examples of irrational numbers

• π (pi) ≈ 3.141592653589793... is the ratio of the circumference of a circle to its diameter and is famously irrational. Its decimal representation is non-terminating and non-repeating.
• √2 ≈ 1.414213562373095... is irrational because it cannot be written as a fraction, and its decimal representation goes on infinitely without repeating.
• Euler's number (e) ≈ 2.718281828459045... is irrational. It is the base of the natural logarithm and has a decimal representation that goes on infinitely without repeating.

The symbol for Irrational Number

In mathematical notation, irrational numbers are typically represented by a specific symbol Q’. They are often defined as the complement of rational numbers within the set of real numbers (R). This means that the set of irrational numbers is all real numbers (R) minus the set of rational numbers (Q). This operation can be symbolically represented as R - Q or R\Q.

Q’ = R – Q

In simpler terms, irrational numbers are all the real numbers that are not rational.

Arithmetic Operation on Irrational Numbers

Performing operations like addition, subtraction, multiplication, and division with two irrational numbers follows the same rules as working with any real numbers. The result of these operations can be either irrational or rational, depending on the specific numbers and the type of operation used.

• The sum of two or more irrational numbers can be either irrational or rational.
• Subtracting one irrational number from another can also give either an irrational or a rational result.
• The product of two irrational numbers, the result can be irrational or rational.
• Dividing one irrational number by another can give either an irrational or a rational result.

How to Identify Irrational Numbers?

Let’s learn how we can distinguish irrational numbers and other numbers.

• Non-integer square roots, cube roots, or roots of non-perfect powers are irrational.
• Numbers with non-repeating, non-terminating decimal expansions are irrational.
• Some numbers are proven to be irrational through mathematical proofs.
• Transcendental numbers, like π and e, are also irrational.

Theorem of Irrational Numbers

Statement:

Prove that the square root of prime numbers are irrational numbers.

Proof:

Suppose that √p is a rational number, where p is a prime number. This means that we can write √p as a fraction in the lowest terms, that is:

√p = a/b

Where a and b are integers without any common factors (i.e., 'a’ and ‘b’ are relatively prime).

p = a2/b2

Multiplying both sides by b² gives:

pb2 = a2 ______ (eq.1)

This means that p divides a². Since p is a prime number, this implies that p divides a. Therefore, we can write a = pc for some integer c. Substituting this expression for ‘a’ into the equation (eq.1) gives:

pb2 = (pc)2

Simplifying, we get:

b2 = pc2

This means that p divides b², and therefore p divides b. This contradicts the fact that a and b have no common factors, which we assumed at the beginning. Our assumption that √p is a rational number leads to a contradiction, and we conclude that √p must be an irrational number.

As a result, the square root of every prime number is irrational.

Different Properties of Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers. They have some distinct properties:

• Non-Repeatability: Irrational numbers have non-repeating, non-terminating decimal expansions.
• Non-Integer Roots: The square root of a non-perfect square (e.g., √2 or √3) is irrational. This means that the length of the diagonal of a square with sides of length 1 cannot be expressed as a simple fraction.
• Non-Rational Equivalents: Irrational numbers cannot be expressed as a fraction (a/b), where "a" and "b" are integers with no common factors other than 1.
• Infinite Decimal Places: The decimal representation of an irrational number goes on infinitely without repeating. For example, π (pi) is an irrational number with a decimal representation of 3.141592653589793... And so on.
• Uncountable: The set of irrational numbers is uncountable, meaning that there are more irrational numbers than there are natural numbers (1, 2, 3, ...), rational numbers, or integers.
• Transcendental Numbers: Some irrational numbers, such as π and e (Euler's number), are also transcendental, which means they are not the roots of any non-zero polynomial equation with integer coefficients.
• Existence: The real number line contains infinitely irrational numbers between any two rational numbers. This density property is related to the fact that irrational numbers are uncountable.

Numerical Solved Examples Related to Irrational Numbers

Example 1.

Find two Irrational Numbers Between 1.5 and 2.

Solution:

Irrational number decimal expansion never ends or repeats. Two irrational numbers between 1.5 and 2 could be:

• 732050807568877...
• 879643482356382...

Example 2

Identifying Irrational Numbers from the following real numbers:

• √25
• 4/9
• 284754383…
• √21

Solution:

• √25 = 5, which is a rational number.
• 4/9 = 0.4444..., is a repeating decimal, so it is a rational number.
• 284754383... is a non-repeating decimal, so it is an irrational number.
• Sqrt (21) is an irrational number.

Example 3

Add (4√5 + 7√2) and (3√5 - √2)

Solution:

(4√5 + 7√2) + (3√5 - √2)

= (4√5 + 3√5) + (7√2 - √2)

= (4 + 3) √5 + (7 - 1) √2

= 7√5 + 6√2

Example 4

Find the product of 3√7 and 5√2.

Solution

3√7 × 5√2

= (3 × 5) × (√7 × √2)

= 15√14

Conclusion

In this article, we have defined irrational numbers with examples. We have highlighted their unique properties. We have seen examples of famous irrational numbers throughout our discussion. We explored the theorem that states that the square root of a prime number is irrational. We solved examples related to irrational numbers to help our readers understand this concept better.