Irrational numbers are some of the most fascinating numbers in mathematics. This guide explains their meaning, properties, and real-world examples in simple language.
By reading this blog, you will clearly understand what irrational numbers are, how they are written, and where they are used.
You will also discover interesting facts like the accidental invention of √2 and why π and e are so special.
Let’s start by learning what makes a number irrational and how to identify one.
Definition of Irrational Numbers
An irrational number is a number that cannot be written as p/q, where p and q are integers and q ≠0.
These numbers have non-terminating and non-repeating decimals.
Examples: √2, π, √3, e
Symbol for Irrational Numbers
Irrational numbers are usually represented by Q′ (read as Q-prime), where Q represents the set of rational numbers.
So, Q′ = Real numbers – Rational numbers.
Properties of Irrational Numbers
- They cannot be expressed as a fraction p/q.
- Their decimal expansion is non-terminating and non-repeating.
- They are a subset of real numbers.
- Adding a rational and an irrational number always gives an irrational number.
- Multiplying a non-zero rational number by an irrational number always gives an irrational number.
List of Irrational Numbers
- √2, √3, √5 (non-perfect square roots)
- π (Pi)
- e (Euler’s number)
- Golden Ratio (φ)
- Non-repeating decimals like 0.1010010001…
Are Irrational Numbers Real Numbers?
Yes, Irrational numbers are part of the real number system. Real numbers include both rational and irrational numbers.
Sum and Product of Irrational Numbers
- Sum: The sum of two irrational numbers can be rational or irrational.
Example: (√2 + -√2) = 0 (rational) - Product: The product of two irrational numbers can also be rational.
Example: (√2 × √2) = 2 (rational)
Proof of Irrational Numbers
Example: Proof that √2 is Irrational
- Assume √2 is rational, so √2 = p/q (q ≠0).
- Squaring both sides: 2 = p²/q² → p² = 2q².
- p² is even, so p must be even.
- Let p = 2k. Then p² = 4k² → 4k² = 2q² → q² = 2k².
- q² is also even, so q is even.
- p and q have a common factor 2, which contradicts the assumption that p/q is in simplest form.
Hence, √2 is irrational.
How to Find Irrational Numbers
- Take any non-perfect square and find its square root (√7, √11).
- Use non-repeating, non-terminating decimals.
- Look at constants like π, e, φ.
Interesting Facts About Irrational Numbers
1. Accidental Invention of √2
Ancient Greek mathematicians discovered √2 while studying diagonals of squares. They were shocked to find it could not be expressed as a fraction.
2. The Value of π
π represents the ratio of a circle’s circumference to its diameter. Its decimal value is endless: 3.14159…
3. Euler’s Number (e)
The number e (~2.71828) is key in exponential growth, finance, and probability.
Examples of Irrational Numbers
- √2 = 1.414213…
- π = 3.141592…
- e = 2.718281…
- √5 = 2.236067…
- Golden Ratio φ = 1.618033…
Difference Between Rational and Irrational Numbers
| Feature | Rational Numbers | Irrational Numbers |
|---|---|---|
| Form | p/q form | Cannot be written as p/q |
| Decimal Expansion | Terminating or repeating | Non-terminating, non-repeating |
| Examples | 3/4, 5, 0.75 | √2, π, e, √3 |
Importance of Irrational Numbers
- Used in geometry (Ï€ for circles).
- Applied in finance (e for compound interest).
- Found in nature (golden ratio in plants, shells).
Conclusion
Irrational numbers are a fascinating part of mathematics because they cannot be written as simple fractions.
They have non-repeating, non-terminating decimal expansions and appear in many areas of life from circles and spirals in nature to complex formulas in science and engineering.
Knowing their properties and how to find them helps students strengthen their understanding of the real number system.
By learning about √2, π, and e, we connect maths to history, science, and real-world applications.
Keep practicing examples to identify irrational numbers easily and build a strong maths foundation.
Read More What are rational Numbers? Properties & Examples
FAQs
What makes a number irrational?
It cannot be written as a fraction and has a non-terminating, non-repeating decimal expansion.
Is 0.333… irrational?
No, it repeats, so it is rational (equal to 1/3).
Is every square root irrational?
No, only non-perfect square roots are irrational (√4 = 2 is rational).
Is π irrational?
Yes, π is irrational because it never ends and never repeats.
Can irrational numbers be negative?
Yes, just like rational numbers, they can be negative.
Are irrational numbers infinite?
Yes, there are infinitely many irrational numbers between any two numbers.
Is √16 irrational?
No, √16 = 4, which is a rational number.
Are irrational numbers used in science?
Yes, they are widely used in physics, engineering, and finance.
What is the symbol for irrational numbers?
Q′ is often used to represent irrational numbers.
Are all prime numbers irrational?
No, prime numbers are whole numbers, so they are rational.