How to Identify and Prove a Number is Irrational
The concept of irrational numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Irrational numbers are important for understanding the complete number system and help students solve a variety of questions in classes 8–12 and competitive exams.
What Is Irrational Number?
An irrational number is defined as a real number that cannot be expressed as a simple fraction \(\dfrac{p}{q}\), where p and q are integers and \(q \neq 0\). Its decimal expansion goes on forever without repeating or terminating—meaning the digits never form a pattern or end. Examples of irrational numbers include π, √2, and e. You’ll find this concept applied in identifying non-repeating decimals, working with roots and powers, and understanding sets inside a Venn diagram.
Key Features and Properties of Irrational Numbers
Here are the standard properties that make a number “irrational”:
- Cannot be written as a fraction p/q (q ≠ 0)
- Decimal expansion is non-terminating and non-repeating
- Lie on the number line and are part of real numbers
- Examples include √3, √5, π, e, and φ (Golden Ratio)
| Property | Irrational Number Example |
|---|---|
| Non-terminating, non-repeating decimal | 3.14159265... (π) |
| Non-fractional root | √2 = 1.4142... |
| Result of irrational × rational (not zero) | 2 × √7 = 2√7 |
Step-by-Step Illustration: How to Identify Irrational Numbers
- Check if the number is a root or decimal.
Example: Is √8 irrational? - If it’s a root: Is it a perfect square?
No, since 8 is not a perfect square. - Write the decimal value:
√8 = 2.8284271… (decimal is non-terminating and non-repeating) - Conclusion: √8 is irrational.
List of Common Irrational Numbers
| Number | Decimal Approximation |
|---|---|
| π | 3.14159265… |
| e | 2.7182818… |
| √2 | 1.4142135… |
| √3 | 1.7320508… |
| Golden Ratio (φ) | 1.6180339… |
Difference Between Rational and Irrational Numbers
| Rational Number | Irrational Number |
|---|---|
| Can be written as p/q | Cannot be written as p/q |
| Terminating or repeating decimal | Non-terminating, non-repeating decimal |
| Eg: 1/2, 0.75, 0.333… | Eg: π, √5, e |
Solved Example: Prove √7 is Irrational
Let’s see how to prove √7 is irrational:
1. Assume √7 is rational, so it can be written as \(\dfrac{p}{q}\), with p and q in simplest form and \(q \neq 0\).2. Squaring both sides: \(7 = \dfrac{p^2}{q^2}\) ⇒ \(p^2 = 7q^2\).
3. So p2 is divisible by 7, which means p is divisible by 7. Let p = 7k.
4. Substituting: \(p^2 = (7k)^2 = 49k^2\), so \(49k^2 = 7q^2\), which means \(q^2 = 7k^2\), so q is also divisible by 7.
5. But then p and q have a common factor of 7, contradicting our assumption.
6. Thus, √7 is irrational.
Try These Yourself
- Write five irrational numbers between 0 and 10.
- Is 0.141592653… rational or irrational?
- Find two irrational numbers between 2 and 3.
- Is 5.123123123… an irrational number? Why or why not?
Frequent Errors and Misunderstandings
- Confusing irrational numbers with non-integers (not all decimals are irrational).
- Thinking all roots are irrational (roots of perfect squares like √16 = 4 are rational).
- Assuming that if a decimal doesn’t end, it’s always irrational (repeating decimals are rational).
Relation to Other Concepts
The idea of irrational numbers connects closely with rational numbers, real numbers, and the number system. Mastering this helps with understanding square roots, surds, and decimal number systems in future chapters.
Classroom Tip
A quick way to remember irrational numbers is: “If the decimal never ends and never repeats, it’s irrational.” Vedantu’s teachers often draw a Venn diagram to show irrational and rational numbers as subsets of real numbers, making the concept easier to remember.
We explored irrational numbers—from definition, properties, examples, and mistakes, to their close connection with rational and real numbers. Continue practicing with Vedantu to become confident in identifying and working with irrational numbers, and master all future chapters in mathematics!
Discover more:
- Rational Numbers: Compare with irrational numbers and master fractions and decimals.
- Real Numbers: See where irrational numbers fit into the bigger picture.
- Decimal Number System: Understand non-terminating and non-repeating decimals.
- Surds: Learn more about this special form of irrational numbers.
FAQs on Irrational Numbers Explained: Definition, Properties & Examples
1. What is the basic definition of an irrational number?
An irrational number is a real number that cannot be expressed as a simple fraction in the form p/q, where 'p' and 'q' are integers and 'q' is not zero. In decimal form, irrational numbers are both non-terminating (they never end) and non-repeating (they do not have a recurring pattern).
2. What are some common examples of irrational numbers found in the Maths syllabus?
Several key irrational numbers are frequently used in the Maths curriculum. Five important examples include:
- Pi (π): Approximately 3.14159..., used for circle calculations.
- The square root of 2 (√2): Approximately 1.41421..., representing the diagonal of a unit square.
- The square root of 3 (√3): Approximately 1.73205...
- Euler's Number (e): Approximately 2.71828..., fundamental in calculus and growth concepts.
- The Golden Ratio (Φ): Approximately 1.61803..., found in geometry, art, and nature.
3. What is the main difference between how rational and irrational numbers are written as decimals?
The main difference lies in their decimal representation. A rational number has a decimal form that either terminates (like 0.5) or repeats in a predictable pattern (like 0.333...). In contrast, an irrational number's decimal form is both non-terminating and non-repeating, meaning it goes on forever without any repeating sequence.
4. How can you quickly identify if the square root of a whole number is rational or irrational?
You can identify it by checking if the number under the square root sign is a perfect square. If it is a perfect square (e.g., 4, 9, 25), its square root is a whole number and therefore rational (e.g., √25 = 5). If the number is not a perfect square (e.g., 7, 10, 15), its square root is irrational.
5. Why is Pi (π) considered an irrational number if we use the fraction 22/7 for it?
This is a common point of confusion. The fraction 22/7 is a rational approximation of Pi, used for convenience in calculations, but it is not its exact value. The true value of π is approximately 3.14159..., and its decimals continue infinitely without repeating. The value of 22/7 is approximately 3.142857..., which is a repeating decimal and therefore rational. The actual value of π cannot be written as a simple fraction.
6. Can the sum or product of two irrational numbers ever result in a rational number?
Yes, it is possible. While the sum or product of a rational and an irrational number is always irrational, the same is not true for two irrationals. For example:
- Sum: Adding √3 and -√3 (both irrational) gives 0, which is a rational number.
- Product: Multiplying √5 by √5 (both irrational) gives 5, which is a rational number.
7. How is it proven that a number like the square root of 2 is irrational?
The irrationality of √2 is proven using a method called 'proof by contradiction', a key concept in the CBSE syllabus. The method works by first assuming the opposite is true—that √2 is rational and can be written as a fraction p/q in its simplest form. Through a series of logical steps, this assumption is shown to lead to a contradiction (that both p and q must be even, which violates the 'simplest form' condition). Since the initial assumption leads to a false conclusion, the assumption itself must be false, proving that √2 is irrational.
8. Are all irrational numbers also considered 'surds'?
No, not all irrational numbers are surds. A surd is a specific type of irrational number that can be expressed as a root of a rational number, such as √2 or ³√5. However, there is another class of irrational numbers called transcendental numbers, like Pi (π) and Euler's number (e). These numbers are not the root of any polynomial equation with rational coefficients. Therefore, while all surds are irrational, famous irrationals like π and e are not surds.
9. What happens when you add or multiply a rational number with an irrational number?
According to the properties of irrational numbers, the outcome is always irrational.
- The sum or difference of a rational number and an irrational number (e.g., 5 + √2) is always irrational.
- The product or division of a non-zero rational number and an irrational number (e.g., 3 × π) is also always irrational.
10. What is the importance of irrational numbers in real-world applications?
Irrational numbers are crucial in many fields beyond the classroom. In engineering and physics, Pi (π) is essential for calculations involving circles, cylinders, waves, and signals. In architecture and design, the diagonal of a square (which involves √2) is fundamental for construction, and the Golden Ratio (Φ) is often used to create aesthetically pleasing proportions. In finance, Euler's number (e) is used to calculate compound interest.
