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 an irrational number in Maths?
In Maths, an irrational number is a real number that cannot be expressed as a simple fraction or ratio of two integers (p/q, where q ≠ 0). Its decimal representation is non-terminating and non-repeating.
2. Give 5 examples of irrational numbers.
Five common examples of irrational numbers include: π (pi), √2, e (Euler's number), √3, and √5. These numbers have decimal expansions that neither terminate nor repeat.
3. Is 0.33333... (repeating) irrational?
No, 0.33333... is rational. It can be expressed as the fraction 1/3. Rational numbers have terminating or repeating decimal expansions.
4. Is √81 an irrational number?
No, √81 = 9, which is a rational number. It can be expressed as a fraction (9/1).
5. How do you identify an irrational number?
An irrational number is identified by its decimal representation: it's non-terminating and non-repeating. Alternatively, it cannot be expressed as a fraction p/q where p and q are integers and q ≠ 0. Look for non-perfect square roots (like √2, √3, etc.) or the constants π and e.
6. Can two irrational numbers be added to get a rational number?
Yes. For example, (√2 + -√2) = 0, which is a rational number.
7. Are all square roots irrational?
No. Only the square roots of non-perfect squares are irrational. The square roots of perfect squares (e.g., √16 = 4) are rational.
8. What happens when you multiply two irrational numbers?
The product of two irrational numbers can be either rational or irrational. For example, √2 x √2 = 2 (rational), but √2 x √3 = √6 (irrational).
9. How to prove a number is irrational?
One common method is proof by contradiction. Assume the number is rational (p/q), then manipulate the equation to show a contradiction, proving the initial assumption was false and the number must be irrational. This technique is frequently used for proving the irrationality of √2, √3, etc.
10. What are the differences between rational and irrational numbers?
Rational numbers can be expressed as fractions (p/q, where p and q are integers, and q ≠ 0), while irrational numbers cannot. Rational numbers have decimal expansions that terminate or repeat, whereas irrational numbers have decimal expansions that are non-terminating and non-repeating.
11. What are some real-world applications of irrational numbers?
Irrational numbers appear in many real-world contexts, particularly involving circles and geometry. π is crucial for calculating the circumference and area of circles, while the golden ratio (approximately 1.618) appears in art, architecture, and natural phenomena.
