Rational vs Irrational Numbers
Rational numbers and irrational numbers together form the set of real numbers. Understanding the difference between these two types is a foundational concept in the NCERT Class 9 Number Systems chapter.
A rational number can be expressed as p/q, where p and q are integers and q ≠ 0. An irrational number cannot be written in this form.
Every point on the number line represents either a rational or an irrational number. There are no gaps — the real number line is complete.
What is Rational vs Irrational Numbers?
Definition: A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.
Definition: An irrational number is a real number that cannot be expressed in the form p/q for any integers p and q.
Key distinction:
- Rational numbers have decimal expansions that are either terminating or non-terminating repeating.
- Irrational numbers have decimal expansions that are non-terminating and non-repeating.
Examples of rational numbers:
- 3/4 = 0.75 (terminating)
- 1/3 = 0.333... (non-terminating repeating)
- −7 = −7/1
- 0 = 0/1
- 2.5 = 5/2
Examples of irrational numbers:
- √2 = 1.41421356... (non-terminating, non-repeating)
- √3 = 1.73205080...
- π = 3.14159265...
- 0.1010010001... (pattern but no repeating block)
Rational vs Irrational Numbers Formula
Key Facts and Formulas:
1. Rational number form:
p/q, where p, q ∈ ℤ and q ≠ 0
2. Decimal expansion test:
Terminating or repeating decimal ⇒ Rational
Non-terminating, non-repeating decimal ⇒ Irrational
3. Sum/Product rules:
- Rational + Rational = Rational
- Irrational + Irrational = may be Rational or Irrational
- Rational + Irrational = Irrational (provided the rational number ≠ 0 for products)
- Rational × Irrational = Irrational (if rational ≠ 0)
- Irrational × Irrational = may be Rational or Irrational
4. Square root test:
- √n is rational only if n is a perfect square.
- If n is not a perfect square, √n is irrational.
Derivation and Proof
Proof that √2 is irrational (by contradiction):
Step 1: Assume √2 is rational
- Assume √2 = p/q, where p and q are co-prime integers (HCF = 1) and q ≠ 0.
Step 2: Square both sides
- √2 = p/q
- 2 = p²/q²
- p² = 2q²
Step 3: Deduce p is even
- Since p² = 2q², p² is even.
- If p² is even, then p must be even (the square of an odd number is always odd).
- Let p = 2k for some integer k.
Step 4: Substitute and simplify
- (2k)² = 2q²
- 4k² = 2q²
- q² = 2k²
Step 5: Deduce q is also even
- Since q² = 2k², q² is even, so q is also even.
Step 6: Contradiction
- Both p and q are even, meaning they share a common factor of 2.
- This contradicts our assumption that p and q are co-prime.
- Therefore, our initial assumption is wrong.
Conclusion: √2 is irrational.
The same method of proof by contradiction can be used to establish that √3, √5, √7, and in general √p for any prime p, are irrational.
Types and Properties
Comparison Table: Rational vs Irrational Numbers
1. Form
- Rational: Can be written as p/q (p, q integers, q ≠ 0)
- Irrational: Cannot be written as p/q for any integers p, q
- Rational: Terminating (e.g., 0.75) or non-terminating repeating (e.g., 0.333...)
- Irrational: Non-terminating and non-repeating (e.g., 1.41421356...)
3. Examples
- Rational: −3, 0, 1/2, 7, 0.6, 2.25, 0.142857142857...
- Irrational: √2, √3, √5, π, e, 0.1010010001...
4. On the Number Line
- Rational: Dense — between any two rationals there is another rational.
- Irrational: Also dense — between any two rationals there is an irrational.
5. Countability
- Rational: Countably infinite (can be listed in a sequence).
- Irrational: Uncountably infinite (cannot be listed in a sequence).
6. Closure under operations
- Rational: Closed under addition, subtraction, multiplication, and division (except by 0).
- Irrational: Not closed — √2 × √2 = 2 (rational).
7. Special numbers
- π (pi), e (Euler's number), and the golden ratio φ = (1 + √5)/2 are all irrational.
- 22/7 is a rational approximation of π, not equal to π.
Solved Examples
Example 1: Example 1: Classify numbers as rational or irrational
Problem: Classify each of the following as rational or irrational: (i) √25 (ii) √7 (iii) 0.3333... (iv) 0.1010010001... (v) 22/7
Solution:
- (i) √25 = 5 = 5/1 → Rational (perfect square)
- (ii) √7 = 2.6457513... (non-terminating, non-repeating) → Irrational (7 is not a perfect square)
- (iii) 0.3333... = 1/3 → Rational (non-terminating repeating)
- (iv) 0.1010010001... → Irrational (non-terminating, non-repeating — the number of zeros keeps increasing)
- (v) 22/7 → Rational (expressed as p/q with integers)
Answer: (i) Rational (ii) Irrational (iii) Rational (iv) Irrational (v) Rational
Example 2: Example 2: Show that 0.999... = 1
Problem: Establish that 0.999... is equal to 1.
Solution:
Method:
- Let x = 0.999...
- Multiply both sides by 10: 10x = 9.999...
- Subtract the first equation from the second: 10x − x = 9.999... − 0.999...
- 9x = 9
- x = 1
Answer: 0.999... = 1, which is rational.
Example 3: Example 3: Convert 0.474747... to p/q form
Problem: Express 0.474747... in the form p/q.
Solution:
- Let x = 0.474747...
- The repeating block "47" has 2 digits, so multiply by 100.
- 100x = 47.474747...
- Subtract: 100x − x = 47.474747... − 0.474747...
- 99x = 47
- x = 47/99
Answer: 0.474747... = 47/99 (rational, since it is non-terminating repeating).
Example 4: Example 4: Determine the nature of the sum
Problem: Find whether the following are rational or irrational: (i) 3 + √5 (ii) √2 + √2 (iii) (2 + √3)(2 − √3)
Solution:
(i) 3 + √5:
- 3 is rational, √5 is irrational.
- Rational + Irrational = Irrational.
- 3 + √5 is irrational.
(ii) √2 + √2:
- √2 + √2 = 2√2
- 2 is rational (non-zero), √2 is irrational.
- 2√2 is irrational.
(iii) (2 + √3)(2 − √3):
- Using (a + b)(a − b) = a² − b²:
- = 4 − 3 = 1
- 1 is rational.
Answer: (i) Irrational (ii) Irrational (iii) Rational
Example 5: Example 5: Product of two irrationals
Problem: Show that the product of two irrational numbers can be rational. Give an example where it is irrational.
Solution:
Case 1: Product is rational
- √3 × √3 = 3 (rational)
- √8 × √2 = √16 = 4 (rational)
Case 2: Product is irrational
- √2 × √3 = √6 (irrational, since 6 is not a perfect square)
Answer: The product of two irrationals is not always rational or always irrational. It depends on the specific numbers.
Example 6: Example 6: Is √4 + √9 rational or irrational?
Problem: Determine whether √4 + √9 is rational or irrational.
Solution:
Given:
- √4 = 2 (since 4 is a perfect square)
- √9 = 3 (since 9 is a perfect square)
Calculate:
- √4 + √9 = 2 + 3 = 5
- 5 = 5/1 (can be written as p/q)
Answer: √4 + √9 = 5, which is rational.
Example 7: Example 7: Find an irrational number between 1/7 and 2/7
Problem: Find an irrational number between 1/7 and 2/7.
Solution:
Given:
- 1/7 = 0.142857...
- 2/7 = 0.285714...
Method: Construct a non-terminating, non-repeating decimal between these values.
- Choose: 0.1501500150001500001... (non-terminating, non-repeating)
- Since 0.142857... < 0.150150... < 0.285714..., this number lies between 1/7 and 2/7.
Alternative: √2/10 = 0.14142... is less than 1/7, so try √2/5 = 0.28284... but this is very close to 2/7. Instead, consider 0.20200200020000... which clearly lies between.
Answer: 0.1501500150001... is an irrational number between 1/7 and 2/7.
Example 8: Example 8: Prove that 3√2 is irrational
Problem: Prove that 3√2 is irrational.
Solution:
Proof by contradiction:
- Assume 3√2 is rational.
- Then 3√2 = p/q for some co-prime integers p, q (q ≠ 0).
- √2 = p/(3q)
- Since p and 3q are integers, p/(3q) is rational.
- This means √2 is rational, which contradicts the fact that √2 is irrational.
Conclusion: Our assumption is wrong. Therefore, 3√2 is irrational.
Example 9: Example 9: Prove that 5 + √3 is irrational
Problem: Prove that 5 + √3 is irrational.
Solution:
Proof by contradiction:
- Assume 5 + √3 is rational.
- Then 5 + √3 = p/q for some co-prime integers p, q.
- √3 = p/q − 5 = (p − 5q)/q
- Since p, q, and 5 are integers, (p − 5q)/q is rational.
- This means √3 is rational, which is a contradiction.
Conclusion: 5 + √3 is irrational.
Example 10: Example 10: Identify from decimal expansion
Problem: Without converting to p/q form, identify each as rational or irrational: (i) 3.141592... (ii) 2.718281... (iii) 1.414213... (iv) 0.123123123... (v) 5.0
Solution:
- (i) 3.141592... = π → non-terminating, non-repeating → Irrational
- (ii) 2.718281... = e → non-terminating, non-repeating → Irrational
- (iii) 1.414213... = √2 → non-terminating, non-repeating → Irrational
- (iv) 0.123123123... → repeating block "123" → Rational (= 123/999 = 41/333)
- (v) 5.0 → terminating decimal → Rational (= 5/1)
Answer: (i) Irrational (ii) Irrational (iii) Irrational (iv) Rational (v) Rational
Real-World Applications
Applications of Rational and Irrational Numbers:
- Measurement: Exact measurements often involve irrational numbers. The diagonal of a unit square is √2, which is irrational. Similarly, the circumference of a circle with diameter 1 is π.
- Construction: Architects and engineers encounter irrational lengths in structures involving diagonals, circular arcs, and inclined surfaces.
- Computing and Approximation: Computers store approximations of irrational numbers. Understanding that π ≈ 3.14159 and √2 ≈ 1.41421 are approximations, not exact, is essential for precision.
- Number Line Completeness: The distinction between rational and irrational numbers underpins the completeness of the real number line, which is essential in calculus and analysis.
- Music and Acoustics: Frequency ratios in equal-tempered musical scales involve irrational numbers (powers of ¹²√2).
- Cryptography: Properties of rational and irrational numbers are used in number theory, which forms the mathematical basis of modern encryption algorithms.
Key Points to Remember
- A rational number can be written as p/q where p, q are integers and q ≠ 0.
- An irrational number cannot be written in p/q form.
- Rational decimals are either terminating or non-terminating repeating.
- Irrational decimals are non-terminating and non-repeating.
- √n is rational only when n is a perfect square; otherwise it is irrational.
- 22/7 is rational — it is an approximation of π, not π itself.
- Rational + Irrational = Irrational.
- Rational × Irrational = Irrational (if the rational ≠ 0).
- Product or sum of two irrationals may be rational or irrational.
- Between any two rational numbers, there exist infinitely many irrational numbers, and vice versa.
Practice Problems
- Classify each as rational or irrational: (a) √49 (b) √11 (c) 0.252525... (d) 3.010010001... (e) −7/3
- Express 0.363636... as a fraction in simplest form.
- Prove that √5 is irrational using the method of contradiction.
- Find two irrational numbers between 3 and 4.
- Determine whether 2√3 + 4√3 is rational or irrational. Justify your answer.
- If x = 3 + 2√5 and y = 3 − 2√5, find x + y and xy. Are they rational or irrational?
- Give an example to show that the difference of two irrational numbers can be rational.
- Is the number 0.12345678910111213... (digits of consecutive natural numbers) rational or irrational? Explain.
Frequently Asked Questions
Q1. What is the main difference between rational and irrational numbers?
A rational number can be expressed as p/q where p and q are integers and q ≠ 0. Its decimal expansion is either terminating or repeating. An irrational number cannot be expressed in p/q form, and its decimal is non-terminating and non-repeating.
Q2. Is π rational or irrational?
π is irrational. Its decimal expansion 3.14159265... never terminates and never repeats. The commonly used value 22/7 is only an approximation, not the exact value of π.
Q3. Is 0 a rational number?
Yes. 0 is rational because it can be expressed as 0/1 (or 0/n for any non-zero integer n).
Q4. Can the sum of two irrational numbers be rational?
Yes. For example, (3 + √2) + (3 − √2) = 6, which is rational. However, this is not always the case — √2 + √3 is irrational.
Q5. How do you prove a number is irrational?
The standard method is proof by contradiction. Assume the number is rational (p/q in lowest terms), derive a contradiction such as the HCF of p and q being greater than 1, and conclude the number is irrational.
Q6. Is every square root irrational?
No. √n is rational when n is a perfect square (e.g., √4 = 2, √9 = 3, √100 = 10). When n is not a perfect square, √n is irrational.
Q7. Are there more rational or irrational numbers?
There are more irrational numbers. The set of rational numbers is countably infinite, while the set of irrational numbers is uncountably infinite. Between any two rationals, there are infinitely many irrationals.
Q8. Is 1.5 rational or irrational?
1.5 is rational. It is a terminating decimal and can be expressed as 3/2.
Q9. Why is 22/7 not equal to π?
22/7 = 3.142857142857... (repeating), while π = 3.14159265... (non-repeating). They agree to only two decimal places. 22/7 is a rational approximation, whereas π is irrational.
Q10. Is Rational vs Irrational Numbers in CBSE Class 9?
Yes. This topic is part of Chapter 1 (Number Systems) in the CBSE Class 9 Mathematics syllabus. Students learn to identify, classify, and prove properties of rational and irrational numbers.
