Real Numbers
Real Numbers form the foundation of higher mathematics and are covered in CBSE Class 10 Chapter 1. Every number used in daily life — for measuring distances, calculating areas, handling money, or reading temperatures — is a real number.
Real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. This chapter builds on the number systems studied in Class 9.
Two foundational results are introduced in this chapter:
- Euclid's Division Lemma — used to compute HCF
- Fundamental Theorem of Arithmetic — every composite number has a unique prime factorisation
By the end of this chapter, students will be able to find HCF and LCM using prime factorisation and Euclid's algorithm, prove the irrationality of certain numbers, and determine the decimal behaviour of rational numbers.
What is Real Numbers?
Definition: Real Numbers are the collection of all rational and irrational numbers. They can be represented on the number line and cover every point on it without any gap.
The set of real numbers is denoted by R. Every real number is either:
- Rational — expressible as p/q where p and q are integers, q ≠ 0
- Irrational — cannot be expressed as p/q; decimal expansion is non-terminating and non-repeating
Hierarchy of Number Systems:
- Natural Numbers (N): {1, 2, 3, 4, ...} — the counting numbers
- Whole Numbers (W): {0, 1, 2, 3, ...} — natural numbers including zero
- Integers (Z): {..., −3, −2, −1, 0, 1, 2, 3, ...} — whole numbers and their negatives
- Rational Numbers (Q): Numbers of the form p/q where p, q are integers, q ≠ 0. Examples: 1/2, −3/7, 0.75, 5
- Irrational Numbers: Cannot be written as p/q. Examples: √2, π, √3
The relationship: N ⊂ W ⊂ Z ⊂ Q ⊂ R, and irrational numbers together with Q form R.
Completeness Property: Every point on the number line corresponds to a unique real number, and vice versa. There are no gaps in the real number system.
In NCERT Class 10 Chapter 1, two foundational results are studied:
- Euclid's Division Lemma — leads to an algorithm for computing HCF
- Fundamental Theorem of Arithmetic — every composite number can be uniquely factorised into primes
These results have consequences for understanding divisibility, HCF, LCM, and decimal expansions of rational numbers.
Real Numbers Formula
Key Formulas and Results:
1. Euclid's Division Lemma:
a = bq + r, where 0 ≤ r < b
For any two positive integers a and b, there exist unique integers q and r satisfying this relation.
2. Fundamental Theorem of Arithmetic:
Every composite number can be expressed as a product of primes, and this factorisation is unique (apart from order).
Example: 60 = 2² × 3 × 5
3. HCF × LCM = Product of two numbers:
HCF(a, b) × LCM(a, b) = a × b
4. Decimal expansion of p/q:
- If the prime factorisation of q (when p/q is in lowest terms) is of the form 2m × 5n, the decimal expansion is terminating.
- Otherwise, it is non-terminating repeating.
5. Irrationality proofs:
Numbers like √2, √3, √5 are irrational. This is proved using the Fundamental Theorem of Arithmetic and proof by contradiction.
Derivation and Proof
Proof that √2 is irrational (using the Fundamental Theorem of Arithmetic):
This proof uses proof by contradiction.
- Assume, on the contrary, that √2 is rational. Then √2 = p/q, where p and q are coprime integers (HCF = 1), q ≠ 0.
- Squaring both sides: 2 = p²/q², which gives p² = 2q².
- This means p² is divisible by 2. Since 2 is prime and 2 divides p², by the Fundamental Theorem, 2 must divide p.
- Let p = 2k for some integer k. Substituting: (2k)² = 2q², giving 4k² = 2q², so q² = 2k².
- This means q² is also divisible by 2. By the same argument, q is divisible by 2.
- Both p and q are divisible by 2. This contradicts the assumption that p and q are coprime.
- Therefore, √2 is irrational. QED.
Note: The same technique proves that √3, √5, √7, and √p (for any prime p) are irrational. Replace 2 with the prime in question throughout.
Proof that p/q terminates if and only if q = 2m × 5n:
- A decimal expansion terminates when it can be written as a fraction with a denominator that is a power of 10. Example: 0.375 = 375/1000 = 375/10³.
- Since 10 = 2 × 5, any power of 10 has the form 2k × 5k. A denominator with only factors of 2 and 5 can always be converted to a power of 10.
- Example: 7/8 = 7/2³ = 7 × 5³/(2³ × 5³) = 875/1000 = 0.875 (terminating).
- Conversely, if q has a prime factor other than 2 or 5 (say 3), then q cannot divide any power of 10. Hence p/q cannot have a terminating decimal.
- Example: 1/6 — q = 6 = 2 × 3. Since q has factor 3, the decimal is non-terminating repeating: 1/6 = 0.1666...
Types and Properties
Classification of Real Numbers:
| Type | Description | Examples |
|---|---|---|
| Natural Numbers (N) | Positive counting numbers | 1, 2, 3, 17, 100 |
| Whole Numbers (W) | Natural numbers + zero | 0, 1, 2, 3, 50 |
| Integers (Z) | Whole numbers + negatives | −5, −1, 0, 3, 12 |
| Rational Numbers (Q) | Expressible as p/q (q ≠ 0) | 1/2, −3/4, 0.6, 7 |
| Irrational Numbers | Not expressible as p/q | √2, √7, π, e |
Properties of Real Numbers:
- Closure: Real numbers are closed under addition, subtraction, and multiplication. Closed under division except when dividing by zero.
- Commutativity: a + b = b + a and a × b = b × a.
- Associativity: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
- Distributive Property: a × (b + c) = a × b + a × c.
- Density: Between any two real numbers, there exist infinitely many rational and irrational numbers.
Decimal Expansions of Rational Numbers:
- Terminating decimals: End after a finite number of digits. Example: 3/8 = 0.375.
- Non-terminating repeating decimals: Continue forever with a repeating block. Example: 1/3 = 0.333... = 0.3̄.
Key distinction: Every terminating decimal is rational. Every non-terminating repeating decimal is rational. A number is irrational if and only if its decimal expansion is non-terminating and non-repeating.
Solved Examples
Example 1: Checking if a number is rational or irrational
Problem: Classify as rational or irrational: (i) √25, (ii) √18, (iii) 0.4545...
Solution:
(i) √25:
- √25 = 5, which is an integer.
- Every integer is rational (5 = 5/1).
- √25 is rational.
(ii) √18:
- √18 = √(9 × 2) = 3√2.
- Since √2 is irrational and 3 is rational, their product is irrational.
- √18 is irrational.
(iii) 0.4545...:
- This is a non-terminating repeating decimal (0.4̄5̄).
- All non-terminating repeating decimals are rational.
- 0.4545... = 45/99 = 5/11.
- 0.4545... is rational.
Example 2: Applying Euclid's Division Lemma
Problem: Apply Euclid's Division Lemma to divide 455 by 42.
Solution:
Given:
- a = 455, b = 42
Using Euclid's Division Lemma (a = bq + r):
- 455 ÷ 42 = 10 with remainder 35
- Check: 42 × 10 + 35 = 420 + 35 = 455 ✓
- Verify: 0 ≤ 35 < 42 ✓
Answer: 455 = 42 × 10 + 35, so q = 10 and r = 35.
Example 3: Finding HCF using Euclid's Division Algorithm
Problem: Find HCF of 867 and 255 using Euclid's Division Algorithm.
Solution:
Using Euclid's Division Algorithm:
- 867 = 255 × 3 + 102 (remainder ≠ 0, continue)
- 255 = 102 × 2 + 51 (remainder ≠ 0, continue)
- 102 = 51 × 2 + 0 (remainder = 0, stop)
The last non-zero remainder is the HCF.
Answer: HCF(867, 255) = 51.
Example 4: Finding HCF and LCM by prime factorisation
Problem: Find HCF and LCM of 96 and 404 by prime factorisation.
Solution:
Prime factorisations:
- 96 = 2⁵ × 3
- 404 = 2² × 101
HCF = product of smallest powers of common factors = 2² = 4
LCM = product of greatest powers of all factors = 2⁵ × 3 × 101 = 32 × 3 × 101 = 9696
Verification:
- HCF × LCM = 4 × 9696 = 38784
- 96 × 404 = 38784 ✓
Answer: HCF = 4, LCM = 9696.
Example 5: Determining terminating or non-terminating decimal
Problem: Without actual division, determine the decimal type: (i) 13/3125, (ii) 17/90.
Solution:
(i) 13/3125:
- Denominator: 3125 = 5⁵
- Form: 2⁰ × 5⁵ (only 2s and 5s)
- Decimal is terminating.
- 13/3125 = 13 × 2⁵ / (5⁵ × 2⁵) = 416/100000 = 0.00416
(ii) 17/90:
- Denominator: 90 = 2 × 3² × 5
- Contains factor 3 (other than 2 and 5)
- Decimal is non-terminating repeating.
Answer: (i) Terminating (= 0.00416). (ii) Non-terminating repeating.
Example 6: Proving a number is irrational
Problem: Prove that √3 is irrational.
Solution:
- Assume √3 is rational. Then √3 = p/q, where p and q are coprime (HCF = 1), q ≠ 0.
- Squaring: 3 = p²/q², so p² = 3q².
- 3 divides p². Since 3 is prime, 3 divides p. Let p = 3m.
- Substituting: (3m)² = 3q² → 9m² = 3q² → q² = 3m².
- 3 divides q², so 3 divides q.
- Both p and q are divisible by 3 — contradicts HCF(p, q) = 1.
Answer: The assumption is wrong. Therefore, √3 is irrational.
Example 7: Proving an expression is irrational
Problem: Prove that 3 + 2√5 is irrational.
Solution:
- Assume 3 + 2√5 is rational. Then 3 + 2√5 = a/b, where a, b are integers, b ≠ 0.
- Rearranging: 2√5 = (a − 3b)/b.
- So √5 = (a − 3b)/(2b).
- Since a, b are integers, (a − 3b)/(2b) is rational. This implies √5 is rational.
- But √5 is irrational (proved using the same method as √2 or √3). Contradiction.
Answer: Therefore, 3 + 2√5 is irrational.
Example 8: Finding HCF and LCM of three numbers
Problem: Find HCF and LCM of 12, 15, and 21 by prime factorisation.
Solution:
Prime factorisations:
- 12 = 2² × 3
- 15 = 3 × 5
- 21 = 3 × 7
HCF = product of smallest powers of common factors. Only common factor: 3¹.
HCF(12, 15, 21) = 3
LCM = product of greatest powers of all factors = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420
Answer: HCF = 3, LCM = 420.
Example 9: Word problem on HCF
Problem: A hall measures 24 m × 18 m. Square tiles of the largest possible size cover the floor completely. Find the tile size and number of tiles.
Solution:
Given:
- Hall dimensions: 24 m × 18 m
The tile side must divide both 24 and 18 exactly. The largest such length is HCF(24, 18).
Finding HCF:
- 24 = 2³ × 3
- 18 = 2 × 3²
- HCF = 2¹ × 3¹ = 6
Side of each tile = 6 m.
Number of tiles:
- Area of hall / Area of tile = (24 × 18) / (6 × 6) = 432 / 36 = 12
Answer: Tile side = 6 m, number of tiles = 12.
Example 10: Word problem on LCM
Problem: Three bells toll at intervals of 9, 12, and 15 minutes. If they toll together at 8:00 AM, when will they next toll together?
Solution:
Given:
- Intervals: 9, 12, 15 minutes
They toll together again after LCM(9, 12, 15) minutes.
Finding LCM:
- 9 = 3²
- 12 = 2² × 3
- 15 = 3 × 5
- LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 minutes = 3 hours
8:00 AM + 3 hours = 11:00 AM.
Answer: They next toll together at 11:00 AM.
Real-World Applications
Measurement:
- All physical measurements — length, mass, temperature, time — are expressed as real numbers.
- Measuring cloth as 2.35 m uses a rational number; calculating circumference using πd uses an irrational number.
Engineering and Architecture:
- HCF is used when designing gear systems, synchronising signals, and planning tile patterns.
- Finding the largest tile that fits a floor perfectly requires HCF.
Computer Science:
- Euclid's Division Algorithm is the basis of many algorithms in cryptography (e.g., RSA encryption).
- It is one of the oldest algorithms still in active use.
Music:
- Musical intervals and harmonics are based on rational number ratios.
- The irrational number 2^(1/12) divides an octave into 12 equal semitones in equal temperament tuning.
Finance:
- Interest calculations, amortisation schedules, and stock price analysis all use real number arithmetic.
Scheduling:
- LCM is used to find when traffic lights, bus schedules, or production cycles will coincide.
Key Points to Remember
- Real numbers = Rational numbers + Irrational numbers. Every point on the number line is a real number.
- Euclid's Division Lemma: a = bq + r (0 ≤ r < b) — basis for Euclid's Division Algorithm to find HCF.
- Fundamental Theorem of Arithmetic: Every composite number has a unique prime factorisation (up to order).
- HCF × LCM = Product of two numbers. This relation does NOT directly extend to three or more numbers.
- HCF = product of smallest powers of common prime factors. LCM = product of greatest powers of all prime factors.
- A rational number p/q (in lowest terms) has a terminating decimal if and only if q = 2m × 5n.
- If the denominator has any prime factor other than 2 or 5, the decimal is non-terminating repeating.
- To prove √p (p prime) is irrational, use proof by contradiction with the Fundamental Theorem of Arithmetic.
- Every integer is rational (n = n/1). Every rational number is real. But not every real number is rational.
- For CBSE board exams, irrationality proofs of √2, √3, √5 and expressions like 3 + 2√5 are frequently asked (4-mark questions).
Practice Problems
- Use Euclid's Division Algorithm to find the HCF of 4052 and 12576.
- Find the LCM and HCF of 6, 72, and 120 by prime factorisation method.
- Prove that √5 is irrational.
- Prove that 7 − 3√2 is irrational.
- Without performing long division, state whether the following will have a terminating or non-terminating repeating decimal expansion: (i) 23/200, (ii) 29/343, (iii) 77/210.
- Check whether 6ⁿ can end with the digit 0 for any natural number n.
- Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker an exact number of times.
- Find the largest number that divides 245 and 1029, leaving remainder 5 in each case.
Frequently Asked Questions
Q1. What are real numbers in Class 10 Maths?
Real numbers are the collection of all rational and irrational numbers. They include natural numbers, whole numbers, integers, fractions, decimals (terminating and non-terminating repeating), and irrational numbers like √2 and π. In CBSE Class 10 Chapter 1, students study Euclid's Division Lemma, the Fundamental Theorem of Arithmetic, and decimal behaviour of rational numbers.
Q2. What is the difference between rational and irrational numbers?
A rational number can be expressed as p/q (p, q integers, q ≠ 0). Its decimal expansion either terminates or repeats. An irrational number cannot be expressed as p/q; its decimal expansion is non-terminating and non-repeating. Examples of irrational numbers: √2, √3, π.
Q3. How do you prove that √2 is irrational?
Assume √2 = p/q where p and q are coprime. Squaring gives p² = 2q², so 2 divides p. Writing p = 2k and substituting gives q² = 2k², so 2 divides q. This contradicts p and q being coprime. Therefore √2 is irrational.
Q4. What is Euclid's Division Lemma?
For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b. This is the foundation of Euclid's Division Algorithm for finding HCF.
Q5. What is the Fundamental Theorem of Arithmetic?
Every composite number (integer > 1 that is not prime) can be expressed as a product of prime numbers, and this factorisation is unique apart from the order of factors. Example: 84 = 2 × 2 × 3 × 7 — no other set of primes multiplies to 84.
Q6. How to find HCF and LCM by prime factorisation?
Write each number as a product of prime powers. HCF = product of smallest powers of common prime factors. LCM = product of greatest powers of all prime factors. Example: 36 = 2² × 3² and 48 = 2⁴ × 3. HCF = 2² × 3 = 12, LCM = 2⁴ × 3² = 144.
Q7. When does a rational number have a terminating decimal expansion?
A rational number p/q (in lowest terms) has a terminating decimal if and only if the prime factorisation of q contains no prime factors other than 2 and 5. That is, q must be of the form 2<sup>m</sup> × 5<sup>n</sup>, where m and n are non-negative integers.
Q8. Is the product of a rational and irrational number always irrational?
The product of a non-zero rational number and an irrational number is always irrational. Example: 3 × √2 is irrational. However, 0 × √2 = 0, which is rational. The statement holds only when the rational number is non-zero.
Q9. What is the relationship between HCF and LCM of two numbers?
For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b. If you know the HCF and one number, you can find the LCM, and vice versa. This formula applies directly only to two numbers, not three or more.
Q10. What are the important formulas in Real Numbers Class 10?
The key formulas are: (1) Euclid's Division Lemma: a = bq + r with 0 ≤ r < b, (2) HCF × LCM = product of two numbers, (3) A rational p/q terminates if q = 2<sup>m</sup> × 5<sup>n</sup>, (4) Every composite number has unique prime factorisation. These are used for HCF/LCM problems, decimal expansion questions, and irrationality proofs.
