Real Numbers Class 10
Download the Real Numbers Class 10 PDF containing notes and a mind map for concept revision. Use the explanations, important results and worked examples below to strengthen your understanding of prime factorisation, HCF, LCM and irrational numbers.
Real numbers include the rational and irrational numbers that can be represented on a number line. In Class 10 Mathematics, the chapter develops your understanding of prime factors, divisibility and the reasoning behind irrationality proofs. These ideas help you explain why a result is true, rather than simply calculate an answer.
These Real Numbers Class 10 notes provide a structured revision guide. Read the explanations first, use the mind map to recall connections between concepts, and then practise questions independently. A mind map helps with quick recall, while detailed notes help you understand the steps needed in a written solution.
Real Numbers Class 10 Notes and Key Concepts
| Concept | Meaning | Example |
|---|---|---|
| Rational number | Can be written as p/q, where p and q are integers and q ≠ 0. | −3, 2/5, 0.75 |
| Irrational number | Cannot be written as a ratio of two integers. | √2, √3, π |
| Prime number | A natural number greater than 1 with exactly two positive factors. | 2, 3, 5, 7 |
| Composite number | A natural number greater than 1 with more than two positive factors. | 4, 6, 8, 9 |
| Co-prime numbers | Two positive integers whose HCF is 1. | 8 and 15 |
A rational number has a terminating or non-terminating recurring decimal expansion. An irrational number has a non-terminating, non-recurring decimal expansion. For example, 0.333… is rational because it equals 1/3, whereas √2 is irrational.
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes. This prime factorisation is unique apart from the order of the factors. For example, writing 60 as 2 × 2 × 3 × 5 or 5 × 3 × 2 × 2 gives the same prime factors.
Example 1: Prime Factorisation of 360
360 = 2 × 180
360 = 2 × 2 × 90
360 = 2 × 2 × 2 × 45
360 = 2 × 2 × 2 × 3 × 15
360 = 2 × 2 × 2 × 3 × 3 × 5
360 = 23 × 32 × 5
Continue factorising until every factor is prime. Leaving 45 in the final product would make the prime factorisation incomplete.
HCF and LCM by Prime Factorisation
- HCF: Multiply the common prime factors, taking the smallest power of each.
- LCM: Multiply all prime factors appearing in either number, taking the greatest power of each.
Example 2: HCF and LCM of 72 and 120
72 = 23 × 32
120 = 23 × 3 × 5
HCF = 23 × 3
HCF = 24
LCM = 23 × 32 × 5
LCM = 360
Verification:
HCF × LCM = 24 × 360
HCF × LCM = 8640
72 × 120 = 8640
Real Numbers Class 10 Important Results
| Result | Condition or Explanation |
|---|---|
| HCF(a, b) × LCM(a, b) = a × b | Applies to two positive integers. |
| If HCF(a, b) = 1, then LCM(a, b) = ab. | Applies to co-prime positive integers. |
| If p is prime and p divides a2, then p divides a. | A useful result in irrationality proofs. |
| Terminating decimal condition | For p/q in lowest terms, q has no prime factors other than 2 and 5. |
Check the Conditions Before Applying a Result
The familiar HCF–LCM product relation is for two positive integers. Do not extend it unchanged to three numbers. When checking a fraction’s decimal expansion, reduce it to lowest terms before examining its denominator.
How to Prove That √2 Is Irrational
An irrationality proof often uses contradiction: assume the number is rational and show that this conflicts with the assumption that its numerator and denominator have no common factor.
Example 3: Irrationality Proof
Assume √2 = a/b, where a and b are integers, b ≠ 0,
and a/b is in lowest terms.
Squaring gives 2 = a2/b2.
Therefore, a2 = 2b2.
Hence a2 is even, so a is even.
Write a = 2k for some integer k.
Substituting gives 4k2 = 2b2.
Therefore, b2 = 2k2.
Hence b is also even.
Thus a and b have a common factor of 2.
This contradicts a/b being in lowest terms.
Therefore, √2 is irrational.
Real Numbers Class 10 Solved Application Questions
Example 4: When Will Two Bells Ring Together?
Question: Two bells ring at intervals of 12 minutes and 18 minutes. If they ring together now, after how long will they next ring together?
12 = 22 × 3
18 = 2 × 32
LCM = 22 × 32
LCM = 36 minutes
Use LCM because the required time is the smallest positive common multiple of both intervals.
Example 5: Is 3 + 2√5 Rational?
Suppose 3 + 2√5 = r, where r is rational.
Then 2√5 = r − 3.
Therefore, √5 = (r − 3)/2.
The right-hand side is rational.
This contradicts the irrationality of √5.
Therefore, 3 + 2√5 is irrational.
Using the Real Numbers Class 10 Mind Map
Use the Real Numbers Class 10 mind map after reading the notes. Recall how the main branches connect: real numbers split into rational and irrational numbers; prime factorisation supports HCF and LCM; divisibility results support irrationality proofs.
For active revision, cover a section of the map and explain it aloud. Then write one example without looking at the notes. A visual summary is most effective when you can explain the reasoning behind each keyword, formula and connection.
Quick Practice Questions with Answers
-
Express 156 as a product of primes.
Answer: 22 × 3 × 13. -
Find the HCF and LCM of 18 and 24.
Answer: HCF = 6; LCM = 72. -
If HCF(26, 91) = 13, find their LCM.
Answer: (26 × 91) ÷ 13 = 182. -
Can 6n end in zero for a positive integer n?
Answer: No. Its prime factors are only 2 and 3; a number ending in zero must also have a factor of 5. -
Are 8 and 15 co-prime?
Answer: Yes, because their HCF is 1.
Practise After Revising the Notes
Apply these concepts using the Real Numbers Class 10 Worksheet with Solutions and MCQs . Attempt the questions independently, then compare your factorisation, calculations and proof steps with the solutions.
Explore more chapters in the CBSE Class 10 Maths Notes and Mind Maps collection .
Frequently Asked Questions
Does this Real Numbers Class 10 PDF contain notes and a mind map?
Yes. The PDF combines notes for understanding the chapter with a mind map for visual revision. Read the notes first, then use the map to recall the main concepts and their connections.
What is the difference between rational and irrational numbers?
Rational numbers can be written as p/q with integers p and q, where q ≠ 0. Irrational numbers cannot. For example, 0.25 = 1/4 is rational, while √2 is irrational.
Are all square roots irrational?
No. The square root of a perfect-square integer is rational: √49 = 7. The square root of a positive integer that is not a perfect square is irrational, such as √7.
Is 1 a prime number?
No. A prime number has exactly two positive factors. The number 1 has only one, so it is neither prime nor composite.
How do I decide whether a word problem needs HCF or LCM?
Use HCF when finding the greatest equal size that divides quantities exactly, such as the longest equal pieces cut from ropes. Use LCM when finding the smallest common quantity or the next shared repetition time, such as bells ringing together.
Can two composite numbers be co-prime?
Yes. For example, 8 and 15 are both composite, but share no prime factor. Their HCF is 1, so they are co-prime.
Can the sum or product of two irrational numbers be rational?
Yes. √2 + (−√2) = 0 and √2 × √2 = 2 are rational. Other pairs can give irrational results. Therefore, there is no rule that every sum or product of irrational numbers is irrational.
How do I check whether a rational number has a terminating decimal?
Reduce the fraction to lowest terms. Its decimal terminates if the denominator contains only prime factors 2 and 5. For example, 7/40 terminates because 40 = 23 × 5. The decimal for 1/6 recurs because 6 also contains a factor of 3.
Is π equal to 22/7?
No. π is irrational, while 22/7 is rational. The fraction is an approximation used in calculations when specified; it is not the exact value of π.
Can two numbers have HCF 16 and LCM 380?
No. For positive integers, their HCF must divide their LCM. Since 380 is not divisible by 16, these cannot be the HCF and LCM of the same pair.
How should I use these notes with NCERT questions?
Revise a concept, study its example and then solve a matching textbook question without looking at the solution. For proofs, write the assumption and contradiction explicitly. Check exercise wording because numbering and topic coverage can differ by edition.
Where can I practise Real Numbers Class 10 questions with solutions?
Visit the Real Numbers Class 10 worksheet with solutions and MCQs . Use it after revising the notes and mind map to check your understanding through independent practice.
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