Real Numbers is one of the most important chapters in Class 10 Maths because it builds the base for divisibility, factorisation, HCF, LCM, irrational numbers, and many board-level concept questions. Even though this chapter looks short, it carries high exam value because the questions are often concept-based and students can score very well when their basics are clear.
This chapter is especially useful for quick revision because once the key definitions, formulas, and number classifications are understood properly, most questions become direct. Students mainly need to be confident about rational numbers, irrational numbers, decimal expansions, prime factorisation, and HCF-LCM rules.
At Deeksha Vedantu, we always encourage students to revise Real Numbers through classification first, formulas next, and examples after that. This makes the chapter easier to retain before exams.
Real Numbers at a Glance
This quick table helps students revise the full chapter faster.
Quick Concept Table
| Topic | Key idea |
| Real numbers | Include both rational and irrational numbers |
| Rational numbers | Can be written in the form p/q, where q ≠ 0 |
| Irrational numbers | Cannot be written in the form p/q |
| Terminating decimal | Rational |
| Non-terminating repeating decimal | Rational |
| Non-terminating non-repeating decimal | Irrational |
| Prime factorisation | Writing a number as a product of prime numbers |
| HCF | Common prime factors with lowest powers |
| LCM | All prime factors with highest powers |
What Are Real Numbers
Real numbers are the collection of both rational and irrational numbers.
This means every number that can be placed on the number line comes under real numbers.
Types of Real Numbers
Real numbers are mainly divided into two categories.
| Type | Meaning |
| Rational numbers | Numbers that can be written in the form p/q, where q ≠ 0 |
| Irrational numbers | Numbers that cannot be written in the form p/q |
Rational Numbers
A rational number is a number that can be written in the form:
p/q
Where:
- p and q are integers
- q is not equal to zero
Important Idea About Rational Numbers
Rational numbers may appear in different forms.
| Form | Example |
| Integer | 5, -7 |
| Fraction | 1/2, 3/4 |
| Terminating decimal | 0.25 |
| Non-terminating repeating decimal | 0.3333… |
Examples of Rational Numbers
- 1/2
- 3/4
- 5
- -7
- 0.25
- 0.3333…
Irrational Numbers
An irrational number is a number that cannot be written in the form p/q.
Its decimal expansion is:
- non-terminating
- non-repeating
Examples of Irrational Numbers
- √2
- √3
- √5
- π
These numbers never end and do not show a repeating pattern.
Decimal Expansion and Number Type
This is one of the easiest and most important revision areas in the chapter.
Decimal Expansion Summary Table
| Type of decimal expansion | Number type |
| Terminating | Rational |
| Non-terminating repeating | Rational |
| Non-terminating non-repeating | Irrational |
Important Examples for Revision
Example 1
1/2 = 0.5
This is a terminating decimal, so it is rational.
Example 2
1/3 = 0.3333…
This is non-terminating repeating, so it is rational.
Example 3
√2 = 1.414213…
This is non-terminating non-repeating, so it is irrational.
Example 4
π is an irrational number because it is non-terminating and non-repeating.
Properties of Rational and Irrational Numbers
These combinations are very important for quick revision and board-level concept questions.
Property Summary Table
| Combination | Result | Example |
| Rational + Rational | Rational | 2 + 3 = 5 |
| Rational + Irrational | Irrational | 2 + √3 |
| Rational × Irrational | Irrational | 2 × √3 |
| Irrational + Irrational | Usually irrational | √2 + √3 |
| Irrational × Irrational | May be rational or irrational | √2 × √2 = 2, √2 × √3 = √6 |
Important Note
Students should be especially careful with this result:
Irrational × Irrational may be rational or irrational.
It is not fixed in every case.
Prime Factorisation Method
Prime factorisation means writing a number as a product of its prime factors.
Example of Prime Factorisation
Take 36:
36 = 2 × 2 × 3 × 3
So it can be written as:
36 = 2² × 3²
This is called prime factorisation.
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of prime numbers, and this factorisation is unique apart from the order of the prime factors.
This theorem is one of the core ideas of the chapter and is used in HCF-LCM questions.
HCF and LCM Using Prime Factorisation
This is a very common exam area in Real Numbers.
HCF and LCM Rule Table
| Quantity | Rule |
| HCF | Take the common prime factors with the lowest powers |
| LCM | Take all prime factors with the highest powers |
Solved Example: Find HCF and LCM of 98 and 28
Given
- Number 1 = 98
- Number 2 = 28
Step 1: Prime Factorisation of 98
98 = 2 × 7 × 7
So:
98 = 2 × 7²
Step 2: Prime Factorisation of 28
28 = 2 × 2 × 7
So:
28 = 2² × 7
Step 3: Find HCF
Take the common factors with lowest powers:
- 2¹
- 7¹
So:
HCF = 2 × 7 = 14
Step 4: Find LCM
Take all prime factors with highest powers:
- 2²
- 7²
So:
LCM = 4 × 49 = 196
Answer
- HCF = 14
- LCM = 196
Important Formula Connecting HCF and LCM
For any two integers:
HCF × LCM = Product of the numbers
This is one of the most important formulas from the chapter.
Verification Using 98 and 28
HCF = 14
LCM = 196
So:
14 × 196 = 2744
Now check the product of the numbers:
98 × 28 = 2744
Both are equal.
So the formula is verified.
Board-Style Example: Value of N – 7M
If HCF of 98 and 28 is M and LCM is N, find the value of N – 7M.
Given
- M = 14
- N = 196
Solution
N – 7M = 196 – 7 × 14
= 196 – 98
= 98
Answer
The value of N – 7M is 98.
Formula Sheet for Real Numbers
Students should revise these formulas and rules carefully.
Formula and Rule Summary Table
| Formula or rule | Meaning |
| Rational number = p/q, where q ≠ 0 | Standard form of a rational number |
| HCF × LCM = Product of the two numbers | Important relation between HCF and LCM |
| Terminating decimal = Rational number | Decimal classification rule |
| Non-terminating repeating decimal = Rational number | Decimal classification rule |
| Non-terminating non-repeating decimal = Irrational number | Decimal classification rule |
| HCF → lowest powers | Prime factorisation rule |
| LCM → highest powers | Prime factorisation rule |
Common Mistakes Students Make in Real Numbers
These mistakes are very common in quick board revision.
Common Mistakes Table
| Mistake | Correct idea |
| Thinking every non-terminating decimal is irrational | Non-terminating repeating decimals are rational |
| Taking highest power in HCF | HCF uses the lowest common powers |
| Taking lowest power in LCM | LCM uses the highest powers |
| Forgetting q ≠ 0 in p/q | In a rational number, q can never be zero |
| Assuming irrational × irrational is always irrational | It may be rational or irrational |
Quick Revision Points Before Exam
This section is useful for last-minute recall.
Remember These Clearly
- real numbers include both rational and irrational numbers
- rational numbers can be written as p/q
- irrational numbers cannot be written as p/q
- terminating decimals are rational
- non-terminating repeating decimals are rational
- non-terminating non-repeating decimals are irrational
- HCF uses lowest powers
- LCM uses highest powers
- HCF × LCM = product of the two numbers
Last-Minute Revision Table
| Topic | What to remember |
| Real numbers | Include rational and irrational numbers |
| Rational number | Can be written as p/q |
| Irrational number | Cannot be written as p/q |
| Decimal expansion | Helps classify the number type |
| Prime factorisation | Needed for HCF and LCM |
| HCF | Lowest common powers |
| LCM | Highest powers |
Study Strategy for Real Numbers
This chapter becomes much easier when revised in the right order.
Step-by-Step Revision Table
| Step | What to do |
| Step 1 | Revise number classification first |
| Step 2 | Memorise decimal expansion rules |
| Step 3 | Practise prime factorisation |
| Step 4 | Revise the HCF-LCM formula |
| Step 5 | Solve previous year questions |
Practice Questions for Students
Important Practice Questions
- Classify 0.25, 0.6666…, and √7 as rational or irrational.
- Find the prime factorisation of 72.
- Find the HCF and LCM of 84 and 126 using prime factorisation.
- Verify that HCF × LCM = product of the two numbers for any pair you choose.
- Decide whether √2 × √8 is rational or irrational.
FAQs
Q1. What are real numbers in Class 10 Maths?
Real numbers are the collection of rational and irrational numbers.
Q2. What is a rational number?
A rational number is a number that can be written in the form p/q where p and q are integers and q is not zero.
Q3. What is an irrational number?
An irrational number is a number that cannot be written in the form p/q.
Q4. Is a non-terminating repeating decimal rational?
Yes. A non-terminating repeating decimal is rational.
Q5. What is the rule for finding HCF using prime factorisation?
Take the common prime factors with the lowest powers.
Q6. What is the rule for finding LCM using prime factorisation?
Take all prime factors with the highest powers.
Q7. What is the relation between HCF and LCM of two numbers?
HCF multiplied by LCM is equal to the product of the two numbers.
Q8. Can the product of two irrational numbers be rational?
Yes. For example, √2 multiplied by √2 gives 2, which is rational.
Conclusion
Real Numbers is one of the most basic yet powerful chapters in Class 10 Maths. It helps students build clarity about number types, decimal expansions, prime factorisation, and HCF-LCM concepts. Since most questions from this chapter are direct and concept-based, students can score very well if they revise the classification rules and formulas carefully.
The best way to master this chapter is to understand the difference between rational and irrational numbers first, then move to prime factorisation and HCF-LCM rules. At Deeksha Vedantu, we always encourage students to revise chapters like Real Numbers through simple concepts and repeated examples, because that makes board preparation much more confident and effective.
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