📘 CBSE Class 10 Maths Chapter 1: Real Numbers
Chapter 1 Real Numbers is an important chapter in Class 10 Mathematics. It introduces you to deeper properties of numbers that you have already learned in earlier classes — natural numbers, whole numbers, integers, rational and irrational numbers.
You will also study:
✅ Fundamental Theorem of Arithmetic
✅ Prime Factorization
✅ Finding HCF and LCM
✅ Irrational Numbers & Their Proofs
✅ Decimal Representation of Rational Numbers
🔑 Key Concepts in Real Numbers
1️⃣ Fundamental Theorem of Arithmetic
This theorem says that:
👉 Every composite number can be written as a product of prime numbers in a unique way (except for the order of the primes).
For example:
140 = 2² × 5 × 7
156 = 2² × 3 × 13
2️⃣ HCF and LCM using Prime Factorization
You can easily find HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two or more numbers using their prime factorizations.
- HCF is the product of common prime factors with lowest powers.
- LCM is the product of all prime factors with their highest powers.
👉 Important Formula:
HCF × LCM = Product of the two numbers
3️⃣ Irrational Numbers
- Irrational numbers are those which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
- Examples: √2, √3, √5, π
Proving numbers are irrational:
In this chapter, you will learn how to prove that √2, √3 and √5 are irrational — these are very common Board exam questions.
4️⃣ Decimal Expansions of Rational Numbers
Every rational number has either:
- A terminating decimal (finite number of digits after the decimal point),
Example: 1/2 = 0.5 - A non-terminating but repeating decimal (recurring digits after the decimal point),
Example: 1/3 = 0.333…
👉 The type of decimal depends on the prime factors of the denominator when the rational number is expressed in lowest terms.
📚 NCERT Exercises — What to Practice?
- Ex 1.1: Prime Factorization, HCF, LCM
- Ex 1.2: Proving irrational numbers
📝 Important Board Exam Questions
Here are some questions often asked in CBSE Class 10 Board Exams:
1️⃣ Find the HCF and LCM of 60 and 72 using prime factorization.
2️⃣ Prove that √2 is irrational.
3️⃣ Show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
4️⃣ Write the decimal expansion of 13/50.
5️⃣ Prove that √5 is irrational.
🎯 Tips to Score Full Marks in Chapter 1
✅ Practice proofs of irrationality carefully — these are almost always asked.
✅ Be perfect in prime factorization and calculating HCF/LCM.
✅ Memorize the form of terminating and non-terminating decimals.
✅ Solve previous years’ questions and NCERT Exemplar problems.
✅ Attempt sample papers and test yourself regularly.
Conclusion
Real Numbers is one of the easiest and most scoring chapters in Class 10 Maths. A strong grip on the basic properties of numbers, HCF, LCM and irrationality proofs will help you score full marks.
For more free Class 10 Notes, Worksheets and MCQs, visit: [www.morescorecbse.in]
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