📘 Real Numbers – Class 10 Summary (CBSE/NCERT)
1. Real Numbers include:
○ Rational numbers (like 1/2, 0.75)
○ Irrational numbers (like √2, π)
○ Together, they make the set of real numbers (ℝ).
2. Euclid’s Division Lemma:
○ For any two positive integers a and b, there exist integers q and r such that:
👉 a = bq + r, where 0 ≤ r < b.
3. HCF using Euclid’s Algorithm:
○ Apply repeated division to find the Highest Common Factor (HCF) of two
numbers.
4. Fundamental Theorem of Arithmetic:
○ Every composite number can be written as a product of prime numbers in a
unique way (except for order).
5. LCM and HCF using prime factorization:
○ Multiply common and unique prime factors appropriately.
6. Irrational Numbers:
○ Numbers that cannot be expressed as a fraction (like √2, √3).
○ Proofs include showing that √2 is irrational.
7. Decimal Expansions:
○ Rational numbers have either terminating or non-terminating repeating
decimals.
○ Use prime factorization of the denominator to check:
■ If denominator = 2m×5n2^m × 5^n2m×5n, it's terminating.
■ Otherwise, non-terminating repeating.
1. Real Numbers include:
○ Rational numbers (like 1/2, 0.75)
○ Irrational numbers (like √2, π)
○ Together, they make the set of real numbers (ℝ).
2. Euclid’s Division Lemma:
○ For any two positive integers a and b, there exist integers q and r such that:
👉 a = bq + r, where 0 ≤ r < b.
3. HCF using Euclid’s Algorithm:
○ Apply repeated division to find the Highest Common Factor (HCF) of two
numbers.
4. Fundamental Theorem of Arithmetic:
○ Every composite number can be written as a product of prime numbers in a
unique way (except for order).
5. LCM and HCF using prime factorization:
○ Multiply common and unique prime factors appropriately.
6. Irrational Numbers:
○ Numbers that cannot be expressed as a fraction (like √2, √3).
○ Proofs include showing that √2 is irrational.
7. Decimal Expansions:
○ Rational numbers have either terminating or non-terminating repeating
decimals.
○ Use prime factorization of the denominator to check:
■ If denominator = 2m×5n2^m × 5^n2m×5n, it's terminating.
■ Otherwise, non-terminating repeating.
