Schooling · Class 10 · Mathematics · Real Numbers · Chapter on a page
CBSE Class 10 · Mathematics · NCERT Mathematics
Real Numbers: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- Every positive integer a, when…
- Euclid's Division Algorithm uses the…
- The Fundamental Theorem of…
- HCF of two or more numbers equals…
- For any two positive integers a and b
- A number is irrational if it cannot…
- A rational number p/q (in lowest…
- Formulas and facts
- a = bq + r
- HCF × LCM = product of the two…
- HCF from prime factors
- LCM from prime factors
- Terminating decimal condition
- √p is irrational whenever p is a…
- Watch out for
- Stopping Euclid's algorithm one step…
- Using greatest powers for HCF…
- Forgetting to reduce the fraction to…
- Assuming every square root is…
- Confusing non-terminating repeating…
What this chapter is about
This chapter builds on what you already know about natural numbers, whole numbers, integers and rational numbers from earlier classes. Now you study the complete number line, which includes both rational and irrational numbers together — these form the real numbers.
Key ideas
- Every positive integer a, when divided by a positive integer b, gives a unique quotient q and remainder r such that a = bq + r, where 0 ≤ r < b. This is Euclid's Division Lemma.
- Euclid's Division Algorithm uses the lemma repeatedly to find HCF(a, b): divide a by b, then divide b by the remainder, and continue until the remainder is 0; the last non-zero remainder is the HCF.
- The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the factors.
- HCF of two or more numbers equals the product of the smallest powers of all common prime factors; LCM equals the product of the greatest powers of all prime factors appearing in any of the numbers.
- For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b.
- A number is irrational if it cannot be written as p/q where p and q are integers with q ≠ 0. Numbers like √2, √3, √5 are irrational.
- A rational number p/q (in lowest terms) has a terminating decimal expansion if and only if the prime factorisation of q contains no primes other than 2 and 5.
- If q has a prime factor other than 2 or 5, the decimal expansion of p/q is non-terminating and repeating.
Formulas and facts to remember
- Euclid's Division Lemma: a = bq + r, where 0 ≤ r < b. (Basis for finding HCF by repeated division.)
- HCF × LCM = product of the two numbers. (Useful shortcut once one of HCF or LCM is known.)
- HCF from prime factors: take common primes with smallest exponents.
- LCM from prime factors: take all primes with greatest exponents.
- Terminating decimal condition: denominator (in lowest terms) = 2^m × 5^n for non-negative integers m, n.
- √p is irrational whenever p is a prime number.
Common mistakes
- Stopping Euclid's algorithm one step early → continue until the remainder is exactly 0; the last non-zero remainder is the HCF.
- Using greatest powers for HCF instead of smallest → HCF needs smallest powers of common primes; LCM needs greatest powers of all primes.
- Forgetting to reduce the fraction to lowest terms before checking termination → always simplify p/q first, then examine the denominator.
- Assuming every square root is irrational → √4 = 2, √9 = 3 are rational; only square roots of non-perfect-square integers are irrational.
- Confusing non-terminating repeating decimals with irrational numbers → repeating decimals are still rational; irrational decimals neither terminate nor repeat.
Quick revision
- Euclid's Division Lemma: a = bq + r (0 ≤ r < b); repeat to get HCF.
- Every composite number = unique product of primes (Fundamental Theorem of Arithmetic).
- HCF × LCM = product of the two numbers.
- √p is irrational for any prime p.
- Terminating decimal ⇔ denominator (lowest terms) has only 2 and 5 as prime factors.
- Non-terminating repeating decimals are rational, not irrational.