Real Numbers — Quick Notes & Formula Sheet
Class 10 CBSE Mathematics · Chapter 1 · Complete Revision Guide
1. Euclid's Division Lemma
For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:
a = bq + r, where 0 ≤ r < b
This is simply the everyday idea of division ("dividend = divisor × quotient + remainder"), but stated formally so it can be used to build an algorithm.
⚠ Trap: The remainder range
Students often write 0 ≤ r ≤ b (with r allowed to equal b). This is wrong — r must be strictly less than b. If r equals b, the division isn't complete.
2. Euclid's Division Algorithm (to find HCF)
- To find HCF(a, b) where a > b, apply the lemma: a = bq + r
- If r = 0, then b is the HCF. Stop here.
- If r ≠ 0, apply the lemma again to the pair (b, r) — i.e., divide b by r.
- Keep repeating with the new pair (previous divisor, previous remainder) until the remainder becomes 0.
- The last non-zero divisor is the HCF.
Worked Example: Find HCF(96, 404)
404 = 96×4 + 20 → 96 = 20×4 + 16 → 20 = 16×1 + 4 → 16 = 4×4 + 0
Last non-zero remainder = 4, so HCF(96, 404) = 4
404 = 96×4 + 20 → 96 = 20×4 + 16 → 20 = 16×1 + 4 → 16 = 4×4 + 0
Last non-zero remainder = 4, so HCF(96, 404) = 4
⚠ Trap: Dividing the wrong way round
Always divide the larger number by the smaller one first. A common error is swapping a and b, which throws off every step that follows.
3. Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of prime numbers, and this factorization is unique, apart from the order in which the prime factors occur.
Example: 240 = 2⁴ × 3 × 5 — no matter how you factor it, you always land on these same primes.
⚠ Trap: Is 1 a prime number?
No. 1 is neither prime nor composite — it's a special case. Don't include it while listing prime factors.
4. HCF-LCM Relationship
HCF(a, b) × LCM(a, b) = a × b
Use this to find one value when the other three are known — very common in exam word problems.
⚠ Trap: This formula does NOT extend directly to 3+ numbers
HCF(a,b,c) × LCM(a,b,c) is not generally equal to a × b × c. This identity only works cleanly for exactly two numbers — a favourite Assertion-Reason trap.
| Given | Find using |
|---|---|
| HCF and one number + product | LCM = Product ÷ HCF |
| LCM and one number + product | HCF = Product ÷ LCM |
| Two numbers in ratio m:n with known LCM | HCF = LCM ÷ (m×n), if m,n co-prime |
5. Proving a Number is Irrational
Standard method: Proof by contradiction
- Assume the number (say √2) is rational: √2 = a/b, where a and b are integers with no common factor other than 1 (in lowest terms), b ≠ 0.
- Square both sides and manipulate to show that a and b must both share a common factor (usually the number itself, e.g., 2).
- This contradicts the assumption that a/b was in lowest terms.
- Hence, the original assumption is false — the number is irrational.
⚠ Trap: Skipping "in lowest terms"
CBSE's marking scheme specifically checks whether you stated a/b are co-prime (no common factor other than 1) at the start. Skipping this line loses marks even if your final answer is correct.
⚠ Trap: "Irrational + Irrational = Irrational" is FALSE
Counterexample: (2+√3) + (2−√3) = 4, a rational number. Similarly, √2 × √2 = 2 shows that irrational × irrational isn't always irrational either. Only "rational ± irrational = irrational" is always guaranteed true.
6. Terminating vs Non-Terminating Decimal Expansions
For a rational number p/q in lowest terms, the decimal expansion:
- Terminates if q is of the form 2ᵐ × 5ⁿ (only 2s and/or 5s as prime factors)
- Non-terminates and repeats if q has any other prime factor
⚠ Trap: Forgetting to reduce the fraction first
The 2ᵐ×5ⁿ rule only applies once the fraction is in its lowest terms. E.g., 21/30 looks non-terminating by its denominator 30, but reduces to 7/10 — which does terminate. Always simplify first.
✓ Quick check tip
To predict the number of decimal places for a terminating fraction, make the powers of 2 and 5 in the denominator equal by multiplying — the higher power tells you the decimal place count.
7. Full Recap — Every Trap in One Place
| Trap | Correct Understanding |
|---|---|
| 0 ≤ r ≤ b in Euclid's Lemma | Should be 0 ≤ r < b (strict inequality) |
| 1 is a prime number | 1 is neither prime nor composite |
| HCF×LCM=product works for 3+ numbers | Only guaranteed for exactly two numbers |
| Irrational + Irrational = Irrational | False — can be rational (e.g., 2+√3 and 2−√3) |
| Irrational × Irrational = Irrational | False — √2×√2 = 2 (rational) |
| Co-prime numbers must be prime | False — e.g., 8 and 9 are co-prime, neither is prime |
| Checking 2ᵐ5ⁿ rule without simplifying fraction first | Always reduce to lowest terms before checking |
| Skipping "a, b co-prime" in irrationality proofs | Always state this — it's a scored step |
Practice What You've Just Revised
Test yourself with the full Real Numbers quiz series — 65 questions across 4 parts, from basic concepts to board-level Assertion-Reason questions.
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