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.
You will learn two powerful results about integers: Euclid's Division Lemma and the Fundamental Theorem of Arithmetic. The first helps you find the highest common factor (HCF) of two positive integers systematically. The second says every composite number breaks into prime factors in exactly one way, which lets you find HCF and LCM using prime factorisation.
The chapter also proves that certain numbers like √2, √3 and √5 are irrational. Finally, you revisit decimal expansions to see precisely when a rational number terminates and when it repeats. After this chapter, you should be able to apply Euclid's algorithm, factorise integers into primes, prove specific irrationality results, and predict the decimal behaviour of any rational number.
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.
Worked examples
Example 1: Use Euclid's algorithm to find HCF of 252 and 105.
Step 1: Divide 252 by 105. 252 = 105 × 2 + 42.
Step 2: Divide 105 by 42. 105 = 42 × 2 + 21.
Step 3: Divide 42 by 21. 42 = 21 × 2 + 0.
The remainder is now 0. The last non-zero remainder is 21. Therefore, HCF(252, 105) = 21.
Example 2: Find HCF and LCM of 72 and 120 using prime factorisation.
Prime factorise each number: 72 = 2³ × 3² 120 = 2³ × 3 × 5
HCF: Take common primes with smallest powers. Common primes are 2 and 3. Smallest powers: 2³ and 3¹. HCF = 2³ × 3 = 8 × 3 = 24.
LCM: Take all primes with greatest powers. Primes involved: 2, 3, 5. Greatest powers: 2³, 3², 5¹. LCM = 8 × 9 × 5 = 360.
Check: HCF × LCM = 24 × 360 = 8640. Also, 72 × 120 = 8640. ✓
Example 3: Prove that √3 is irrational.
Assume √3 is rational. Then √3 = p/q, where p and q are integers with no common factor and q ≠ 0.
Squaring: 3 = p²/q², so p² = 3q².
This means p² is divisible by 3, so p itself is divisible by 3. Write p = 3k for some integer k.
Substituting: (3k)² = 3q² gives 9k² = 3q², so q² = 3k².
Thus q² is divisible by 3, so q is divisible by 3.
Now both p and q are divisible by 3, contradicting our assumption that they share no common factor.
Therefore, √3 is irrational.
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.