Real Numbers · Lesson 5 of 5
Chapter Summary and Practice
“Prime factors, irrational numbers and decimal expansions come together for one final numerical adventure.”
• Revise the major ideas from all four lessons. • Recall the essential definitions, theorems and formulas. • Recognise common mistakes and improve exam presentation. • Solve questions of gradually increasing difficulty. • Practise MCQs and assertion–reason questions.
Chapter at a Glance
Real numbers include both rational and irrational numbers. Natural Numbers ⊂ Whole Numbers ⊂ Integers ⊂ Rational Numbers ⊂ Real Numbers Irrational Numbers ⊂ Real Numbers A rational number can be written in the form p/q, where p and q are integers and q ≠ 0. An irrational number cannot be written in this form.
For any two positive integers a and b, there exist unique whole numbers q and r such that: a = bq + r, where 0 ≤ r < b. The lemma can be repeatedly applied to find the HCF of two positive integers.
Every composite number can be expressed as a product of prime numbers, and this prime factorisation is unique except for the order of the factors. HCF uses the common prime factors with their smallest powers. LCM uses all required prime factors with their greatest powers.
If p is a prime number and p divides a², then p divides a. This theorem is used in contradiction proofs showing that numbers such as √2 and √3 are irrational. The square root of a perfect square is rational, while the square root of a positive integer that is not a perfect square is irrational.
Key Formula and Concept Sheet
Prime factorisation is the expression of a composite number as a product of prime numbers. By the Fundamental Theorem of Arithmetic, this factorisation is unique except for the order of the prime factors.
Two positive integers are coprime if their only common factor is 1. Equivalently, their HCF is 1.
An irrational number cannot be expressed in the form p/q, where p and q are integers and q ≠ 0. Its decimal expansion is non-terminating and non-repeating.
Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.
| Concept | What to Select | Memory Trick |
|---|---|---|
| HCF | Common prime factors with the smallest powers | HCF means common and smallest |
| LCM | All required prime factors with the greatest powers | LCM means collect all and greatest |
| Euclid's algorithm | Continue division until the remainder becomes 0 | The last non-zero remainder is the HCF |
| Irrationality proof | Assume rational, derive a contradiction | Assume, simplify, contradict, conclude |
Key Takeaways
• Every natural number is a whole number, integer, rational number and real number. • Every integer is rational because it can be written with denominator 1. • Rational and irrational numbers together form the real numbers. • Euclid's Division Lemma has the form a = bq + r, where 0 ≤ r < b. • Euclid's algorithm finds the HCF by repeated division. • A prime number has exactly two positive factors: 1 and itself. • A composite number has more than two positive factors. • Every composite number has a unique prime factorisation apart from factor order. • HCF uses common prime factors with their smallest powers. • LCM uses all required prime factors with their greatest powers. • For two positive integers, HCF × LCM equals their product. • Coprime numbers have HCF 1. • An irrational number has a non-terminating, non-repeating decimal expansion. • If a prime p divides a², then p divides a. • The proof that √2 is irrational uses contradiction. • Square roots of perfect squares are rational. • Square roots of positive non-perfect-square integers are irrational.
• Writing r ≤ b instead of r < b in Euclid's Division Lemma. • Stopping Euclid's algorithm before the remainder becomes 0. • Treating 1 as a prime number. • Using the greatest powers while calculating HCF. • Using only common factors while calculating LCM. • Forgetting that HCF × LCM = product applies directly to two positive integers. • Assuming every square root is irrational. • Forgetting to write p/q in lowest terms in an irrationality proof. • Saying p is divisible by a prime without using the theorem. • Reaching a contradiction but failing to state the final conclusion.
Graded Practice Questions
The following questions increase gradually in difficulty. Attempt them in order.
Practice Problems
- Classify each number as rational or irrational: 7, −3/5, √16, √7 and 0.121221222… .
- State Euclid's Division Lemma and identify the dividend, divisor, quotient and remainder in 47 = 6 × 7 + 5.
- Express 756 as a product of prime factors.
- Use Euclid's algorithm to find the HCF of 405 and 252.
- Find the HCF and LCM of 96 and 144 using prime factorisation. Verify that HCF × LCM equals the product of the two numbers.
- Two bells ring at intervals of 18 minutes and 24 minutes. They ring together at 9:00 a.m. At what time will they next ring together?
- Show that every odd positive integer is of the form 6q + 1, 6q + 3 or 6q + 5 for some whole number q.
- Prove that √3 is irrational.
- Prove that 5√2 is irrational.
- Using the Fundamental Theorem of Arithmetic, determine whether 6ⁿ can end with the digit 0 for any positive integer n. Justify your answer.
Multiple-Choice Questions
Quiz
Which of the following is irrational?
In a = bq + r, which condition must the remainder satisfy?
The HCF of two coprime numbers is:
The prime factorisation of 72 is:
To calculate the HCF using prime factorisation, we select:
If HCF(a, b) = 6, LCM(a, b) = 180 and a = 30, then b equals:
Which statement is correct?
If a prime p divides a², then:
The contradiction in the standard proof of the irrationality of √2 is that:
Which pair has LCM equal to the product of the two numbers?
Assertion–Reason Questions
Choose the correct option: A. Both Assertion and Reason are true, and Reason correctly explains Assertion. B. Both Assertion and Reason are true, but Reason does not correctly explain Assertion. C. Assertion is true, but Reason is false. D. Assertion is false, but Reason is true.
- Assertion: The HCF of 14 and 25 is 1. Reason: The numbers 14 and 25 have no common prime factor.
- Assertion: √36 is irrational. Reason: 36 is a perfect square.
- Assertion: The prime factorisation of a composite number is unique apart from factor order. Reason: This is stated by the Fundamental Theorem of Arithmetic.
- Assertion: If 2 divides p², then 2 divides p. Reason: The number 2 is prime.
- Assertion: The LCM of two coprime positive integers equals their product. Reason: Their HCF is 1 and HCF × LCM equals the product of the integers.
Solutions
Both the Assertion and the Reason are true, and the Reason correctly explains the Assertion. Since 14 and 25 have no common prime factor, their HCF is 1.
The Assertion is false, but the Reason is true. Since 36 is a perfect square, √36 = 6, which is rational.
Both the Assertion and the Reason are true, and the Reason correctly explains the Assertion. The uniqueness of prime factorisation is stated by the Fundamental Theorem of Arithmetic.
Both the Assertion and the Reason are true, and the Reason correctly explains the Assertion. Since 2 is prime, the theorem applies: if 2 divides p², then 2 divides p.
Both the Assertion and the Reason are true, and the Reason correctly explains the Assertion. For coprime numbers, HCF = 1. Therefore, LCM equals the product of the two numbers.
You are ready to move to the next chapter when you can: • Explain the difference between rational and irrational numbers. • Apply Euclid's Division Lemma correctly. • Find HCF using Euclid's algorithm. • Write the unique prime factorisation of a composite number. • Calculate HCF and LCM using prime powers. • Solve simple real-life HCF and LCM problems. • State and apply the prime divisibility theorem. • Reproduce the proof that √2 is irrational without skipping steps.
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Revisiting Irrational Numbers
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