In this chapter, you will learn
- —Understand Euclid's division algorithm and its applications in finding HCF
- —State and apply the fundamental theorem of arithmetic
- —Classify numbers into rational and irrational categories
- —Prove that certain numbers like √2 are irrational
- —Understand decimal expansions of rational and irrational numbers
- —Apply HCF and LCM in real-world problems
Euclid's Division Lemma
Euclid's Division Lemma is a fundamental theorem used to find the Highest Common Factor (HCF) of two positive integers.
Euclid's Division Lemma: For any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that:
a = bq + r
where 0 ≤ r < b
Here, a = dividend, b = divisor, q = quotient, r = remainder
Example: Divide 13 by 5
- 13 = 5 × 2 + 3
- Here, a = 13, b = 5, q = 2, r = 3
- Since 0 ≤ 3 < 5, the division is valid
Finding HCF using Euclid's Division Lemma:
Example: Find HCF of 456 and 336
- 456 = 336 × 1 + 120
- 336 = 120 × 2 + 96
- 120 = 96 × 1 + 24
- 96 = 24 × 4 + 0
Therefore, HCF(456, 336) = 24
When remainder becomes 0, the divisor at that step is the HCF.
Key Points:
- Euclid's algorithm is used to find HCF efficiently without prime factorization
- It works for any two positive integers
- The remainder must always satisfy: 0 ≤ r < b
- This algorithm is the basis for the Euclidean algorithm in computer science
Exam Tip
Euclid's division lemma questions are common in board exams. Practice finding HCF step-by-step. Remember: when remainder = 0, the divisor is the HCF.
Common Mistake
Students forget to check that r < b. The remainder must always be less than the divisor. Also, don't confuse with simple division—here we're tracking the quotient and remainder.
Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way (regardless of the order).
Fundamental Theorem of Arithmetic: Every composite number can be factorized as a product of prime numbers, and this factorization is unique except for the order of factors.
Examples of Prime Factorization:
- 60 = 2² × 3 × 5 (unique prime factorization)
- 100 = 2² × 5²
- 140 = 2² × 5 × 7
Applications in HCF and LCM:
Example: Find HCF and LCM of 60 and 100
- 60 = 2² × 3 × 5
- 100 = 2² × 5²
- HCF = 2² × 5 = 20 (product of common prime factors with lowest powers)
- LCM = 2² × 3 × 5² = 300 (product of all prime factors with highest powers)
Verify: HCF × LCM = 20 × 300 = 6000 = 60 × 100 ✓
Key Points:
- Every composite number is a unique product of primes (ignoring order)
- The theorem guarantees uniqueness, which is fundamental to number theory
- Prime numbers themselves cannot be factorized further (they are the building blocks)
- Using prime factorization, we can easily find HCF and LCM
- For any two numbers a and b: HCF(a,b) × LCM(a,b) = a × b
Exam Tip
Prime factorization questions appear frequently. Always use the relationship: HCF × LCM = a × b to verify your answers.
Common Mistake
Students confuse HCF and LCM. HCF uses lowest powers of common factors; LCM uses highest powers of all factors. Also remember: 1 is not prime.
Rational and Irrational Numbers
Numbers can be classified into two main categories: Rational and Irrational.
Rational Numbers:
- Can be expressed in the form p/q where p and q are integers and q ≠ 0
- Their decimal expansion is either terminating (e.g., 0.5, 0.25) or non-terminating repeating (e.g., 1/3 = 0.333...)
- Examples: 3/4, 5, -2/7, 0.125
Irrational Numbers:
- Cannot be expressed in the form p/q
- Their decimal expansion is non-terminating and non-repeating
- Examples: √2 ≈ 1.414..., π ≈ 3.14159..., √3 ≈ 1.732...
Real Numbers = Rational Numbers + Irrational Numbers
Key Points:
- The sum or difference of a rational and irrational number is irrational
- The product of a non-zero rational and irrational number is irrational
- Between any two rational numbers, there are infinitely many irrational numbers and vice versa
Exam Tip
Understand the difference between rational and irrational clearly. Questions often ask to identify or distinguish between them. Repeating decimals = rational, non-repeating = irrational.
Common Mistake
Students think √4 is irrational. NO! √4 = 2, which is rational. Only √2, √3, √5 (non-perfect squares) are irrational.
Proving Irrationality
One of the most important techniques in Class 10 Mathematics is proving that certain numbers are irrational using the method of contradiction.
Theorem: √2 is irrational
Proof by Contradiction:
- Assume the opposite: √2 is rational
- Then √2 = p/q where p and q are coprime integers (HCF(p,q) = 1) and q ≠ 0
- Squaring both sides: 2 = p²/q²
- Therefore: p² = 2q² ... (equation 1)
- This means p² is even, so p must be even. Let p = 2m for some integer m
- Substituting in equation 1: (2m)² = 2q²
- 4m² = 2q²
- 2m² = q² ... (equation 2)
- This means q² is even, so q must be even
- Contradiction: Both p and q are even, so they are not coprime!
- Conclusion: Our assumption was wrong. Therefore, √2 is irrational
How to Prove √p is irrational (for prime p):
- Assume √p = a/b (where HCF(a,b) = 1)
- Square: p = a²/b²
- So: a² = pb²
- This means p divides a², so p divides a. Let a = pk
- Then: (pk)² = pb² → p²k² = pb² → pk² = b²
- This means p divides b², so p divides b
- Contradiction: p divides both a and b
- Therefore, √p is irrational
Key Points:
- √2, √3, √5, √7, √11 etc. (square roots of primes) are all irrational
- More generally, √n is irrational if n is not a perfect square
- The proof technique (method of contradiction) is essential to learn
- This same method can be used to prove many other numbers are irrational
Exam Tip
The proof of √2 is irrational is frequently asked (5 marks). Learn it step-by-step. The key insight is: if p² = 2q², then both p and q are even, contradicting coprimality.
Common Mistake
Students don't follow the logic properly. Remember: if p² is even, then p is even (not just divisible). Also, coprime means HCF = 1.
Decimal Expansions
The decimal expansion of a number reveals whether it's rational or irrational.
Rational Numbers:
Decimal expansion is either terminating or non-terminating repeating
Examples of Rational Numbers:
- 1/2 = 0.5 (terminating)
- 1/4 = 0.25 (terminating)
- 1/3 = 0.333... = 0.3̄ (non-terminating repeating)
- 1/6 = 0.1666... = 0.16̄ (non-terminating repeating)
- 1/7 = 0.142857142857... = 0.1̄4̄2̄8̄5̄7̄ (non-terminating repeating)
Irrational Numbers:
Decimal expansion is non-terminating and non-repeating
Examples of Irrational Numbers:
- √2 = 1.414213562373095...
- √3 = 1.732050807568877...
- π = 3.141592653589793...
- e = 2.718281828459045...
Key Points About Decimal Expansions:
- Terminating decimals can be written as fractions: 0.25 = 25/100 = 1/4
- Repeating decimals: The bar notation shows which digits repeat. 0.3̄ means 0.333...
- A fraction p/q in lowest form has a terminating decimal if and only if q has only factors of 2 and 5
- Otherwise, the decimal is non-terminating repeating
Exam Tip
If asked to identify rational/irrational from decimal expansion: terminating or repeating = rational, non-repeating = irrational.
Common Mistake
Students confuse π with 22/7. Remember: 22/7 is an approximation of π (and rational), but π itself is irrational.
Operations and HCF-LCM Applications
Real-world applications of HCF and LCM appear frequently in problem-solving questions.
Example 1: HCF Application
Problem: Two pieces of cloth are 36 m and 48 m long. They need to be cut into equal-length pieces with no wastage. What is the maximum length of each piece?
Solution:
- We need the maximum length that divides both 36 and 48
- This is HCF(36, 48)
- 36 = 2² × 3²
- 48 = 2⁴ × 3
- HCF = 2² × 3 = 12 m
- Number of pieces: 36/12 + 48/12 = 3 + 4 = 7 pieces
Example 2: LCM Application
Problem: Two bells ring at intervals of 15 minutes and 20 minutes. They ring together at 9:00 AM. At what time will they ring together again?
Solution:
- We need the least time when both bells ring together
- This is LCM(15, 20)
- 15 = 3 × 5
- 20 = 2² × 5
- LCM = 2² × 3 × 5 = 60 minutes = 1 hour
- They will ring together again at 10:00 AM
When to use HCF vs LCM:
- HCF: Maximum, largest, greatest, biggest, maximum length/size
- LCM: Minimum, least, smallest, next time, repeating events
Relationship Between HCF and LCM:
HCF(a, b) × LCM(a, b) = a × b
Key Points:
- Always use this relationship to verify your HCF and LCM calculations
- For three numbers, find HCF and LCM step by step (pairwise)
- Real numbers can be added, subtracted, multiplied, and divided (except by 0)
Exam Tip
Always verify HCF × LCM = a × b. Word problems require identifying whether to find HCF or LCM. Look for keywords: 'maximum' → HCF, 'minimum/next time' → LCM.
Common Mistake
Mixing up HCF and LCM applications. If asked for maximum length → HCF. If asked when events repeat → LCM.
Chapter Summary
Real Numbers form the foundation of higher mathematics. This chapter covers:
- Euclid's Division Lemma: Used to find HCF using the division algorithm
- Fundamental Theorem of Arithmetic: Every composite number has a unique prime factorization
- Rational Numbers: Can be expressed as p/q with terminating or repeating decimals
- Irrational Numbers: Cannot be expressed as p/q with non-repeating, non-terminating decimals
- Proving Irrationality: Using the method of contradiction for √2 and similar numbers
- Applications: HCF and LCM in real-world problems like cutting cloth, ringing bells, etc.
Exam Focus: Euclid's algorithm, proving √2 is irrational, HCF-LCM applications, identifying rational/irrational numbers.