Welcome to Class 10 MP Board Mathematics: Real Numbers
Welcome to Section 1.1 of Chapter 1, Real Numbers! In Class 9, you began your journey into the world of real numbers and learned about irrational numbers. In Class 10, we deepen our understanding by studying fundamental properties of integers, including prime factorization, HCF, LCM, and proofs of irrationality.
Important Formulas, Rules & Key Points
- Real Numbers (&mathbb;R): The collection of both rational and irrational numbers.
- Natural Numbers (N): 1, 2, 3, 4, ...
- Whole Numbers (W): 0, 1, 2, 3, ...
- Integers (Z): ..., -3, -2, -1, 0, 1, 2, 3, ...
- Rational Numbers (Q): Numbers in the form p/q, where p and q are integers and q ≠ 0.
- Irrational Numbers: Numbers that cannot be expressed as p/q (e.g., √2, √3, π).
- Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed (factored) as a product of primes, apart from the order in which the prime factors occur.
- Key Formula for HCF and LCM: For any two positive integers a and b:
HCF(a, b) × LCM(a, b) = a × b
Step-by-Step Introductory Exercise Solutions
Question 1: Express the number 140 as a product of its prime factors.
Solution:
Step 1: Divide 140 by the smallest prime number, 2.
140 ÷ 2 = 70
Step 2: Divide 70 by 2 again.
70 ÷ 2 = 35
Step 3: 35 is not divisible by 2 or 3. Divide 35 by 5.
35 ÷ 5 = 7
Step 4: 7 is a prime number, so 7 ÷ 7 = 1.
Therefore, the prime factorization of 140 is:
140 = 2 × 2 × 5 × 7 = 22 × 5 × 7
Question 2: Find the HCF and LCM of 6 and 20 by the prime factorization method, and verify that HCF(6, 20) × LCM(6, 20) = 6 × 20.
Solution:
Step 1: Find the prime factorization of each number.
6 = 21 × 31
20 = 2 × 2 × 5 = 22 × 51
Step 2: Find HCF (product of the smallest power of each common prime factor involved in the numbers).
Common prime factor is 2. The smallest power of 2 is 21.
HCF(6, 20) = 2
Step 3: Find LCM (product of the greatest power of each prime factor involved in the numbers).
LCM(6, 20) = 22 × 31 × 51 = 4 × 3 × 5 = 60
Step 4: Verification:
HCF × LCM = 2 × 60 = 120
Product of numbers = 6 × 20 = 120
Since 120 = 120, HCF × LCM = Product of the two numbers is verified.
Aptitude Corner & Exam Tips
- Co-prime Trick: If two numbers a and b have no common prime factors (i.e., they are co-prime), then
HCF(a, b) = 1andLCM(a, b) = a × b. - Quick HCF Trick: HCF of two numbers can never be greater than the difference between the two numbers.
- MP Board Tip: Always show full prime factorization trees or division steps in board exams to score full marks!
We hope this introductory lesson on MP Board Class 10 Maths Chapter 1: Real Numbers clarifies your basics! Keep practicing prime factorization and formulas to score top marks in your examinations.
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