π· 1. Euclidβs Division Lemma
β Statement:
For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:
a = bq + r
where
0 \le r < b
π Example:
Find HCF of 135 and 225 using Euclidβs Division Algorithm.
225 = 135 Γ 1 + 90
135 = 90 Γ 1 + 45 ξ
90 = 45 Γ 2 + 0
Since remainder becomes 0, HCF = 45
π· 2. Fundamental Theorem of Arithmetic
β Statement:
Every composite number can be expressed as a product of prime numbers, and this factorization is unique (except for order).
π Example:
360 = 2^3 Γ 3^2 Γ 5
This factorization is unique.
π· 3. Revisiting Irrational Numbers
β Proof that β2 is Irrational (Important for Board)
Assume β2 is rational.
So,
\sqrt{2} = \frac{p}{q}
where p and q are co-prime integers.
After solving:
p^2 = 2q^2
This shows p is even.
If p is even β q is also even.
This contradicts the assumption that p and q are co-prime.
Therefore, β2 is irrational.
π· 4. Decimal Expansion of Rational Numbers
Important Rule:
If in lowest form, denominator of p/q has only prime factors 2 and/or 5, then decimal expansion is terminating.
Otherwise β Non-terminating recurring
π Examples:
40 = 2^3 Γ 5
Only 2 and 5 β Terminating
12 = 2^2 Γ 3
Contains 3 β Non-terminating recurring
π· 5. Relationship Between HCF and LCM
Formula:
\text{HCF} Γ \text{LCM} = Product\ of\ two\ numbers
Example:
Find LCM if HCF = 12 and numbers are 36 and 60.
12 Γ LCM = 36 Γ 60
LCM = \frac{36 Γ 60}{12}
LCM = 180
π· 6. Important Exam Points
β Euclidβs lemma proof is important.
β Proof of β2 irrational is frequently asked.
β Decimal expansion questions are common (2β3 marks).
β Prime factorisation method for HCF & LCM is very important.
π’ 1. Rational Numbers
β Definition:
A rational number is a number that can be written in the form:
\frac{p}{q}
where:
- p and q are integers
- q β 0
π Examples:
- 2 (because )
- 0 (because )
π Properties of Rational Numbers
- Closure Property
Rational numbers are closed under:- Addition
- Subtraction
- Multiplication
- Division (except division by 0)
- Commutative Property
- Associative Property
- Distributive Property
- Decimal Expansion A rational number has:
- Terminating decimal (e.g., 0.25)
- Non-terminating recurring decimal (e.g., 0.333…)
π· 2. Irrational Numbers
β Definition:
An irrational number is a number that cannot be written in the form p/q.
Their decimal expansion is:
- Non-terminating
- Non-repeating
π Examples:
- Ο (Pi)
- 0.1010010001… (non-repeating)
π Important Facts
| Rational Numbers | Irrational Numbers |
|---|---|
| Can be written as p/q | Cannot be written as p/q |
| Decimal expansion terminates or repeats | Decimal expansion never ends and never repeats |
| Examples: 1/2, -3, 0.75 | Examples: β2, Ο |
πΆ 3. Real Numbers
All rational and irrational numbers together form the set of Real Numbers.
\textbf{Real Numbers = Rational Numbers + Irrational Numbers}
They can be represented on the number line.
βοΈ Extra Important Points for Board Exam
β Every integer is a rational number.
β Every rational number is a real number.
β β2 is irrational (proved in NCERT).
β If denominator of p/q (in lowest form) has only 2 and/or 5 as prime factors β decimal is terminating.
Example:
- β terminating
- β non-terminating recurring
π Chapter 1 β Real Numbers
2 & 3 Marks Short Questions (CBSE Class 10)
πΉ 2 MARKS QUESTIONS
- Define a rational number. Give two examples.
Answer: A number that can be written in the form , where p and q are integers and .
Examples: - Write one rational and one irrational number between 1 and 2.
Answer:
Rational:
Irrational: - Is 0.375 a rational number? Justify.
Answer: Yes, because , which is of the form . - State Euclidβs Division Lemma.
Answer: For any two positive integers a and b, there exist unique integers q and r such that
a = bq + r,\; 0 \le r < b
- Write the prime factorisation of 180.
Answer:
180 = 2^2 \times 3^2 \times 5
πΉ 3 MARKS QUESTIONS
- Use Euclidβs Division Algorithm to find the HCF of 306 and 657.
Answer:
657 = 306 \times 2 + 45
306 = 45 \times 6 + 36 ξ
45 = 36 \times 1 + 9
36 = 9 \times 4 + 0 ξ
HCF = 9
- Find whether the decimal expansion of will terminate or not.
Answer:
125 = 5^3
- Prove that is irrational.
Answer (Outline):
Assume (in lowest form).
Squaring:
β p divisible by 3 β q also divisible by 3
Contradiction.
Hence, is irrational.
- Find the LCM of 24 and 90 using prime factorisation.
Answer:
24 = 2^3 \times 3
90 = 2 \times 3^2 \times 5 ξ
LCM = 2^3 \times 3^2 \times 5 = 360
- State the Fundamental Theorem of Arithmetic.
Answer: Every composite number can be expressed as a product of primes, and this factorisation is unique except for the order of primes.
π Chapter 1 β Real Numbers
π Case Study Based Questions (CBSE Class 10)
π· Case Study 1: School Sports Day Arrangement (HCF & LCM)
During Sports Day, a school has 306 boys and 657 girls. The sports teacher wants to arrange them in groups such that:
- Each group has the same number of boys
- Each group has the same number of girls
- No student is left out.
Questions:
- Which mathematical concept will be used to solve this problem? (1 mark)
- Find the maximum number of groups that can be formed. (2 marks)
- How many boys and girls will be there in each group? (1 mark)
Answers:
- HCF (Highest Common Factor)
- Using Euclidβs Division Algorithm:
657 = 306 Γ 2 + 45
306 = 45 Γ 6 + 36 ξ
45 = 36 Γ 1 + 9
36 = 9 Γ 4 + 0 ξ
HCF = 9
Maximum groups = 9
- Boys in each group =
Girls in each group =
π· Case Study 2: Tile Flooring Problem (Prime Factorisation & LCM)
A rectangular hall is 24 m long and 90 m wide. The owner wants to cover it with square tiles of equal size, such that:
- No tile is cut.
- The tile size is maximum possible.
Questions:
- Which mathematical concept will help to find the tile size? (1 mark)
- Find the side of the largest square tile. (2 marks)
- How many tiles will be required? (1 mark)
Answers:
- HCF (using prime factorisation)
- Prime factorisation:
24 = 2^3 Γ 3
90 = 2 Γ 3^2 Γ 5 ξ
HCF =
Largest tile side = 6 m
- Number of tiles:
\frac{24}{6} Γ \frac{90}{6}
4 Γ 15 = 60
Total tiles = 60
π· Case Study 3: Sweet Distribution (Decimal Expansion Concept)
A shopkeeper packs sweets equally into boxes. He divides 13 kg of sweets into 125 equal packets.
Questions:
- Express the weight of each packet as a fraction. (1 mark)
- Will its decimal expansion terminate? Why? (2 marks)
- Find the weight of each packet in decimal form. (1 mark)
Answers:
Since denominator contains only 5 β decimal expansion terminates.
π Chapter 1 β Real Numbers
β 20 MCQs (CBSE Class 10)
πΉ Multiple Choice Questions
1. The HCF of 306 and 657 is:
A) 3
B) 6
C) 9
D) 18
Answer: C) 9
2. Euclidβs Division Lemma states that for integers a and b:
A) a = b + r
B) a = bq + r
C) a = br + q
D) a = b β r
Answer: B) a = bq + r
3. The decimal expansion of is:
A) Non-terminating recurring
B) Terminating
C) Non-terminating non-recurring
D) Irrational
Answer: B) Terminating
4. The prime factorisation of 210 is:
A) 2 Γ 3 Γ 5 Γ 7
B) 2 Γ 5 Γ 21
C) 3 Γ 70
D) 2 Γ 105
Answer: A) 2 Γ 3 Γ 5 Γ 7
5. Which of the following is irrational?
A) 0.25
B)
C) β5
D) β3
Answer: C) β5
6. If HCF of two numbers is 12 and their product is 1800, their LCM is:
A) 120
B) 150
C) 180
D) 144
Answer: B) 150
7. Which of the following has non-terminating recurring decimal expansion?
A)
B)
C)
D)
Answer: C)
8. The HCF of two consecutive even numbers is:
A) 1
B) 2
C) 4
D) Depends
Answer: B) 2
9. β2 is:
A) Rational
B) Irrational
C) Integer
D) Whole number
Answer: B) Irrational
10. The LCM of 24 and 36 is:
A) 12
B) 48
C) 72
D) 144
Answer: C) 72
11. If a number has prime factors 2 and 5 only, its decimal expansion is:
A) Non-terminating
B) Recurring
C) Terminating
D) Irrational
Answer: C) Terminating
12. The HCF of 45 and 75 is:
A) 5
B) 10
C) 15
D) 25
Answer: C) 15
13. Which is a rational number?
A) β3
B) Ο
C) 0.121121112β¦
D) 0.333β¦
Answer: D) 0.333β¦
14. Fundamental Theorem of Arithmetic is related to:
A) Addition
B) Division
C) Prime factorisation
D) Subtraction
Answer: C) Prime factorisation
15. The LCM of two prime numbers is:
A) 1
B) Their sum
C) Their product
D) Their HCF
Answer: C) Their product
16. The HCF of 17 and 23 is:
A) 1
B) 17
C) 23
D) 391
Answer: A) 1
17. Which of the following is NOT irrational?
A) β7
B) β11
C) β49
D) Ο
Answer: C) β49
18. The decimal expansion of will:
A) Terminate
B) Repeat
C) Be irrational
D) Never end
Answer: A) Terminate
19. If HCF = 6 and LCM = 180, product of numbers is:
A) 30
B) 1080
C) 186
D) 174
Answer: B) 1080
20. Which of the following is a real number?
A) β2
B) β5
C) 3/7
D) All of these
Answer: D) All of these
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