Chapter: Real Numbers (NCERT Class 10 Mathematics)
Introduction to Real Numbers
A real number is any number that can be found on the number line. It includes natural numbers, whole numbers, integers, rational numbers, and irrational numbers. In this chapter, you will learn about divisibility, HCF and LCM, the Fundamental Theorem of Arithmetic, and irrational numbers.
Topics Covered:
- Euclid’s Division Lemma
- Fundamental Theorem of Arithmetic
- Prime Factorization and its Applications
- Rational and Irrational Numbers
- Decimal Expansions
- Exercises and Solutions
1. Euclid’s Division Lemma
The Euclid’s Division Lemma states:
For any two positive integers (a) and (b), there exist two whole numbers (q) and (r) such that:
[
a = b \times q + r, \quad 0 \leq r < b
]
- (a) is the dividend
- (b) is the divisor
- (q) is the quotient
- (r) is the remainder
Example 1:
Divide (45) by (7). Find the quotient (q) and remainder (r).
Solution:
[
45 = 7 \times 6 + 3
]
Here, the quotient (q = 6) and the remainder (r = 3).
This lemma is the basis for finding HCF (Highest Common Factor).
2. Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states:
Every composite number can be expressed (factorized) as a product of prime numbers in a unique way, except for the order of the factors.
Example 2:
Express (180) as a product of prime factors.
Solution:
[
180 = 2^2 \times 3^2 \times 5
]
This unique factorization helps in calculating HCF and LCM.
3. HCF and LCM Using Prime Factorization
Finding HCF (Highest Common Factor):
- Take the lowest power of all common prime factors.
Finding LCM (Lowest Common Multiple):
- Take the highest power of all prime factors involved.
Example 3:
Find the HCF and LCM of (60) and (72).
Solution:
Prime factorization of (60 = 2^2 \times 3 \times 5)
Prime factorization of (72 = 2^3 \times 3^2)
- HCF = (2^2 \times 3 = 12)
- LCM = (2^3 \times 3^2 \times 5 = 360)
[
\text{HCF} \times \text{LCM} = 60 \times 72 = 4320
]
The product of the two numbers is equal to the product of their HCF and LCM.
4. Rational and Irrational Numbers
Rational Numbers:
- A rational number can be expressed as (\frac{p}{q}), where (p) and (q) are integers, and (q \neq 0).
- Example: ( \frac{1}{2}, -3, 0.75 )
Irrational Numbers:
- An irrational number cannot be expressed as a ratio of two integers.
- Example: ( \sqrt{2}, \pi ), etc.
- These numbers have non-terminating, non-repeating decimal expansions.
5. Decimal Expansions of Real Numbers
Terminating Decimal Expansion:
A fraction has a terminating decimal expansion if, in its simplest form, the denominator has only 2 or 5 as prime factors.
Example:
[
\frac{1}{4} = 0.25
]
Non-terminating, Repeating Decimal Expansion:
If the decimal expansion repeats after some digits, it is called non-terminating, repeating.
Example:
[
\frac{1}{3} = 0.\overline{3}
]
Irrational Numbers:
- Have non-terminating, non-repeating decimal expansions.
Example:
[
\sqrt{2} = 1.41421356\ldots
]
Exercise Problems (NCERT Questions)
Exercise 1: Euclid’s Division Lemma
- Use Euclid’s Division Lemma to find the HCF of 56 and 72.
Solution:
72 = 56 1 + 16
[
56 = 16 \times 3 + 8
]
[
16 = 8 \times 2 + 0
]
HCF = 8
Exercise 2: Prime Factorization
- Express 504 as a product of prime factors.
Solution:
[
504 = 2^3 \times 3^2 \times 7
]
Exercise 3: HCF and LCM
- Find the HCF and LCM of 24 and 36.
Solution:
Prime factorization of (24 = 2^3 \times 3)
Prime factorization of (36 = 2^2 \times 3^2)
- HCF = (2^2 \times 3 = 12)
- LCM = (2^3 \times 3^2 = 72)
Exercise 4: Rational and Irrational Numbers
- Classify the following numbers as rational or irrational:
(a) ( \sqrt{25} )
(b) ( \sqrt{3} )
(c) ( 0.3333\ldots )
(d) ( \pi )
Solution:
(a) Rational (since ( \sqrt{25} = 5 ))
(b) Irrational (since ( \sqrt{3} ) is non-terminating)
(c) Rational (repeating decimal)
(d) Irrational
Conclusion
In this chapter, you learned about real numbers, Euclid’s division lemma, the Fundamental Theorem of Arithmetic, and HCF and LCM using prime factorization. You also explored the differences between rational and irrational numbers and learned how to interpret decimal expansions.
This chapter builds a strong foundation for understanding numbers and their properties, which is essential for higher-level mathematics.
Suggested Visuals:
- Prime Factor Tree for 180
- Number Line with Rational and Irrational Points
- Flowchart: Steps to Find HCF and LCM
- Examples of Terminating and Non-terminating Decimals
This content provides a comprehensive explanation of the chapter “Real Numbers” along with solved examples and exercises. The visuals can help students understand the concepts better and develop problem-solving skills.
