Class 10 Maths Real Numbers Notes

Class 10 Maths Notes – Real Numbers

Introduction

Real numbers are the foundation of much of mathematics and are essential for solving various types of problems. They can be broadly classified into rational and irrational numbers. Understanding real numbers is crucial, not just for exams, but also for grasping the deeper concepts in mathematics. In this blog, we will explore the key concepts from the “Real Numbers” chapter of Class 10 Maths, including the Fundamental Theorem of Arithmetic and irrational numbers.

1.1 Introduction to Real Numbers

Real numbers include all the numbers that can be found on the number line, which means they encompass both rational and irrational numbers.

Rational Numbers:

Rational numbers are numbers that can be written as the ratio of two integers, where the denominator is not zero. They can be expressed in the form $\frac{p}{q}$ qp, where $p$ p and $q$ q are integers and $q \neq 0$ q=0. Examples of rational numbers include $2, \frac{3}{4}, -5, 0.25$ 2,43,−5,0.25, etc.

Irrational Numbers:

Irrational numbers are numbers that cannot be written as a simple fraction. Their decimal form goes on forever without repeating. Examples include $\sqrt{2}, \pi, e$ 2,π,e, etc.

Real Numbers:

The real number system is the combination of both rational and irrational numbers. This means any number that can be plotted on the number line is a real number.

1.2 The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic is one of the most important concepts in number theory. It states that:

“Every integer greater than 1 is either a prime number or can be factored uniquely into prime numbers (except for the order of the factors).”

In other words, every composite number can be broken down into prime factors, and this factorization is unique for every number.

Prime Numbers:

A prime number is a number greater than 1 that has no divisors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, 13, etc.

Composite Numbers:

A composite number is a number that can be factored into smaller integers other than 1 and itself. For example, 6 can be factored into $2 \times 3$ 2×3, 12 can be factored into $2 \times 2 \times 3$ 2×2×3, etc.

Prime Factorization:

The prime factorization of a number is the representation of that number as the product of prime numbers. For instance:

$18 = 2 \times 3 \times 3$ 18=2×3×3 or $18 = 2 \times 3^2$ 18=2×32
$36 = 2^2 \times 3^2$ 36=22×32

This factorization is unique, as guaranteed by the Fundamental Theorem of Arithmetic. The uniqueness of prime factorization is essential in various areas of mathematics, such as finding the greatest common divisor (GCD) and least common multiple (LCM).

1.3 Revisiting Irrational Numbers

Irrational numbers play a significant role in mathematics. These numbers cannot be written as fractions and their decimal expansion is non-terminating and non-repeating.

Examples of Irrational Numbers:

$\sqrt{2}$ 2: The square root of 2 is irrational because it cannot be expressed as a fraction, and its decimal representation goes on forever without repeating.
$\pi$ π: The value of pi (π) is approximately 3.14159, but it is irrational, meaning it has an infinite number of non-repeating digits after the decimal point.
$e$ e: The base of natural logarithms is another example of an irrational number.

Important Properties:

Irrational numbers are dense on the number line. This means between any two real numbers, there exists an irrational number.
When adding or multiplying a rational number with an irrational number, the result is always irrational, except in certain cases like 0.

Approximating Irrational Numbers:

Although we cannot express irrational numbers exactly as fractions, we can approximate them. For example, $\pi$ π is often approximated as 3.14 or $\frac{22}{7}$ 722. Similarly, $\sqrt{2}$ 2 is approximated as 1.414.

1.4 Summary

In this chapter, we have covered the basic concepts of real numbers, including their classification into rational and irrational numbers. We explored the importance of the Fundamental Theorem of Arithmetic, which ensures the uniqueness of prime factorization for integers. Furthermore, we revisited the properties of irrational numbers, which are integral to our understanding of real numbers.

Key takeaways:

Real numbers are the set of rational and irrational numbers.
The Fundamental Theorem of Arithmetic ensures the uniqueness of prime factorization.
Irrational numbers, though non-repeating and non-terminating, are a critical part of the real number system.

1. Which of the following is a rational number?

a) $\pi$ π
b) $\sqrt{3}$ 3
c) $\frac{5}{7}$ 75
d) $\sqrt{2}$ 2

Answer: c) $\frac{5}{7}$ 75

2. The number 5\sqrt{5}5 is:

a) Rational
b) Irrational
c) Integer
d) Natural

Answer: b) Irrational

3. Which of the following numbers is a prime number?

a) 15
b) 23
c) 39
d) 51

Answer: b) 23

4. The square root of 9 is:

a) 2
b) 3
c) $\sqrt{3}$ 3
d) 4

Answer: b) 3

5. Which of the following is the prime factorization of 36?

a) $2 \times 3$ 2×3
b) $2^2 \times 3^2$ 22×32
c) $3^2 \times 2^2$ 32×22
d) $6 \times 6$ 6×6

Answer: b) $2^2 \times 3^2$ 22×32

6. The Fundamental Theorem of Arithmetic states that every number greater than 1 is:

a) A composite number
b) A prime number
c) Either a prime or composite number
d) Either prime or a unique product of prime factors

Answer: d) Either prime or a unique product of prime factors

7. Which of the following is an irrational number?

a) $\frac{22}{7}$ 722
b) 0.5
c) $\pi$ π
d) 7

Answer: c) $\pi$ π

8. Which of the following numbers has a non-terminating, non-repeating decimal expansion?

a) 3.5
b) $\frac{1}{3}$ 31
c) $\sqrt{7}$ 7
d) 0.25

Answer: c) $\sqrt{7}$ 7

9. Which of the following is the correct prime factorization of 56?

a) $2^3 \times 7$ 23×7
b) $2^4 \times 7$ 24×7
c) $2 \times 3 \times 7$ 2×3×7
d) $7^2 \times 2$ 72×2

Answer: a) $2^3 \times 7$ 23×7

10. The decimal expansion of an irrational number is:

a) Non-terminating and non-repeating
b) Terminating
c) Repeating
d) Finite

Answer: a) Non-terminating and non-repeating

11. Which of the following is not an irrational number?

a) $\sqrt{5}$ 5
b) $\pi$ π
c) $\frac{7}{9}$ 97
d) $\sqrt{2}$ 2

Answer: c) $\frac{7}{9}$ 97

12. What is the prime factorization of 63?

a) $2 \times 3^2$ 2×32
b) $3^2 \times 7$ 32×7
c) $2^2 \times 3 \times 7$ 22×3×7
d) $3 \times 7$ 3×7

Answer: b) $3^2 \times 7$ 32×7

13. Which of the following is a composite number?

a) 13
b) 17
c) 19
d) 21

Answer: d) 21

14. Which of the following is the correct prime factorization of 72?

a) $2^2 \times 3^2$ 22×32
b) $2^3 \times 3$ 23×3
c) $2^3 \times 3^2$ 23×32
d) $3^2 \times 4$ 32×4

Answer: c) $2^3 \times 3^2$ 23×32

15. The least common multiple (LCM) of two numbers is:

a) Always a prime number
b) A product of their prime factors
c) The greatest common divisor (GCD) of the two numbers
d) The smallest number that both numbers divide into exactly

Answer: d) The smallest number that both numbers divide into exactly

16. Which of the following is an example of a terminating decimal?

a) $\frac{5}{7}$ 75
b) $\frac{4}{9}$ 94
c) $\frac{1}{8}$ 81
d) $\pi$ π

Answer: c) $\frac{1}{8}$ 81

17. What is the prime factorization of 120?

a) $2^3 \times 3 \times 5$ 23×3×5
b) $2^2 \times 3 \times 5$ 22×3×5
c) $2^3 \times 3^2 \times 5$ 23×32×5
d) $2 \times 3^2 \times 5$ 2×32×5

Answer: a) $2^3 \times 3 \times 5$ 23×3×5

18. Which of the following numbers is neither a rational nor an irrational number?

a) $\sqrt{5}$ 5
b) 0
c) 1
d) Infinity

Answer: d) Infinity

19. Which of the following is true about irrational numbers?

a) They can always be written as a fraction
b) They are always whole numbers
c) They cannot be written as a fraction and have a non-terminating, non-repeating decimal
d) They are always negative numbers

Answer: c) They cannot be written as a fraction and have a non-terminating, non-repeating decimal

20. Which of the following is not a prime number?

a) 2
b) 3
c) 5
d) 9

Answer: d) 9