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Real Numbers Class 10 Notes Maths Chapter 1

Real Numbers

Real numbers are all the numbers that can be found on the number line. This includes both rational numbers (like 7, -3, 0.5, and 4/3) and irrational numbers (like √2) . They encompass integers, fractions, and decimals, representing a continuous, unbroken set of values. 

Real Numbers Class 10 Notes Maths Chapter 1Rational numbers such as integers (-2, 0, 1), fractions (1/2, 2.5) and irrational numbers such as √3, π(22/7), etc., are all real numbers. 

Classification of Real Numbers

  1. Natural Numbers: Natural Numbers are a set of counting numbers. They are denoted by N.
    N = {1, 2, 3, 4..........∞}
  2. Whole Numbers: Whole numbers are a set of natural numbers plus zero.
    W = {0, 1, 2, 3 ......... ∞}
  3. Integers: Integers is a set of whole numbers and negative of all natural numbers.
    Z = { -3, -2, -1, 0, 1, 2, 3}
  4. Rational Numbers: All the numbers that can be written in the p/q form where p and q are integers and q ≠ 0 are called rational numbers.
    E.g. 8/11, -3/17
  5. Irrational Numbers: All the numbers that cannot be written in the p/q form are called irrational numbers. All the non-terminating and non-repeating decimal numbers are irrational numbers.
    E.g. √5, √3, √5 + √3, π

Fundamental Theorem of Arithmetic

To understand the fundamental theorem of Arithmetic, first, it is important to know what are composite numbers and prime numbers.

Composite Number

Composite Numbers are those numbers that have at least one factor other than one and the number itself.
Consider a number, 10. Now, the factors of 10 are 1, 2, 5 and 10. So it is a composite number.

Prime Number

Prime Numbers are those numbers that have exactly two factors: 1 and the number itself.
Let us take one more number 23. Now, the factors of 23 are 1 and 23. That means it has two factors 1 and the number itself, which is called a prime number.

Prime and Composite NumbersPrime and Composite Numbers

Theorem:  Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Any composite number can be written as a product of primes in one way only as long as we are not particular about the order in which the primes occur.
Let us consider an example here: we will use a tree diagram to show the factors of 270.
Real Numbers Class 10 Notes Maths Chapter 1270 = 2 x 33 x 5
Here, in the prime factorization of 270, the prime numbers appearing in both cases are the same, only the order in which they appear is different.
Therefore, the prime factorization of 270 is unique except for the order in which the primes occur.

Example 1:  Check whether 15n can end with the digit zero for any natural number n.

Sol: 

Real Numbers Class 10 Notes Maths Chapter 1

Example 2: Explain, why (7 × 6 × 5 × 4 × 3 × 2 × 1) + 5 and (3 × 5 × 13 × 46) + 23 is a composite number?

Sol:

Real Numbers Class 10 Notes Maths Chapter 1

HCF and LCM by Prime Factorisation Method

In this method, we first express the given numbers as a product of prime factors separately. Then, HCF is the product of the smaller power of each common prime factor in the numbers, and LCM is the product of the greatest power of each prime factor involved in the numbers.

For any two positive integers a and b,

HCF (a, b) × LCM (a, b) = a × b


Example 3: Find the LCM and HCF of 120 and 144 by the fundamental arithmetic theorem.

Sol:

120 = 23 × 3 × 5
144 = 2× 32
Now, HCF is the product of the smallest power of each common prime factor in the numbers.
Real Numbers Class 10 Notes Maths Chapter 1HCF (120, 144) = 23 × 3 = 8 × 3 = 24
Real Numbers Class 10 Notes Maths Chapter 1LCM is the product of the greatest power of each prime factor involved in the numbers.
LCM (120,144) = 24 × 32 × 5 = 16 × 9 × 5 = 720

Example 4: If two positive integers p and q can be expressed as p = ab2 and q = a3b, where a, b are prime numbers, find the LCM (p, q).

Sol:

Given: p = ab2 and q = a3b
Real Numbers Class 10 Notes Maths Chapter 1LCM is the product of the greatest power of each prime factor involved in the numbers.
LCM (p, q) = a3 × b2 = a3b2

Example 5: Write the HCF and LCM of the smallest odd composite number and the smallest odd prime number.

Sol:

The smallest odd composite number is 9, and the smallest odd prime number is 3.
9 = 32
3 = 31
Now, the smallest power of the common prime factor is 31.
HCF (9, 3) = 3
The greatest power of the common prime factor is 32.
LCM (9, 3) = 32 = 9

Example 6: If HCF (253,440) = 11 and LCM (253,440)= 253 × R. Find the value of R.

Sol:

We know that,
HCF (a, b) × LCM (a, b) = a × b
∴ HCF (253, 440) × LCM (253, 440) = 253 × 440
11 × 253 × R = 253 × 440
R = 253 x 440 / 253 x 11
R = 40

Example 7: Ravi and Shikha drive around a circular sports field. Ravi takes 16 min to complete one round, while Shikha completes the round in 20 min. If both start at the same point, at the same time, and go in the same direction, then how much time will they meet at the starting point?

Sol:

Time taken by Ravi to drive one round of the circular field = 16 min. Time taken by Shikha to drive one round of the circular field =20 min.
The time after which they will again meet at the starting point will be equal to the LCM of 16 min and 20 min.
16 = 24
20 = 22 × 5
LCM(16, 20) = 24 × 5 = 16 × 5 = 80
Therefore, Ravi and Shikha will meet again at the starting point after 80 min.

Question for Chapter Notes: Real Numbers
Try yourself:According to the Fundamental Theorem of Arithmetic, which of the following statements is true?
View Solution

Revisiting Irrational Numbers

Irrational numbers are those numbers that cannot be written in the form p/q, where p and q are integers and q ≠0. E.g.,√2, √3, √15

The square roots of all the numbers do not give an irrational number.

Real Numbers Class 10 Notes Maths Chapter 1For example, √2 is an irrational number, but √4 = 2, which is rational.
Therefore, the square roots of all prime numbers are irrational.
If p is a prime number, then √p is an irrational number.

Theorem : If a prime number p divides a2, then p divides a, where a is a positive integer.

Real Numbers Class 10 Notes Maths Chapter 1

Theorem: Prove that √2  is an irrational number.

Sol:

Real Numbers Class 10 Notes Maths Chapter 1

Example 8: Show that 3√2 is an irrational number.

Sol:

Real Numbers Class 10 Notes Maths Chapter 1

Example 9: Show that 5 − √3 is irrational.

Sol:

Real Numbers Class 10 Notes Maths Chapter 1

The document Real Numbers Class 10 Notes Maths Chapter 1 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Real Numbers Class 10 Notes Maths Chapter 1

1. What are the different classifications of real numbers?
Ans. Real numbers can be classified into several categories: 1. <b>Natural Numbers</b>: These are the counting numbers starting from 1 (1, 2, 3, ...). 2. <b>Whole Numbers</b>: These include all natural numbers and zero (0, 1, 2, ...). 3. <b>Integers</b>: This set includes whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3, ...). 4. <b>Rational Numbers</b>: Numbers that can be expressed as a fraction of two integers, where the denominator is not zero (e.g., 1/2, -3/4). 5. <b>Irrational Numbers</b>: Numbers that cannot be expressed as fractions, such as √2 or π. 6. <b>Real Numbers</b>: The complete set that includes all rational and irrational numbers.
2. What is the Fundamental Theorem of Arithmetic?
Ans. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed uniquely as a product of prime numbers, apart from the order of the factors. This means that any composite number can be broken down into prime factors, and this factorization is unique for each number, which is crucial for understanding number theory.
3. How can we find the HCF and LCM using the Prime Factorisation Method?
Ans. To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) using the Prime Factorisation Method: 1. <b>Prime Factorisation</b>: Factor each number into its prime factors. 2. <b>HCF</b>: Take the product of the lowest powers of all common prime factors. 3. <b>LCM</b>: Take the product of the highest powers of all prime factors present in both numbers. For example, for numbers 12 (2² × 3¹) and 18 (2¹ × 3²): - HCF = 2¹ × 3¹ = 6 - LCM = 2² × 3² = 36
4. Can you explain irrational numbers with examples?
Ans. Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They have non-repeating, non-terminating decimal expansions. Examples include: 1. <b>Square roots of non-perfect squares</b>: √2, √3, √5, etc. 2. <b>Mathematical constants</b>: π (approximately 3.14159...), e (approximately 2.71828...). These numbers cannot be written as simple fractions, which differentiates them from rational numbers.
5. Why is understanding real numbers important in mathematics?
Ans. Understanding real numbers is fundamental in mathematics because they form the basis for various mathematical concepts and operations. Real numbers are used in algebra, geometry, calculus, and applied mathematics. They help in modeling real-world situations, solving equations, and performing calculations. Additionally, the classification of real numbers into rational and irrational helps in deepening the understanding of number systems and their properties.
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