Chapter 1: Real Numbers Mathematics – Class 10 Reprint Edition: 2025–26

 

Real Numbers

Mathematics – Class 10
Reprint Edition: 2025–26

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Chapter 1: Real Numbers

1.1 Introduction

In Class IX, you began your exploration of the world of real numbers and encountered irrational numbers. We continue our discussion on real numbers in this chapter. We begin with two very important properties of positive integers in Sections 1.2, namely:

1. Euclid’s Division Algorithm

2. Fundamental Theorem of Arithmetic

Euclid’s Division Algorithm, as the name suggests, deals with the divisibility of integers. Stated simply, it means that for any two positive integers $a$ and $b$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that:

$a = bq + r \quad \text{where} \quad 0 \leq r < b$

This is essentially the long division process you have already used. Despite its simplicity, this algorithm has many powerful applications. One important application is computing the Highest Common Factor (HCF) of two positive integers.

The Fundamental Theorem of Arithmetic focuses on multiplication. It states that:

"Every composite number can be expressed (factorised) as a product of prime numbers, and this factorisation is unique, except for the order in which the prime factors occur."

This result, though simple, has deep implications in mathematics. We will explore its applications in two main areas:

1. Proving irrationality of certain numbers such as $\sqrt{2}, \sqrt{3}, \sqrt{5}$ etc.

2. Determining the nature of decimal expansions of rational numbers ($\frac{p}{q}$).

By studying the prime factorisation of the denominator $q$, we can determine whether the decimal expansion of $\frac{p}{q}$ is:

• Terminating, or

• Non-terminating repeating.

So let us begin our detailed exploration of these fascinating concepts.

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1.2 Euclid’s Division Lemma

Let us recall the division algorithm you are familiar with:

If you divide an integer $a$ by a non-zero integer $b$, you can write: $a = bq + r \quad \text{where} \quad 0 \leq r < b$

This result is formally called Euclid’s Division Lemma. It is a lemma because it is a proven statement used to prove other results.

Statement:

For any two positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that:

$a = bq + r \quad \text{where} \quad 0 \leq r < b$

Example:

Divide $87$ by $6$:

$87 = 6 \times 14 + 3 \quad \Rightarrow q = 14, \ r = 3$

Hence, Euclid’s Division Lemma holds.

Use of Euclid’s Division Lemma to find HCF

Euclid’s Division Lemma can be used to find the HCF (Highest Common Factor) of two positive integers. This method is known as Euclid’s Algorithm.

Euclid’s Algorithm Steps:

Let’s say we want to find the HCF of two positive integers $a$ and $b$, where $a > b$:

1. Apply the division lemma to $a$ and $b$: $a = bq + r$

2. If $r = 0$, then $b$ is the HCF of $a$ and $b$.

3. If $r \ne 0$, apply the division lemma again with $b$ and $r$.

4. Repeat the process until the remainder is 0. The divisor at this stage will be the HCF of $a$ and $b$.

Example:

Find the HCF of $455$ and $42$:

• Step 1: $455 = 42 \times 10 + 35$

• Step 2: $42 = 35 \times 1 + 7$

• Step 3: $35 = 7 \times 5 + 0$

Since remainder is 0, the HCF is $7$.

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1.3 The Fundamental Theorem of Arithmetic

This theorem is one of the most important results in number theory.

Statement:

Every composite number can be expressed (or factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Example:

Let’s take 90: $90 = 2 \times 3 \times 3 \times 5 = 2 \times 3^2 \times 5$

Another number: 315 $315 = 3 \times 3 \times 5 \times 7 = 3^2 \times 5 \times 7$

Application 1: Proving Irrationality

We use this theorem to prove that certain numbers are irrational. For example:

Proving $\sqrt{2}$ is irrational:

Assume $\sqrt{2}$ is rational. Then it can be written as $\frac{a}{b}$, where $a$ and $b$ are co-prime integers.

Squaring both sides: $2 = \frac{a^2}{b^2} \Rightarrow a^2 = 2b^2$ So $a^2$ is even $\Rightarrow a$ is even. Let $a = 2k$. Then:

But if both \(a\) and \(b\) are even, they have a common factor 2, contradicting the assumption that they are co-prime. Hence, \( \sqrt{2} \) is irrational. (Similar proofs can be done for \( \sqrt{3}, \sqrt{5} \), etc.) ### Application 2: Nature of Decimal Expansion Let \( \frac{p}{q} \) be a rational number, where \(p\) and \(q\) are co-prime and \(q \ne 0\). **The decimal expansion of \( \frac{p}{q} \) is terminating if and only if the prime factorisation of \(q\) is of the form \(2^n \times 5^m\), where \(n, m \geq 0\).** ### Example: - \( \frac{7}{8} = 0.875 \) (terminating, because 8 = \(2^3\)) - \( \frac{3}{11} = 0.272727\ldots \) (non-terminating repeating, because 11 is not 2 or 5) --- **End of Chapter 1**