# Class 10 Mathematics – Chapter 1: Real Numbers
**[Opening]**
Hello students!
Welcome to Class 10 Mathematics.
In this chapter, we are going to learn about **Real Numbers**.
Don't worry—this chapter is quite easy if we understand the concepts step by step.
We will mainly learn about **Euclid's Division Algorithm, Fundamental Theorem of Arithmetic, HCF and LCM, irrational numbers, and decimal expansions of rational numbers.**
Let's begin!
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## Concept 1: Euclid's Division Lemma
Suppose we have two positive integers, **a** and **b**.
Euclid's Division Lemma says:
**a = bq + r**
where **r is greater than or equal to 0 and less than b**.
Here:
* **a** is the dividend
* **b** is the divisor
* **q** is the quotient
* **r** is the remainder
For example:
**225 = 135 × 1 + 90**
Here, 225 is the dividend, 135 is the divisor, 1 is the quotient, and 90 is the remainder.
This idea is used in **Euclid's Division Algorithm to find the HCF of two numbers.**
---
## Concept 2: Euclid's Division Algorithm
Let's find the HCF of **225 and 135**.
First:
225 = 135 × 1 + 90
Now divide 135 by 90:
135 = 90 × 1 + 45
Again:
90 = 45 × 2 + 0
When the remainder becomes zero, the **last non-zero remainder is the HCF**.
Therefore:
**HCF = 45**
So remember:
**Keep dividing until the remainder becomes zero. The last non-zero remainder is the HCF.**
---
## Concept 3: Fundamental Theorem of Arithmetic
Now let's talk about prime factorisation.
Every composite number can be expressed as a product of prime numbers, and this factorisation is unique, apart from the order of the factors.
For example:
60 can be written as:
**60 = 2 × 2 × 3 × 5**
or
**60 = 2² × 3 × 5**
This is called the **prime factorisation** of 60.
This concept is very useful when finding the **HCF and LCM**.
---
## Concept 4: HCF and LCM
Suppose:
**24 = 2³ × 3**
and
**36 = 2² × 3²**
For HCF, we take the **smallest powers of the common prime factors**.
So:
**HCF = 2² × 3 = 12**
For LCM, we take the **greatest powers of all prime factors**.
So:
**LCM = 2³ × 3² = 72**
There is also an important relationship:
**HCF × LCM = Product of the two numbers**
So for 24 and 36:
**12 × 72 = 24 × 36**
Both sides are equal to 864.
---
## Concept 5: Irrational Numbers
Now let's understand irrational numbers.
A rational number can be written in the form:
**p/q**
where p and q are integers and q is not zero.
Numbers that cannot be expressed in this form are called **irrational numbers**.
For example:
**√2, √3, √5 and π**
are irrational numbers.
In this chapter, you will also learn how to **prove that a number is irrational**, usually using proof by contradiction.
---
## Concept 6: Decimal Expansion
The last important concept is the decimal expansion of rational numbers.
Suppose we have a rational number:
**p/q**
where p and q have no common factor.
If the denominator q has only the prime factors **2 and 5**, then the decimal expansion will be **terminating**.
For example:
**3/8 = 0.375**
Since:
**8 = 2³**
the decimal expansion terminates.
But if the denominator contains a prime factor other than 2 or 5, the decimal expansion will be **non-terminating recurring**.
For example:
**5/6 = 0.8333...**
Since:
**6 = 2 × 3**
and it contains the prime factor 3, the decimal expansion is non-terminating recurring.
---
## Quick Revision
Let's quickly revise what we learned.
First, **Euclid's Division Algorithm** helps us find the HCF.
Second, **Fundamental Theorem of Arithmetic** tells us about unique prime factorisation.
Third, we can use **prime factorisation to find HCF and LCM**.
Fourth, numbers such as **√2 and √3 are irrational numbers**.
And finally, the prime factors of the denominator help us determine whether a rational number has a **terminating or non-terminating recurring decimal expansion**.
That's it for the basic concepts of **Real Numbers**.
In the next lesson, we'll solve questions based on these concepts step by step.
**Keep practising, and remember—understanding the concept is more important than memorising the formula!**
