📘 Real Numbers – Class 10 NCERT

🔢 1. Euclid's Division Lemma

Euclid's Division Lemma describes the relationship between two positive integers and their quotient and remainder. According to this lemma, for positive integers \(a\) and \(b\), there exist unique integers \(q\) and \(r\) such that:

Here, \(q\) represents the quotient, while \(r\) represents the remainder. Furthermore, the remainder always satisfies:

Thus, the divisor must always be greater than the remainder.

💡 Example

Consider the division of 17 by 5. In this case, the relationship can be written as:

Consequently, the quotient is 3 and the remainder is 2. ✅

🧮 2. Euclid's Division Algorithm

Euclid's Division Algorithm provides a convenient method for finding the HCF of two positive integers. It works by applying the division lemma repeatedly.

First, divide the larger number by the smaller number. Next, use the remainder as the new divisor. Continue the process until the remainder becomes zero.

📝 Steps to Find HCF

1. Divide the larger number by the smaller number.
2. Use the remainder in the next division.
3. Repeat the divisions until the remainder becomes zero.
4. Finally, identify the last non-zero remainder.

🔐 3. Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers.

More importantly, this prime factorisation is unique apart from the order in which the prime factors are written.

🔍 Example of Prime Factorisation

For instance, consider the number 60. Its prime factorisation is:

As a result, prime factorisation becomes useful when finding the HCF and LCM of two or more numbers.

🔗 4. HCF Using Prime Factorisation

Prime factorisation offers another method for finding the HCF. First, express each number as a product of prime factors.

After that, identify the prime factors common to all the numbers. Then select the smallest power of each common prime factor.

🔗 5. LCM Using Prime Factorisation

Similarly, prime factorisation can be used to determine the LCM. Start by writing every number as a product of prime factors.

Subsequently, select the greatest power of every prime factor that appears in the given numbers.

⭐ Important HCF–LCM Relationship

For any two positive integers \(a\) and \(b\), the following relationship is very important:

Therefore, if three of the four quantities are known, the fourth quantity can be calculated easily.

🚫 6. Irrational Numbers

An irrational number is a number that cannot be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).

Common examples include √2, √3, √5 and π. Unlike rational numbers, their decimal expansions do not terminate or repeat in a fixed pattern.

🧠 7. Proof of Irrationality

Class 10 Mathematics includes proofs that numbers such as √2, √3 and √5 are irrational.

Usually, these proofs use the method of contradiction. First, assume that the given number is rational.

Next, represent the number in the form \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers.

The mathematical steps then produce a contradiction. Hence, the original assumption is false, proving that the number is irrational. ✅

🔢 8. Rational Numbers

A rational number is any number that can be represented in the form:

Here, both \(p\) and \(q\) are integers. Rational numbers can be positive, negative or zero.

📌 Examples

• \(\frac{1}{2}\)
• \(\frac{3}{4}\)
• \(-\frac{5}{7}\)
• \(8 = \frac{8}{1}\)

💻 9. Decimal Expansion of Rational Numbers

A rational number can have either a terminating decimal expansion or a non-terminating recurring decimal expansion.

To identify the type of decimal expansion, first express \(\frac{p}{q}\) in its lowest form.

After simplification, examine the prime factors of the denominator. The result can then be determined using the following rule.

Denominator After Simplification	Decimal Expansion
Only 2 and/or 5 as prime factors	Terminating decimal ✅
Any prime factor other than 2 or 5	Non-terminating recurring decimal 🔁

✅ Example: Terminating Decimal

Consider the rational number \(\frac{3}{8}\). Since \(8 = 2^3\), its only prime factor is 2.

Therefore, the decimal expansion terminates.

🔁 Example: Non-Terminating Recurring Decimal

Now consider \(\frac{2}{15}\). Here, \(15 = 3 \times 5\), so the denominator contains the prime factor 3.

Thus, the decimal expansion is non-terminating and recurring.

📊 10. Rational vs Irrational Numbers

The following comparison makes the difference between rational and irrational numbers easier to remember.

Rational Numbers	Irrational Numbers
Can be written as \(\frac{p}{q}\)	Cannot be written as \(\frac{p}{q}\)
Decimal may terminate	Decimal never terminates
Decimal may be recurring	Decimal does not repeat in a fixed pattern
Example: \(\frac{3}{4}\)	Example: √2

⭐ 11. Important Results

⚡ 12. Quick Revision

❓ 13. Frequently Asked Questions

🎯 Final Exam Preparation