Real Numbers

Quick Revision Notes: Class 10 Mathematics

Chapter 1: Real Numbers (वास्तविक संख्याएँ)

1. Introduction & Classification of Numbers

Natural Numbers (प्राकृतिक संख्याएँ): Counting numbers starting from $1, 2, 3, 4, \dots$
Whole Numbers (पूर्ण संख्याएँ): Natural numbers including zero: $0, 1, 2, 3, \dots$
Integers (पूर्णांक): All whole numbers and their negatives: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$
Rational Numbers (परिमेय संख्याएँ): Numbers that can be expressed in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$. (e.g., $3/4, -5, 0.25$)
Irrational Numbers (अपरिमेय संख्याएँ): Numbers that cannot be expressed in the form $p/q$. Their decimal expansion is non-terminating and non-recurring. (e.g., $\sqrt{2}, \sqrt{3}, \pi$)
Real Numbers (वास्तविक संख्याएँ): The collection of all rational and irrational numbers.

2. Euclid's Division Lemma (यूक्लिड विभाजन प्रमेयिका)

For any two given positive integers $a$ and $b$, there exist unique whole numbers $q$ and $r$ satisfying: a = bq + r where 0 ≤ r < b

a: Dividend (भाज्य)
b: Divisor (भाजक)
q: Quotient (भागफल)
r: Remainder (शेषफल)

3. Euclid's Division Algorithm (यूक्लिड विभाजन एल्गोरिथ्म)

It is a technique to compute the Highest Common Factor (HCF / म.स.) of two given positive integers.

Step 1: Apply Euclid's division lemma to $a$ and $b$ ($a > b$), to find $q$ and $r$ such that $a = bq + r$, $0 \le r < b$.
Step 2: If $r = 0$, $b$ is the HCF of $a$ and $b$. If $r \neq 0$, apply Euclid's division lemma to $b$ and $r$.
Step 3: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

4. The Fundamental Theorem of Arithmetic (अंकगणित की आधारभूत प्रमेय)

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

Composite Number (भाज्य संख्या): Numbers having more than two factors.
Prime Number (अभाज्य संख्या): Numbers having exactly two factors (1 and itself).

5. Important Relation between HCF and LCM

For any two positive integers $a$ and $b$: HCF (a, b) × LCM (a, b) = a × b

HCF (Highest Common Factor / महत्तम समापवर्तक): Product of the smallest power of each common prime factor in the numbers.
LCM (Lowest Common Multiple / लघुत्तम समापवर्त्य): Product of the greatest power of each prime factor involved in the numbers.

Note: This property (HCF × LCM = Product of two numbers) does NOT generally hold true for three or more numbers.

6. Revisiting Irrational Numbers (अपरिमेय संख्याओं का पुनر्विमर्श)

If $p$ is a prime number and $p$ divides $a^2$, then $p$ divides $a$, where $a$ is a positive integer.
Common Irrational Numbers: Numbers like $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $3 + 2\sqrt{5}$, etc., can be proven irrational using the method of contradiction (विरोधाभास विधि).
Key Rules:
- Sum or difference of a rational and an irrational number is always irrational.
- Product or quotient of a non-zero rational and an irrational number is always irrational.

7. Decimal Expansions of Rational Numbers (परिमेय संख्याओं के दशमलव प्रसार)

Let $x = \frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is of the form $2^n \cdot 5^m$, where $n$ and $m$ are non-negative integers. Then $x$ has a terminating decimal expansion (शांत दशमलव प्रसार).

If the prime factorization of $q$ is not of the form $2^n \cdot 5^m$ (where $q$ has prime factors other than 2 and 5), then $x$ has a non-terminating repeating (recurring) decimal expansion (अशांत आवर्ती दशमलव प्रसार).

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