Real Numbers

Quick Revision Notes: Class 10 Mathematics

Chapter 1: Real Numbers (वास्तविक संख्याएँ)

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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.

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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)

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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.

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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).

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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.

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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.

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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).