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Real Numbers — Revision Notes

Gujarat Board · Class 10 · Mathematics

Real Numbers revision notes for Gujarat Board Class 10 Mathematics: 3 topics in quick points. Part of the Gujarat Board Class 10 Mathematics syllabus.

45 questions24 flashcards7 formulas & key relations5 concepts

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A nested Venn diagram illustrating the relationships between natural numbers (N), integers (Z), rational numbers (Q), and real numbers (R) as subsets of each other.
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Key Topics to Revise

1

The Fundamental Theorem of Arithmetic

  • Every composite number can be expressed as a product of prime numbers, and this factorisation is UNIQUE (except for the order of factors).
  • This is called the Fundamental Theorem of Arithmetic because it is the most basic and important fact about integers.
  • The theorem was first recorded in Euclid's Elements (Proposition 14, Book IX) and correctly proved by Carl Friedrich Gauss in 'Disquisitiones Arithmeticae'.
2

Revisiting Irrational Numbers and Proofs

  • A number is irrational if it CANNOT be written as p/q, where p and q are integers and q ≠ 0.
  • Examples of irrationals: √2, √3, √5, √7, π, 0.10110111011110...
  • The proof that √2 is irrational uses a method called 'Proof by Contradiction'.
3

Step-by-Step: Proof that √2 is Irrational

  • STEP 1 — Assume the Opposite: Assume that √2 is rational.
  • STEP 2 — Write in Fraction Form: Since √2 is rational, we can write √2 = a/b, where a and b are integers, b ≠ 0, and HCF(a, b) = 1 (i.e., a and b are coprime).
  • STEP 3 — Square Both Sides: Squaring gives 2 = a²/b², so a² = 2b².

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Full Notes

Key Concepts

Every composite number can be expressedThis theorem states that every compositeA number is irrational if itThis is a logical technique whereA prime number is a natural

Frequently Asked Questions

What are the important topics in Real Numbers for Gujarat Board Class 10 Mathematics?
Key topics in Real Numbers include The Fundamental Theorem of Arithmetic, Revisiting Irrational Numbers and Proofs, Step-by-Step: Proof that √2 is Irrational. Study these first, then practise questions on each for the Gujarat Board Class 10 board exam.
How should I revise Real Numbers for the Gujarat Board Class 10 board exam?
Learn the core ideas first, then work through the 45 practice questions on Real Numbers. Revise definitions regularly and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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All Gujarat Board Class 10 Mathematics Revision Notes

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