Real Numbers Class 9
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Questions and Answers

What does the Fundamental Theorem of Arithmetic state?

  • Every integer can be expressed as a sum of primes.
  • Every composite number can be factorized uniquely into primes. (correct)
  • Every prime number can be expressed as a sum of composites.
  • All integers are either prime or composite.

    • Define a composite number.

    A composite number is a positive integer that has at least one positive divisor other than one or itself.

    If a prime number divides a square, it must also divide the original number.

    True

    Match the following terms with their definitions:

    <p>Prime Number = A number greater than 1 that has no positive divisors other than 1 and itself. Composite Number = A number that has at least one positive divisor other than 1 and itself. Rational Number = Any number that can be expressed as the quotient of two integers. Irrational Number = A number that cannot be expressed as a fraction of two integers.</p> Signup and view all the answers

    The number ____ is an example of a prime number.

    <p>7</p> Signup and view all the answers

    Which of the following is an example of a composite number?

    <p>4</p> Signup and view all the answers

    The number 1 is considered a composite number.

    <p>False</p> Signup and view all the answers

    Give an example of a number that is both prime and irrational.

    <p>None, as all prime numbers are rational.</p> Signup and view all the answers

    What does the Fundamental Theorem of Arithmetic state?

    <p>Every composite number can be expressed as a product of primes uniquely.</p> Signup and view all the answers

    The first correct proof of the Fundamental Theorem of Arithmetic was given by Archimedes.

    <p>False</p> Signup and view all the answers

    What unique property does the factorization of a composite number have according to the Fundamental Theorem of Arithmetic?

    <p>It is unique apart from the order of the prime factors.</p> Signup and view all the answers

    The prime factorization of 32760 is __________.

    <p>2^3 × 3^2 × 5 × 7 × 13</p> Signup and view all the answers

    Match the mathematicians to their contributions.

    <p>Carl Friedrich Gauss = Fundamental Theorem of Arithmetic Euclid = First recorded version of the theorem Archimedes = One of the greatest mathematicians Isaac Newton = Influenced mathematics and science</p> Signup and view all the answers

    Which of the following numbers is a composite number?

    <p>9</p> Signup and view all the answers

    List one composite number and its prime factorization.

    <p>Example: 12, Prime factorization: 2^2 × 3</p> Signup and view all the answers

    The numbers 3803 and 3607 are primes.

    <p>True</p> Signup and view all the answers

    What does the Fundamental Theorem of Arithmetic state?

    <p>Every composite number can be expressed as a product of primes in a unique way.</p> Signup and view all the answers

    The Euclid’s division algorithm can be used to compute the LCM of two numbers.

    <p>False</p> Signup and view all the answers

    What is a composite number?

    <p>A composite number is a positive integer that has at least one positive divisor other than one and itself.</p> Signup and view all the answers

    Every prime number greater than 2 is __________.

    <p>odd</p> Signup and view all the answers

    Match the type of number with its definition:

    <p>Prime Number = A number greater than 1 that has no positive divisors other than 1 and itself. Composite Number = A number greater than 1 that is not prime. Irrational Number = A number that cannot be expressed as a fraction of two integers. Rational Number = A number that can be expressed as the quotient of two integers.</p> Signup and view all the answers

    Which of the following is a prime number?

    <p>7</p> Signup and view all the answers

    All prime numbers are also composite numbers.

    <p>False</p> Signup and view all the answers

    What is the significance of the number 1 in the context of prime and composite numbers?

    <p>The number 1 is neither prime nor composite; it only has one positive divisor.</p> Signup and view all the answers

    Study Notes

    Introduction to Real Numbers

    • Exploration of real numbers continues, following the introduction to irrational numbers in Class IX.
    • Focus on important properties of positive integers: Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.

    Euclid’s Division Algorithm

    • A positive integer ( a ) can be divided by another positive integer ( b ) such that the remainder ( r ) is less than ( b ).
    • Fundamental for understanding divisibility among integers.
    • Utilized primarily to compute the highest common factor (HCF) of two positive integers.

    Fundamental Theorem of Arithmetic

    • Every composite number can be expressed uniquely as a product of prime numbers, except for the order of factors.
    • Provides a basis for proving the irrationality of certain numbers (e.g., ( \sqrt{2}, \sqrt{3}, \sqrt{5} )).
    • Important for determining whether the decimal expansion of a rational number is terminating or non-terminating repeating.

    Proof of Irrationality

    • To show that ( \sqrt{5} - \sqrt{3} ) is irrational, assume it is rational, leading to a contradiction with the properties of integers.
    • Similar proof structure used to demonstrate ( \sqrt{3}^2 ) is irrational, using the assumption of rationality and ending in contradiction.

    Examples and Exercises

    • Exercises encourage proving various forms of irrational numbers:
      • ( \sqrt{5} )
      • ( 3 + 2\sqrt{5} )
      • Other combinations of irrational numbers.

    Summary of Key Concepts

    • Fundamental Theorem of Arithmetic:
      • Unique prime factorization of composite numbers.
      • Example: ( 32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13 ).
    • Properties of Primes:
      • If ( p ) is prime and divides ( a^2 ), then ( p ) divides ( a ).
    • Historical Context:
      • Theorem first recorded by Euclid and formally proven by Carl Friedrich Gauss, a pivotal figure in mathematics.

    Significance of the Theorem

    • Provides an essential foundation for number theory and integer arithmetic.
    • Highlights the uniqueness of prime factorization and its broad applications in mathematics and science.

    Important Figures

    • Carl Friedrich Gauss (1777 – 1855): Known as the "Prince of Mathematicians," recognized for his extensive contributions to mathematics and science.

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    Real Numbers PDF 10th

    Description

    Explore real numbers in this quiz designed for Class IX students. You will dive deeper into concepts such as Euclid's division algorithm and the properties of positive integers. Test your understanding of both rational and irrational numbers as you enhance your mathematical knowledge.

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