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Dirichlet Theorem Video Lecture | Mathematics for Competitive Exams

Q: 4. How does Dirichlet's Theorem relate to the irrationality of certain numbers?

Ans. Dirichlet's Theorem can be used to prove the irrationality of certain numbers by contradiction. Suppose we have a real number x that is rational. Then, there exists some rational number p/q (where p and q are coprime integers) such that x = p/q. However, Dirichlet's Theorem guarantees that there exist infinitely many rational numbers that are close approximations to x. This contradicts the assumption that x is rational, leading to the conclusion that x must be irrational.

Q: 5. Is Dirichlet's Theorem applicable to all real numbers?

Ans. No, Dirichlet's Theorem is not applicable to all real numbers. It specifically applies to irrational numbers, which are numbers that cannot be expressed as the ratio of two integers. The theorem guarantees the existence of rational approximations to irrational numbers. For rational numbers, which can be expressed as the ratio of two integers, the theorem is not relevant as they can be exactly represented by fractions.

FAQs on Dirichlet Theorem Video Lecture - Mathematics for Competitive Exams

1. What is Dirichlet's Theorem in mathematics?

Ans. Dirichlet's Theorem, also known as Dirichlet's Approximation Theorem, states that for any real number x and any positive integer N, there exist infinitely many pairs of integers (a, b) such that the absolute difference between ax - b and zero is less than 1/N. In simpler terms, the theorem guarantees the existence of infinitely many rational numbers that are very close approximations to any given irrational number.

2. How is Dirichlet's Theorem applied in number theory?

Ans. Dirichlet's Theorem has various applications in number theory. One of its main applications is in the study of Diophantine approximation, where it provides a powerful tool to find rational number approximations to irrational numbers. It is also used to prove the existence of infinitely many prime numbers in certain arithmetic progressions, which is known as Dirichlet's Prime Number Theorem.

3. Can you provide an example illustrating Dirichlet's Theorem?

Ans. Sure! Let's consider the number π. According to Dirichlet's Theorem, there exist infinitely many pairs of integers (a, b) such that |aπ - b| < 1/N for any positive integer N. For example, one such pair is (3, 1) because |3π - 1| is less than 1. Another pair is (22, 7) because |22π - 7| is also less than 1. This theorem guarantees that we can always find rational approximations to π with arbitrarily high precision.

4. How does Dirichlet's Theorem relate to the irrationality of certain numbers?

Ans. Dirichlet's Theorem can be used to prove the irrationality of certain numbers by contradiction. Suppose we have a real number x that is rational. Then, there exists some rational number p/q (where p and q are coprime integers) such that x = p/q. However, Dirichlet's Theorem guarantees that there exist infinitely many rational numbers that are close approximations to x. This contradicts the assumption that x is rational, leading to the conclusion that x must be irrational.

5. Is Dirichlet's Theorem applicable to all real numbers?

Ans. No, Dirichlet's Theorem is not applicable to all real numbers. It specifically applies to irrational numbers, which are numbers that cannot be expressed as the ratio of two integers. The theorem guarantees the existence of rational approximations to irrational numbers. For rational numbers, which can be expressed as the ratio of two integers, the theorem is not relevant as they can be exactly represented by fractions.

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Sep 23, 2025 Last updated

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