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Introduction to Real Numbers

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Questions and Answers

What does Euclid's division algorithm state about two positive integers a and b?

The integer *a* can be divided by *b* and leaves a remainder *r* that is greater than or equal to *b*.

The integer *b* can be divided by *a* leaving a remainder *r*.

The integer *a* can be divided by *b* leaving a remainder *r* smaller than *b*. (correct)

The integer *a* can always be divided by *b* without any remainder.

What is a key application of the Fundamental Theorem of Arithmetic?

It is used to calculate the square roots of irrational numbers.

It simplifies the process of decimal conversion.

It provides methods for solving linear equations.

It demonstrates the uniqueness of the prime factorization of composite numbers. (correct)

How can the Fundamental Theorem of Arithmetic help understand the decimal expansion of a rational number p/q?

It explains the relationship between rational and irrational numbers.

The theorem allows the direct conversion of *p*/*q* into integers.

The prime factorization of *q* reveals the nature of the decimal expansion of *p*/*q*. (correct)

It provides the prime factorization of *p* only.

What notable irrational numbers were studied in Class IX related to the Fundamental Theorem of Arithmetic?

<p>√2, √3, and √5</p> Signup and view all the answers

What is the primary focus of the section discussing positive integers?

<p>The properties of positive integers related to divisibility and multiplication.</p> Signup and view all the answers

Study Notes

Introduction to Real Numbers

Continued exploration of real numbers and irrational numbers.
Focus on two key properties of positive integers: Euclid's Division Algorithm and the Fundamental Theorem of Arithmetic.

Euclid's Division Algorithm

States that any positive integer a can be divided by another positive integer b with a remainder r smaller than b.
Represents the standard long division process.
Important for understanding divisibility properties of integers.
Utilized mainly to compute the Highest Common Factor (HCF) of two positive integers.

Fundamental Theorem of Arithmetic

Asserts every composite number can be uniquely expressed as a product of prime numbers.
Simple to comprehend but has profound implications in mathematics.
Used to prove the irrationality of certain numbers such as √2, √3, and √5.
Helps determine the nature of the decimal expansion for rational numbers of the form p/q, with the prime factorization of q revealing key insights.

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Description

This quiz explores real numbers, focusing on the Euclid's Division Algorithm and the Fundamental Theorem of Arithmetic. Understand the properties of positive integers and their significance in mathematics, including the concepts of divisibility and prime factorization. Perfect for students looking to deepen their knowledge of number theory.