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# Real Numbers Chapter 1: Introduction and Properties

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@StaunchHeptagon

## Questions and Answers

### What is the purpose of using the Fundamental Theorem of Arithmetic in Class IX?

• To prove the rationality of numbers
• To prove the irrationality of numbers (correct)
• To find the prime factorization of decimal numbers
• To explore the properties of prime numbers
• ### What can be obtained by multiplying prime numbers in the Fundamental Theorem of Arithmetic?

• Only rational numbers
• Only irrational numbers
• A large collection of positive integers (correct)
• Only prime numbers
• ### What is the significance of the prime factorization of the denominator in the decimal expansion of a rational number?

• It reveals the nature of the decimal expansion of the rational number (correct)
• It determines the cube root of the rational number
• It determines the square root of the rational number
• It determines the numerator of the rational number
• ### What is the result of multiplying some or all of the prime numbers 2, 3, 7, 11, and 23?

<p>A large collection of positive integers</p> Signup and view all the answers

### What is the assumption made about the collection of primes in the Fundamental Theorem of Arithmetic?

<p>It includes all the possible primes</p> Signup and view all the answers

### What is the purpose of listing the examples of multiplying prime numbers in the Fundamental Theorem of Arithmetic?

<p>To demonstrate the infinite number of positive integers that can be produced</p> Signup and view all the answers

### What does Euclid's division algorithm deal with?

<p>Divisibility of integers</p> Signup and view all the answers

### What is the main purpose of using Euclid's division algorithm in this chapter?

<p>To compute the HCF of two positive integers</p> Signup and view all the answers

### What is the Fundamental Theorem of Arithmetic related to?

<p>Multiplication of positive integers</p> Signup and view all the answers

### What is the main application of the Fundamental Theorem of Arithmetic in this chapter?

<p>To express a composite number as a product of primes in a unique way</p> Signup and view all the answers

### What is the significance of the Fundamental Theorem of Arithmetic?

<p>It has some very deep and significant applications in the field of mathematics</p> Signup and view all the answers

### What did you begin to explore in Class IX?

<p>The world of real numbers</p> Signup and view all the answers

### What is the result of combining all primes in all possible ways?

<p>An infinite collection of numbers, including all primes and products of primes</p> Signup and view all the answers

### What is the goal of factorizing positive integers?

<p>To do the opposite of combining all primes in all possible ways</p> Signup and view all the answers

### What is true about the number 32760?

<p>It can be written as a product of primes</p> Signup and view all the answers

### What is the Fundamental Theorem of Arithmetic?

<p>A theorem that states every composite number can be written as the product of powers of primes</p> Signup and view all the answers

### What is the purpose of checking if 3803 and 3607 are primes?

<p>To verify that they are indeed prime numbers</p> Signup and view all the answers

### What is the result of expressing a number as a product of powers of primes?

<p>A prime factorization of the number</p> Signup and view all the answers

### What is the definition of an irrational number?

<p>A number that cannot be written in the form p/q, where p and q are integers and q ≠ 0</p> Signup and view all the answers

### What is the theorem used in the proof that a prime number is irrational?

<p>The Fundamental Theorem of Arithmetic</p> Signup and view all the answers

### What is the condition for a number to be called irrational?

<p>The number cannot be written in the form p/q, where p and q are integers</p> Signup and view all the answers

### Which of the following numbers is an example of an irrational number?

<p>0.10110111011110...</p> Signup and view all the answers

### What is the theorem that states that if a prime number p divides a^2, then p divides a?

<p>Theorem 1.2</p> Signup and view all the answers

### What is the purpose of the proof in this section?

<p>To prove that 2, 3, 5, and in general, p is irrational, where p is a prime</p> Signup and view all the answers

### What is the technique used in the proof of Theorem 1.3?

<p>Proof by contradiction</p> Signup and view all the answers

### What can be concluded about the integers r and s in the proof of Theorem 1.3?

<p>They have no common factors other than 1</p> Signup and view all the answers

### What is the result of squaring both sides of the equation b² = a in the proof of Theorem 1.3?

<p>2b² = a²</p> Signup and view all the answers

### What is the contradiction that arises in the proof of Theorem 1.3?

<p>2 divides a and b</p> Signup and view all the answers

### What is the conclusion of the proof of Theorem 1.3?

<p>2 is irrational</p> Signup and view all the answers

### What is the result of squaring both sides of the equation b³ = a in the proof of Example 5?

<p>3b² = a²</p> Signup and view all the answers

### What is the significance of the order of prime factors in the factorisation of a composite number according to the Fundamental Theorem of Arithmetic?

<p>The order of prime factors is not significant, and any rearrangement of the prime factors is considered the same factorisation.</p> Signup and view all the answers

### Who is credited with providing the first correct proof of the Fundamental Theorem of Arithmetic?

<p>Carl Friedrich Gauss</p> Signup and view all the answers

### What is the key feature of the factorisation of a composite number, according to the Fundamental Theorem of Arithmetic?

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

### What is the importance of the Fundamental Theorem of Arithmetic in number theory?

<p>It states that every composite number can be expressed as a product of prime numbers in a unique way.</p> Signup and view all the answers

### What is the significance of Euclid's Elements in the context of the Fundamental Theorem of Arithmetic?

<p>An equivalent version of the Fundamental Theorem of Arithmetic was first recorded in Euclid's Elements.</p> Signup and view all the answers

### What is the reputation of Carl Friedrich Gauss in the mathematical community?

<p>He is often referred to as the 'Prince of Mathematicians' and is considered one of the three greatest mathematicians of all time.</p> Signup and view all the answers

### Explain the concept of an irrational number and provide an example.

<p>An irrational number is a number that cannot be written in the form p/q, where p and q are integers and q is not equal to 0. An example of an irrational number is the square root of 2.</p> Signup and view all the answers

### State the theorem that is used to prove that a prime number is irrational.

<p>The theorem used to prove that a prime number is irrational is Theorem 1.2, which states that if a prime number p divides a^2, then p divides a.</p> Signup and view all the answers

### What is the significance of the Fundamental Theorem of Arithmetic in the proof of the irrationality of a prime number?

<p>The Fundamental Theorem of Arithmetic is used to prove that a prime number is irrational by showing that the only prime factors of a^2 are p1, p2,..., pn, which are the prime factors of a.</p> Signup and view all the answers

### How can we prove that the square root of 2 is irrational?

<p>We can prove that the square root of 2 is irrational by using Theorem 1.2 and the Fundamental Theorem of Arithmetic.</p> Signup and view all the answers

### What is the relationship between the prime factors of a and a^2?

<p>The prime factors of a^2 are the same as the prime factors of a, namely p1, p2,..., pn.</p> Signup and view all the answers

### What is the purpose of the proof of the irrationality of a prime number?

<p>The purpose of the proof is to show that certain numbers, such as the square root of 2, cannot be expressed as a ratio of integers and are therefore irrational.</p> Signup and view all the answers

### What is the reasoning behind the conclusion that 3 is irrational in the given proof?

<p>The assumption that 3 is rational leads to a contradiction that a and b have at least 3 as a common factor, which contradicts the fact that a and b are coprime.</p> Signup and view all the answers

### What is the purpose of assuming the contrary in the proof of Example 6, i.e., that 5 – 3 is rational?

<p>To reach a contradiction and conclude that 5 – 3 is indeed irrational.</p> Signup and view all the answers

### What is the significance of the fact that a and b are integers in the proof of Example 6?

<p>It implies that 5 – is rational, which leads to a contradiction that 3 is rational.</p> Signup and view all the answers

### What is the underlying assumption in the proof of Example 7, i.e., that 3 2 is rational?

<p>That we can find coprime a and b (b ≠ 0) such that 3 2 = a/b.</p> Signup and view all the answers

### What is the relationship between the rationality of a and b, and the rationality of a² and b²?

<p>If a and b are rational, then a² and b² are also rational. Conversely, if a² and b² are rational, then a and b are also rational.</p> Signup and view all the answers

### What is the importance of Theorem 1.3 in the context of irrational numbers?

<p>It provides a criterion for determining whether a number is irrational or not.</p> Signup and view all the answers

### What is the significance of listing the prime factors in ascending order in the prime factorization of a composite number?

<p>It guarantees the uniqueness of the prime factorization of the number.</p> Signup and view all the answers

### How does the Fundamental Theorem of Arithmetic help in determining whether a number is divisible by 5 or not?

<p>It helps by checking if 5 is present in the prime factorization of the number.</p> Signup and view all the answers

### What is the importance of the uniqueness of the prime factorization of a number in the Fundamental Theorem of Arithmetic?

<p>It ensures that the prime factorization of a number is unique, making it possible to determine the factors of the number without ambiguity.</p> Signup and view all the answers

### How does the prime factorization of a number help in determining whether it ends with a particular digit or not?

<p>It helps by checking if the prime factors of the number are compatible with the digit.</p> Signup and view all the answers

### What is the significance of the Fundamental Theorem of Arithmetic in finding the HCF and LCM of two positive integers?

<p>It helps in finding the HCF and LCM by providing the prime factorization of the numbers.</p> Signup and view all the answers

### What is the relationship between the prime factorization of a number and its divisibility by other numbers?

<p>The prime factorization of a number determines its divisibility by other numbers.</p> Signup and view all the answers

### What is the significance of the rearrangement of the equation $3^2 = ab$ in the proof of the irrationality of $3^2$?

<p>The rearrangement of the equation $3^2 = ab$ to $2 = 3b/a$ is significant because it allows us to conclude that 2 is rational, which is a contradiction, and hence $3^2$ is irrational.</p> Signup and view all the answers

### What is the importance of the uniqueness of the prime factorization in the Fundamental Theorem of Arithmetic?

<p>The uniqueness of the prime factorization is important because it ensures that the factorization of a composite number is unique, apart from the order in which the prime factors occur.</p> Signup and view all the answers

### How does the proof of the irrationality of $2$ lead to the conclusion that $3^2$ is irrational?

<p>The proof of the irrationality of $2$ is used to show that $3^2$ is irrational by assuming that $3^2$ is rational and reaching a contradiction, thus proving that $3^2$ is irrational.</p> Signup and view all the answers

### What is the relationship between the HCF and LCM of three positive integers p, q, and r?

<p>The HCF and LCM of three positive integers p, q, and r are related by the formulas: LCM(p, q, r) = p × q × r × HCF(p, q, r) and HCF(p, q, r) = p × q × r × LCM(p, q, r).</p> Signup and view all the answers

### What is the purpose of the exercise to prove that $3 + 2rac{1}{2}$ is irrational?

<p>The purpose of the exercise is to apply the concept of irrationality to a more complex number, $3 + 2rac{1}{2}$, and to practice the proof of irrationality.</p> Signup and view all the answers

### What is the significance of the fact that the product of the HCF and LCM of three positive integers p, q, and r is not equal to p × q × r?

<p>The significance of this fact is that it highlights the importance of considering the HCF and LCM separately when working with three positive integers.</p> Signup and view all the answers

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