Real Numbers Chapter 1: Introduction and Properties

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

To prove the irrationality of numbers

What can be obtained by multiplying prime numbers in the Fundamental Theorem of Arithmetic?

A large collection of positive integers

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

What is the result of multiplying some or all of the prime numbers 2, 3, 7, 11, and 23?

A large collection of positive integers Signup and view all the answers

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

It includes all the possible primes Signup and view all the answers

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

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

What does Euclid's division algorithm deal with?

Divisibility of integers Signup and view all the answers

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

To compute the HCF of two positive integers Signup and view all the answers

What is the Fundamental Theorem of Arithmetic related to?

Multiplication of positive integers Signup and view all the answers

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

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

What is the significance of the Fundamental Theorem of Arithmetic?

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

What did you begin to explore in Class IX?

The world of real numbers Signup and view all the answers

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

An infinite collection of numbers, including all primes and products of primes Signup and view all the answers

What is the goal of factorizing positive integers?

To do the opposite of combining all primes in all possible ways Signup and view all the answers

What is true about the number 32760?

It can be written as a product of primes Signup and view all the answers

What is the Fundamental Theorem of Arithmetic?

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

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

To verify that they are indeed prime numbers Signup and view all the answers

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

A prime factorization of the number Signup and view all the answers

What is the definition of an irrational number?

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

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

The Fundamental Theorem of Arithmetic Signup and view all the answers

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

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

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

0.10110111011110... 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?

Theorem 1.2 Signup and view all the answers

What is the purpose of the proof in this section?

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

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

Proof by contradiction Signup and view all the answers

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

They have no common factors other than 1 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?

2b² = a² Signup and view all the answers

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

2 divides a and b Signup and view all the answers

What is the conclusion of the proof of Theorem 1.3?

2 is irrational 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?

3b² = a² 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?

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

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

Carl Friedrich Gauss 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?

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

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

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

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

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

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

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

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

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. Signup and view all the answers

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

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. 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?

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. Signup and view all the answers

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

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

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

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

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

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. Signup and view all the answers

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

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. 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?

To reach a contradiction and conclude that 5 – 3 is indeed irrational. 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?

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

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

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

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

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. Signup and view all the answers

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

It provides a criterion for determining whether a number is irrational or not. 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?

It guarantees the uniqueness of the prime factorization of the number. 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?

It helps by checking if 5 is present in the prime factorization of the number. 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?

It ensures that the prime factorization of a number is unique, making it possible to determine the factors of the number without ambiguity. 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?

It helps by checking if the prime factors of the number are compatible with the digit. 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?

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

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

The prime factorization of a number determines its divisibility by other numbers. 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 $3^2$?

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. Signup and view all the answers

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

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. Signup and view all the answers

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

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. Signup and view all the answers

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

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). Signup and view all the answers

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

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. 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?

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