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Proof of Theorems : Real Numbers | Mathematics (Maths) Class 10 PDF Download

Theorem 1.1 - Fundamental Theorem of Arithmetic

Statement: Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Given: Let N be a composite number. It can be expressed as a product of prime numbers.
To Prove: The prime factorization of N is unique, apart from the order of factors
Proof:

  •  Consider any composite number N, which can be written as a product of primes. 
  •  For example, take 32760: 
  • 32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13
  •  Written in exponent form: 23 × 32 × 5 × 7 × 13
  •  Suppose there exists another way to factorize N into primes. 
  •  By the Fundamental Theorem of Arithmetic, the set of prime factors obtained will always be the same, only differing in order. 
  •  Thus, the prime factorization of a number is unique (except for the arrangement of factors). 

Conclusion: The Fundamental Theorem of Arithmetic ensures that every composite number has a unique prime factorization, apart from the order of the factors

Theorem 1.2

Statement: If a prime number p divides a², then p also divides a, where a is a positive integer.

Given: Let p be a prime number and a be a positive integer such that p divides a².
To Prove: If p divides a², then p must also divide a.
Proof:

  • Let a be any positive integer, and let its prime factorization be:
    a = p1 × p2 × ... × pn
  • Squaring both sides, we get:
    a² = (p1 × p2 × ... × pn)² = p × p2² × ... × pn².
  • If a prime number p divides a², then p must be one of the prime factors of a².
  • Since each prime factor in a² appears as a square, p must already be a factor of itself.
  • This follows directly from the Fundamental Theorem of Arithmetic, which states that prime factorization is unique.
  • Therefore, if p divides a², then p must also divide a.

Conclusion:

The Theorem 1.2 proves that if a prime number divides the square of a number, then it must also divide the number itself. This theorem is widely used in number theory and proofs of irrationality, such as proving that √2 is irrational.

Theorem 1.3 

Statement: √2 is an irrational number.

To Prove: √2 cannot be expressed as a fraction pq, where p and q are integers and q ≠ 0.
Proof :

  • Assume √2 is rational.
  • Then, we can write √2 as pq, where p and q are coprime integers (i.e., they have no common factor other than 1).
  • Squaring both sides:
    2 = ⇒ p² = 2q².
  • This means p² is divisible by 2, so p must also be divisible by 2 (from Theorem 1.2).
  • Let p = 2k for some integer k.
  • Substituting in p² = 2q², we get:
    (2k)² = 2q² ⇒ 4k² = 2q² ⇒ q² = 2k².
  • This means q² is also divisible by 2, so q must also be divisible by 2.
  • But this contradicts our assumption that p and q are coprime (as both are divisible by 2).
  • Therefore, our assumption that √2 is rational must be false and√2 is rational.

Conclusion: Thus, √2 is irrational.

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FAQs on Proof of Theorems : Real Numbers - Mathematics (Maths) Class 10

1. What is the Fundamental Theorem of Arithmetic?
Ans. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely factored into prime numbers, up to the order of the factors. This means that there is one and only one way to express a natural number as a product of primes, which is foundational in number theory.
2. How do Theorems 1.2 and 1.3 relate to the Fundamental Theorem of Arithmetic?
Ans. While Theorem 1.1, the Fundamental Theorem of Arithmetic, provides the uniqueness of prime factorization, Theorems 1.2 and 1.3 likely extend these concepts or explore their implications in the context of real numbers. They may cover properties of real numbers related to factors, divisibility, or applications of prime factorization in broader mathematical contexts.
3. Why is the uniqueness of prime factorization important in mathematics?
Ans. The uniqueness of prime factorization is crucial because it allows mathematicians to work with integers in a structured way. It ensures that many mathematical concepts, such as greatest common divisors and least common multiples, can be defined consistently. This property also plays a significant role in number theory, cryptography, and various algorithms in computer science.
4. Can the Fundamental Theorem of Arithmetic be applied to negative integers or zero?
Ans. No, the Fundamental Theorem of Arithmetic applies only to positive integers greater than 1. Negative integers and zero do not have a prime factorization in the same sense, as the concept of prime numbers is defined only for positive integers.
5. How is the Fundamental Theorem of Arithmetic used in real-world applications?
Ans. The Fundamental Theorem of Arithmetic is used in various real-world applications, including cryptography, where the security of many encryption algorithms relies on the difficulty of factoring large composite numbers into their prime factors. Additionally, it is used in computer algorithms for number theory, data compression, and error detection in digital communications.
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