The Fundamental Theorem of Arithmetic

1.2 The Fundamental Theorem of Arithmetic

Description

Quick Overview

The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of prime numbers.

Standard

This section explores the Fundamental Theorem of Arithmetic, demonstrating that every composite number can be factored into prime numbers in a unique way. It also illustrates the theorem's significance in proving the irrationality of specific numbers and its applications in finding the highest common factor (HCF) and least common multiple (LCM).

Detailed

The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic posits that every natural number greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors. This means that composite numbers, like 12 or 28, can be broken down into prime factors (e.g., 12 = 2 × 2 × 3 and 28 = 2 × 2 × 7). This section illustrates the process of prime factorization using examples and emphasizes the uniqueness of this factorization.

The theorem has profound implications, including its use in identifying rational and irrational numbers. The section details a proof that demonstrates the irrationality of numbers such as 2, 3, and other prime numbers via the theorem's principles. Additionally, it introduces methods for calculating the HCF and LCM of numbers using their prime factorization, reinforcing how the theorem aids in these calculations. Ultimately, understanding the Fundamental Theorem of Arithmetic lays the groundwork for exploring more complex mathematical concepts.

Key Concepts

  • Prime Factorization: Every composite number can be expressed as a product of prime numbers.

  • Uniqueness of Factorization: Each number has a unique factorization into primes, aside from the order.

  • Applications in Mathematics: The theorem is used to find HCF and LCM, and in proofs regarding irrational numbers.

Memory Aids

🎵 Rhymes Time

  • Prime numbers are rare, not many can share, composite can grow with primes in their care.

📖 Fascinating Stories

  • Once in a village of numbers, the composite numbers sought help from friendly primes to unveil their true identities, proving their uniqueness.

🧠 Other Memory Gems

  • P.U.N.C.H: 'Prime Uniqueness, Natural Composite Harvest!' This reminds us about the uniqueness in prime factorization.

🎯 Super Acronyms

P.U.N.

  • 'Prime Understanding of Numbers.' This helps recall the importance of understanding primes.

Examples

  • 60 can be factored as 2 × 2 × 3 × 5 (and also expressed as 2² × 3 × 5).

  • In the case of 28, its prime factorization is 2 × 2 × 7 or 2² × 7.

Glossary of Terms

  • Term: Prime Number

    Definition:

    A natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

  • Term: Composite Number

    Definition:

    A natural number greater than 1 that is not prime; it has factors other than 1 and itself.

  • Term: Prime Factorization

    Definition:

    The process of expressing a composite number as the product of its prime numbers.

  • Term: Fundamental Theorem of Arithmetic

    Definition:

    States that every composite number can be uniquely expressed as a product of prime numbers.

  • Term: Irrational Number

    Definition:

    A number that cannot be expressed as a fraction of two integers.

  • Term: HCF (Highest Common Factor)

    Definition:

    The largest number that divides two or more numbers without leaving a remainder.

  • Term: LCM (Lowest Common Multiple)

    Definition:

    The smallest number that is a multiple of two or more numbers.

Similar Question

Example 5: Find the HCF and LCM of 18, 30, and 42, using the prime factorisation method.

Solution: We have :
$$
18 = 2^1 \times 3^2, \quad 30 = 2^1 \times 3^1 \times 5^1, \quad 42 = 2^1 \times 3^1 \times 7^1
$$

Here, $2^1$ and $3^1$ are the smallest powers of the common factors 2 and 3, respectively.

So,
$$
\text{HCF}(18, 30, 42) = 2^1 \times 3^1 = 6
$$

Next,
$$
2^1, 3^2, 5^1, \text{ and } 7^1 \text{ are the greatest powers of the prime factors 2, 3, 5, and 7 respectively involved in the three numbers.}
$$

Thus,
$$
\text{LCM}(18, 30, 42) = 2^1 \times 3^2 \times 5^1 \times 7^1
$$