Extending the Fundamental Theorem of Arithmetic to the rationals

In summary, the Fundamental Theorem of Arithmetic states that any positive whole number can be written as a unique product of primes, and that no two distinct combinations of primes will produce the same natural number. It is relatively easy to prove that this also applies to positive rational numbers if we allow any integer as an exponent. However, the uniqueness of this extended version is unsure and it is difficult to express succinctly. Additionally, the FTA is not applicable to the rational numbers since there are no irreducible or prime numbers in a field.
  • #36
An application of FTA is something basically different than an extension of FTA.

This is logically relevant as it is algebraically. To throw it all in one pot teaches the wrong motivations. A true statement isn't the same thing as a true classification. All posts above thought to an end would mean: Let's gather all theorems, list them in a book and call it: True. I cannot see how this is doing anyone a favor.
 
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<h2>1. What is the Fundamental Theorem of Arithmetic?</h2><p>The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a unique product of primes. This means that any positive integer can be written as a multiplication of prime numbers in a specific order.</p><h2>2. How does the Fundamental Theorem of Arithmetic relate to the rationals?</h2><p>The Fundamental Theorem of Arithmetic can be extended to the rationals by expressing each rational number as a product of primes in the numerator and denominator. This extension allows for the representation of any rational number as a unique product of primes, similar to the representation of integers.</p><h2>3. Why is extending the Fundamental Theorem of Arithmetic to the rationals important?</h2><p>Extending the Fundamental Theorem of Arithmetic to the rationals allows for a deeper understanding of the properties and relationships between rational numbers. It also provides a more efficient way of representing and manipulating rational numbers.</p><h2>4. Is the extension of the Fundamental Theorem of Arithmetic to the rationals proven?</h2><p>Yes, the extension of the Fundamental Theorem of Arithmetic to the rationals has been proven and is widely accepted by mathematicians. The proof involves using the unique factorization property of integers to extend to the unique factorization of rationals.</p><h2>5. Can the Fundamental Theorem of Arithmetic be extended to other number systems?</h2><p>Yes, the Fundamental Theorem of Arithmetic can be extended to other number systems, such as complex numbers and algebraic numbers. However, the unique factorization property may not hold for all number systems, making the extension more complex and requiring different approaches.</p>

FAQ: Extending the Fundamental Theorem of Arithmetic to the rationals

1. What is the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a unique product of primes. This means that any positive integer can be written as a multiplication of prime numbers in a specific order.

2. How does the Fundamental Theorem of Arithmetic relate to the rationals?

The Fundamental Theorem of Arithmetic can be extended to the rationals by expressing each rational number as a product of primes in the numerator and denominator. This extension allows for the representation of any rational number as a unique product of primes, similar to the representation of integers.

3. Why is extending the Fundamental Theorem of Arithmetic to the rationals important?

Extending the Fundamental Theorem of Arithmetic to the rationals allows for a deeper understanding of the properties and relationships between rational numbers. It also provides a more efficient way of representing and manipulating rational numbers.

4. Is the extension of the Fundamental Theorem of Arithmetic to the rationals proven?

Yes, the extension of the Fundamental Theorem of Arithmetic to the rationals has been proven and is widely accepted by mathematicians. The proof involves using the unique factorization property of integers to extend to the unique factorization of rationals.

5. Can the Fundamental Theorem of Arithmetic be extended to other number systems?

Yes, the Fundamental Theorem of Arithmetic can be extended to other number systems, such as complex numbers and algebraic numbers. However, the unique factorization property may not hold for all number systems, making the extension more complex and requiring different approaches.

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