Show that every positive rational number r can be written as a product: r = p^a1

Question

Show that every positive rational number r can be written as a product: r = p^a1_1 p^a2_2 p^as_s, where p_1 = 2, p_2 = 3... are the prime numbers and each a_k is an integer (positive, negative, zero), using the following information: a. All rules of exponents (how to add/subtract them, what happens when the signs change, etc.). b. The integers are closed under addition/subtraction (adding/subtracting two integers results in another integer). c. Any natural number can be factored uniquely into primes. Prove that the product in the above problem is unique. Prove that between any two rational numbers there exist infinitely many irrational numbers. Define and as the usual operations of addition and multiplication over the real numbers, except that all results are rounded to two decimal places. Show that the distributive law doesn't hold for these operations.

Explanation / Answer

EXISTENCE OF fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1[3] either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors

We need to show that every integer greater than 1 is either prime or a product of primes. For the base case, note that 2 is prime. By induction: assume true for all numbers between 1 and n. If n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a b < n. By the induction hypothesis, a = p1p2...pj and b = q1q2...qk are products of primes. But then n = ab = p1p2...pjq1q2...qk is a product of primes. NOW

We claim that every rational number has a unique prime factorization. Clearly, such a factorization exists - just write qq as a fraction rac abba and take the primes in bbwith negative exponents, i.e. let v_p( a/b)=v_p(a)-v_p(b). Moreover, this factorization is unique because if we had two different factorizations, then we could multiply qq by an appropriate positive integer to obtain two different factorizations of a positive integer, contradicting the Fundamental Theorem of Arithmetic.

If a real number can be written as the product of natural numbers raised to rational exponents, there is indeed a unique way to express it as a product of distinct primes raised to rational exponents. This follows from the unique factorization theorem for integers

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