In Example 3.9 we used unique factorization to show that while and are both equal to to twelve decimal places, they are not the same number.
Proof of Claim 3.10. More generally, suppose for a contradiction that where and . Multiply through by and square to get
Now consider the prime factorizations of each side. In particular, we will look at the exponent of the prime 3. Let be the exponent of 3 in the prime factorization of and let . Similarly, let be the exponent of 3 in the prime factorization of , and let . Since and we have
On the left-hand side the exponent of 3 is and on the right-hand side the exponent of 3 is . This contradicts unique factorization.
Connection with Example 3.9. In Example 3.9 we had and and the equation became
On the right-hand side the exponent of 3 is 2 and on the left-hand side the exponent of 3 is 5. As expected, this contradicts unique factorization. (Of course you could also multiply out both sides, and show that the left-hand side is , whereas the right-hand side is , but looking at the powers of 3 uses much less calculation!)
Question 4. Now try to adapt the proof above to show that is irrational.
Background. The fraction is an unusually good approximation to . Such good approximations can be found using a version of Euclid’s Algorithm, to be seen later in the course. You could also try searching for convergents or continued fractions on the web.