Hard Made Easy II
This is a continuation of the Hard Made Easy series, in which we feature certain questions where simplifying the problem actually helps us approach it.
Below, is a simple case which shows how our understanding of Rational Numbers is based (mainly) on our knowledge of the integers.
This seems hard to approach, because there seem to be a lot of unknowns, and we could be unfamiliar with what rational numbers are. Let’s try the following:
Now, this is more friendly, but still a little tricky. In the spirit of this post, let’s make it even easier.
Now this is almost silly, and seems to be just a play on words. The proper way to prove this statement that doesn’t make my head run around in circles, is to show the contrapositive – To show that , it is equivalent to show the contrapositive which is .
Proof: If is an integer , then , which is the square of the integer .
Now, what is the generalized version?
Proof: Test Yourself 1.
Proof: From Question 4, we know that y must be the square of a rational with . Hence , and the only way for this to be an integer is , so .
Now, back to question 2.
Proof: Suppose is an integer. Consider . Squaring both sides, we obtain that , so this means that is rational. By the corollary above, must be an square. Similarly, must be an square. Hence we are done.
And finally, back to question 1.
Proof: Proof by contradiction. Suppose such that , where are all integers. Then, . This contradicts question 2!
The slightly surprising result, is that in simplifying Question 1 to Question 2, we ended up using Question 2 to prove Question 1. In fact, as is often the case with rational numbers, it is sufficient to consider the integer case, and then clear out denominators by multiplying throughout.
1. Complete the proof of Question 4.
2. Prove that if , and are rational numbers such that is rational, then , and are all rational. How many terms can you show this for?
3. How many ordered triples of integers are there such that ?
4. (**) Prove that if are rational numbers such that is rational, then and are all rational. Note: This is extremely hard, and not approachable by the methods discussed in this post.