Test 3 Review, with answers/links

Test 3 is (re)scheduled for Tuesday 15 November.

The Test 3 Review problems are here. (link fixed)

Answers, and/or links to the relevant posts, are below the fold... I am temporarily working with an old computer which is very slow and has slow internet connection and has a difficult ketboard, so pardon any uncorrected typos (let me know about them!)

Note: in the answers below, I have written fractions using a slant line / instead of writing them vertically like $\frac{1}{2}$. You should NOT use the slant line: always write them vertically. The only reason I used the slant line here is so that people whose phones will not display the math notation correctly will be able to read this without having to translate.

1) A rational number is a number which can be written as a ratio of two integers: it can be written in the form a/b, where a and b are integers and b is not 0.
Comment: You don't have to use these exact words: in fact, it's probably better if you don't. But make sure that you get the essential points across. Link to blog post. Also notice how we use this definition when we prove that the square root of 2 is not a rational number.

2) This was done in class. Make sure that you clearly state your assumption for the proof by contradiction, and that you have imcluded all the important steps of the reasoning.
Note: the test question will ask about the square root of some number, but it may or may not be 3.

3) a) 0.037 = 37/1000
b) 0.135353535...  = 67/495
c) 0.249999999...  = 1/4
I haven't shown the work here, but you will have to! See the videos and other links in this post and the examples I worked in class. And don't forget to reduce to lowest terms and always write your fractions vertically!

4) See this post. It's best to illustrate your proof with an example, as I did. Don't use the same example!

5) See this post. The proof uses the Pigeonhole Principle, whether or not you say so, but it's nicer Thinking Mathematically if you point it out when you use it!

6) See this post, where we outlined the proof and I gave better links. Make sure that your proof covers all the points of the reasoning in that outline! (The outline is NOT the proof: you have to fill in the details of the reasoning.)