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Unformatted text preview: MATH 135 Winter 2009 Lecture II Notes Patterns and Conjectures We often use patterns in mathematics to guess what is happening in general (that is, make a con- jecture), and then try to prove our pattern/conjecture (that is, turn it into a theorem). Consider the following problem: n points are chosen on the circumference of a circle in such a way that when all pairs of points are connected, no three of these lines intersect at a single point. How many regions are formed inside the circle? Lets try some values of n : n # of Regions 1 1 2 2 3 4 4 8 So what does pattern appear to be? # Regions = 2 n- 1 Check: When n = 5, we get 16 regions. Great! Of course, wed need to try to prove that this works. (Or do we need to? Isnt it obvious?) What happens when n = 6? We would bet the farm on 32. But we get 31 regions, no matter what we do. (Proving this fact is a different story, since we cant really try all possible positions for the sixth line.) So the pattern fails. This again emphasizes why proof is important. While we were able to come up with a counterexample relatively quickly, this might not always be the case. Without proving a statement, how can we be absolutely certain? We cant be. This comes back to the precision idea as well starting with the sequence 1, 2, 4, 8, 16, we need to give a rule in order to find the next term. So proof and precision are very important....
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- Winter '07