Second Post – Some Interesting Connections

All this work with the Fibonacci sequence and the golden ratio inspired me to go online and do a little research into other interesting mathematical relationships. In doing so I came across Pascal’s Triangle, which is well know to most, or at least most in this course have an idea of what Pascal’s Triangle is.

One interesting property of this triangle I actually came across last year while working on some other math stuff. The property is that the first 5 rows of the triangle can be constructed by considering each digit obtained by the first 5 powers of 11 (including 11^0). I only mention this because in my research I discovered that in fact this relation holds for all powers of 11. Considering the construction of each row of the triangle by way of the coefficients of the x-values of the binomial expansion of (x+y)^n (binomial theorem), which is another interesting property of the triangle, if the value of x in (x+y)^n is replaced with 10 and y is given a value of 1 for simplicity, the nonzero digits of each number in the binomial expansion (summation) are in fact the numbers for the corresponding row of the triangle. Because this can be done for all values of n in 11^n, I feel as though there must be an algorithm for constructing any given row of the triangle with no knowledge other than the value of 11^n and this particular algorithm. Although in the couple of hours I attempted to discover this algorithm, I was not able to find anything solid from which I could attempt a proof by induction. However I think this will be an interesting problem to work on.

Another interesting relation is that of Pascal’s Triangle and the Fibonacci sequence. If you consider the triangle below and only those numbers in the square parentheses, that follow a diagonal pattern, the sum of those numbers, from top left to bottom right, correspond to the Fibonacci number, F(n), where n is the row number of the last digit added along the diagonal.

1 row 1
1 1 row 2
1 2 1 row 3
[1] 3 3 1 row 4
[1] 4 [6] 4 1 row 5
1 5 [10] 10 [5] 1 row 6
1 6 15 20 [15] 6 [1] row 7, F(7) = 13 = 1 + 6 + 5 + 1
1 7 21 35 35 21 [7] 1 row 8
1 8 28 56 70 56 28 8 [1] row 9, F(9) = 34 = 1 + 10 + 15 + 7 + 1

The majority of this information was obtained form Wikipedia, which is not necessarily a valid source for research but these properties are easily verified. In addition I have provided a link to the page I read, because it provides very interesting resemblance between the fractal Sierpinski’s Triangle and shading various different number patterns within Pascal’s triangle. It seems that by shading the odd numbers found with Pascal’s triangle it begins to appear as the Sierpinski Triangle.

http://en.wikipedia.org/wiki/Pascal_triangle#Other_patterns_and_properties

http://mathforum.org/workshops/usi/pascal/pascal_sierpinski.html

More to come on these relations in the future, but for now it’s a gorgeous sunny Thanksgiving Monday so I’m going to go out side and enjoy the nice weather.

First Post

I’m sitting here thinking of things to write for my first blog and coming up with nothing. I suppose the main purpose of this SLOG is to create discussion about previous or future problems/proof in the class so that’s probably a good place to start.

So far we have covered simple and complete induction and the principle of well ordering. The principle of well ordering, to me, seems like a formal way of saying something that is inherently true about Natural Number based on their construction; it seems no more relevant to me than, when using an array for some program, to say a nonempty array has a first and last element. But having said that, I understand the need to formalize mathematical proofs and this principle provides a means to do that.

As far as simple and complete induction, I’ve seen these types of proofs in several mathematics and computer science courses I’ve taken and it seems that some of these proofs rely on reasoning inside a formal proof which seems to give the reasoning validity, but to me it seems only slightly stronger, if at all, than arguing the proof by reason without the use of induction.

For example P(n): Every set of n elements has exactly 2n subsets, the proof of which uses the fact that every subset of a set S can be considered as those that contain a particular element and those that don’t. Because of this, there is a 1-1 correspondence between the two subsets and thus as set of n+1 elements is composed to two sets of subset, those with element and those without, each of which has 2n elements (by induction hypothesis). I understand this to be true, however there doesn’t seem to be as much validity to a proof of this nature compared to one that employs algebra or one that can be easily verified by the reader, such as the proof of 12n – 1 is a multiple of 11, which I personally find to be logically and algebraically simple and easier to follow therefore making it a better proof.

I suppose the nature of mathematical problems doesn’t allow for every proof to be constructed in a clear concise manner that even people without a mathematical background could grasp. But it does seem that induction is the most useful and efficient tool for proving something for a large (or infinite) set of numbers that would otherwise not be so easily proven, with or without reasoning.