Weekly 4: History of Fibonacci

For this post, let's explore the history of Fibonacci. Leonardo Pisano (Fibonacci) was born on 1170 to Guglielmo Bonacci, a merchant of some kind. Fibonacci was educated in Bejaia, Algeria in North Africa (where his father worked). Fibonacci actually was enrolled in Bejaia's school of accounting. From all of Fibonacci and his fathers travels, Fibonacci learned the Hindu-Arabic number system, the system we use today. He was one of the first people to introduce the number system to Europe. His book on how to do operations in the new number system was called the Liber abbaci. This book convinced many mathematicians, and others, in Europe to start using the new number system. Actually, much of the book had to do with accounting mathematics: price of goods, how to calculate profit, currency conversion, etc... This number system seems very natural to us. In class it was very difficult to perform operations using roman numerals. I tried to do it and I pretty much was fed up after the first problem. I couldn't imagine doing every day operations in those cruel roman numerals. It's amazing that Fibonacci could even understand more than one number system. But, for me the truly astonishing thing Fibonacci did was to actually recognize that the Hindu-Arabic number system was easier to use. As we have seen, any new number system is hard to get used to, but to recognize one is easier than another seems so difficult. Who knows, maybe our grand children will be learning a different number system than the one we learned (hope not). 

Fibonacci presented many problems in his Liber abbaci, including the one about bunnies we studied in class. Out of this problem came the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc...

As we have seen in our math 495 class, the Fibonacci sequence has a lot to do with the golden ratio. For instance, if you take any two successive numbers in the Fibonacci sequence, the ratio gets closer and closer to the golden ratio. The golden ration also has ties to finance, biology, architecture, and many other areas. There are so many interesting things in nature that deal with the golden ratio, but this post is about Fibonacci. If you wan't to learn more about nature and the golden ration, you should read this article

http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

 , 

Fibonacci's book also discussed perfect numbers, rational approximations of square roots of numbers, and sums of arithmetic and geometric series. This may be his most famous book, but another one of his works was Liber quadratorum. Here he discussed number theory including Pythagorean triples, square numbers, and interesting number theory results such as

there is no x, y such that x^2 + y^2 and x^2 - y^2 are both squares.

Since Fibonacci lived before printing, it was difficult to keep some of his original works. Some of his papers were actually lost, including his book on commercial arithmetic Di minor guisa.

Frederick II (the holy roman emperor at the time) actually was aware of Fibonacci because the Emperor's scholars corresponded with him after his return to Pisa in 1200 A.D. One of those scholars, Johannes of Palermo, gave some problems as challenges to Fibonacci. Fibonacci solved three of these problems and gave the solutions in his book Flos which he sent to Frederick II. I never knew mathematicians were so competitive back in the day. I imagine there was trash talking going on all over the intellectual community back then. Actually, when I think about it, we do have math competitions today in colleges and high schools; though I have yet to hear trash talk going on in my math classes. We'll save the trash talk for Richard Sherman. 

Fibonacci was a very important figure in mathematics and history in general. The Hindu-Arabic number system he brought to Europe was so important in the study of mathematics, science, business, and pretty much any discipline that used numbers (which was like all of them). The Fibonacci sequence is probably the most popular of all sequences because of how much of it is related to science and nature. I hope you enjoyed this non-rigorous post on Fibonacci. I usually like to post proofs and other interesting math theorems, but I'm so thankful for Fibonacci that I thought a brief biography would be nice.

References

Leonardo Pisano Fibonacci (JOC/EFR, October 1998 School of Mathematics and Statistics, University of      St. Andrews, Scotland) http://www-groups.dcs.st-and.ac.uk/history/Biographies/Fibonacci.html

Nature, The Golden Ration, and Fibonacci too (Copyright © 2011 MathsIsFun.com)

     http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

Weekly 3: Al Kharaji's formulas

One of my favorite things about mathematics are series. That's why when we started talking about Al Kharaji and his series formulas I couldn't help but explore and prove them. The first one we talked about was an interesting closed form for the sum of squares: (note: I'm using sum( ) instead of the summation symbol)

sum(k^2) = sum(k) + sum(k(k-1)).  

To see why this is true, observe that k^2 can be manipulated in the following way:

k^2 = kk = k(k+1-1) = k(1+(k-1)) = k+k(k-1),

which means

sum(k^2) = sum( k+k(k-1)).

Now the question is, can you distribute the sum ( )? Yes of course; observe that

sum(k+k(k-1)) = (1 + 1(1-1)) + (2 + 2(2-1)) + (3 + 3(3-1)) + ...

and from each expression in the parenthesis, we can group all the first terms together and then all the second terms together to get

sum(k+k(k-1)) = (1 + 2 + 3 + ...) + (1(1-1) + 2(2-1) + 3(3-1) + ...)

where we can see that

sum(k+k(k-1)) = sum(k) + sum(k(k-1)).

Then since sum(k^2) = sum(k+k(k-1)), we have

sum (k^2) = sum(k) + sum(k(k-1)).

Note that I merely explained why this is true, I didn't really prove it for the most part. Now another one of Al Kharaji's claims was

sum(k^3) = (sum(k))^2.

This one I will prove using mathematical induction (one of my favorite proof methods). To make the proof easier to read, I typed it up and put it on a pdf file. You can't really put pdf files on blogger.com so I inserted them as pictures. To see them, just hold the "control" key and push the "+" key a few times to zoom in. Then to zoom out, hold the "control" key and push the "-" key, sorry about that. 

I probably missed on some minor subtitles, but the main part of the proof should be fine. Now proving this wasn't so bad, it was fun. Make sure to observe the "note" at the end of the proof. I had to actually use the sum of integers formula (lemma 1) to prove sum(k^3) = (sum(k))^2 (lemma 1 actually serves as my daily 5 assignment).  I imagine that to prove the sum of squares or sum of cubes formulas, you could use mathematical induction and the proofs would be straightforward; but, let's save those for a different post. I hope we explore more series in the future. If we do, expect more proofs in my blogs. :)

Weekly 2: Wallpaper groups (Revised)

One interesting connection between geometry and algebra is the idea of wallpaper groups. To explain what it is, let's first discuss symmetry groups. Remember in MTH 410 (Modern Algebra II) where we first started discussing groups. A group is a set combined with one operation that satisfy the four group axioms; let me explain.

A group is a set G on which we define one binary operation "*" that satisfy the following axioms:
1. For all a,b in G, a*b is also in G (G is closed under its operation).
2. For all a,b,c in G, (a*b)*c = a*(b*c), (* is associative in G).
3. There exists an element i in G such that for all a in G, a*i = a = i*a (G has an identity element).
4. For all a in G, there exists b in G such that a*b = i = b*a (each element in G has an inverse).

Some examples of a group are: the set of integers over addition, the set of non-zero rational numbers under multiplication, and the set of all n by n matrices with real entries under standard addition. For a symmetry group, the set is a collection of symmetries (rotations, translations, reflections, and glide-reflections) and the operation is function composition. For example, to perform a horizontal reflection composed with a 180 degree rotation on a rectangle, you would first rotate the rectangle 180 degrees and then reflect it about a horizontal line through its center. Some other examples are the group of symmetries of a square, triangle, or hexagon. So then a wallpaper group is a 2 dimensional symmetric group that is a repetitive pattern (like a wallpaper) that is categorized and classified by its symmetries. There are only 17 different types of wallpaper groups. A reason for this is because though many patterns and tessellations may look completely different, they would still fall into the same wallpaper group (essentially they have the same symmetries). The first proof that there are only 17 was actually written pretty recently by Evgraf Fedorov in 1891. Though, knowledge of some of these different groups were already known throughout history. Some of the works by M.C. Escher actually belong in one of the 17 groups. Now let's explore some of the 17 wallpaper groups. An example of the wallpaper group "p1" is

http://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Wallpaper_group-p1-3.jpg/91px-Wallpaper_group-p1-3.jpg

This group has 2 translational symmetries. For this particular tessellation, it can be translated in two ways (up/down and left/right). An example of the wallpaper group "p2" is

http://en.wikipedia.org/wiki/File:Wallpaper_group-p2-2.jpg

This group has translations and rotations only. The tessellation above can be translated  to the left or right and rotated 180 degrees. An example of the wallpaper group "pm" is

http://en.wikipedia.org/wiki/File:Wallpaper_group-pm-3.jpg

This group has a reflection symmetry and a translational symmetry. This tessellation can be reflected about a vertical line through the center and be shifted to the left/right and still keep its design. An example of the wallpaper group "p4" is

http://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/A_tri-colored_Pythagorean_tiling_View_4.svg/120px-A_tri-colored_Pythagorean_tiling_View_4.svg.png

This group has translations and 90 degree rotations. This particular tessellation can translated at about a 150 degree angle and be rotated 90 degrees and still keep its shape. Things get difficult when certain groups start having glide reflections (combination of a reflection and translation). The groups I mentioned above are the more simpler ones. Wallpaper groups are a great way to connect geometry to algebra (or more generally, symmetry groups). Take MTH 410 if you would like to learn more about symmetry groups as a whole. In the future, I would like to explore why there are only 17 wallpaper groups and the proof behind it. I tried to look online and I couldn't find a real good proof. So the best option would probably be to get a book about symmetry groups on amazon.

References:

http://en.wikipedia.org/wiki/Wallpaper_group#The_seventeen_groups

A. Nelson, H. Newman, M. Shipley: 17 PLANE SYMMETRY GROUPS. wordpress.com

     http://caicedoteaching.files.wordpress.com/2012/05/nelson-newman-shipley.pdf

Daily 2: Propositions 5-11

The wording on Euclid's book VII, propoitions 5 through 11 seemed very murky. There were no translations for those propositions on our class google.doc. So for my post, I wanted to "decode" propositions 5 through 11, without looking it up on google or something. That proved to take more than a few hours. So for this post I will be translating proposition 5. Though the rest of the 6 through 11 have very similar wording; so translating 5 can help build on those. Here it is,
(Caution: this is just an attempt, I'm not 100% sure on the translation.)

Proposition 5: If a number is part of a number, and another is the same part of another, then the sum is also the same part of the sum that the one is of the one.

Translation: Let x,y,z,x1,y1,z1 be non-zero real numbers. If xy = z, x1y1 = z1, and x/z = x1/z1, then
(x + x1)/(z + z1) = x/z.

Now let's break down some of Euclid's terms. It seems like if we have a number (x) being a part of another number (z), then the term "part" means z/x. For instance, 2 is a part of 8 because 2 is 8/2 = 4 parts of 8. Also, the phrase "the one is of the one" is referring to (x) of (z), where (x) is z/x parts of (z)
Now using the notation above, I will restate the proposition, adding in notes as necessary.

New Proposition 5: If a number (x) is part of a number (z, where xy = z), and another (x1) is the same part of another (y1, where x1y1 = z1 and z/x = z1/x1), then the sum (x + x1) is also the same part of the sum (z + z1) that the one is of the one ((z + z1)/(x + x1) = z/x).

I had to read the proposition like 10 times to get it in my mind. When Euclid says "the one is of the one," it's very confusing and it's not so clear what he is referring to. I actually found the translation by ignoring the one to one part. The propositions 6-11 follow similar wording make similar algebraic claims. So after translating 5, they don't seem so bad. As part of the main daily 2 work, I wrote a proof for this proposition.

What is math?

Mathematics is the study of how our universe operates. Math is a way to interpret nature using ideas like calculus, rings, topology, or lie Algebras. Math is based on logic; any mathematical proof is a series of logical steps that arrive at a certain conclusion. But, this "definition" of math is probably lacking in some way. Math encompasses so much, it's hard to describe all of what it is. So how about I describe parts of it, from my experience.

For most of the math classes I've taken, I was given a set of elements, a couple of operations, and some basic definitions (for instance a ring). Then the rest of the class was just exploring that set of elements by understanding different properties (if the ring was a field or commutative) and theorems (the isomorphism theorems of rings). That was honestly modern algebra I and II, linear algebra I and II, and topology. But, then you have your calculus and advanced calculus classes. There, you would study limits, sequences, series, continuity, derivatives, integrals, etc... Then of course you have probability and statistics. I could go on and on. But, to better understand what math is, study it! Learn about the different areas of mathematical research (algebra, geometry, differential algebra, algebraic topology, etc...). Learn about the countless properties of the real and complex numbers. The study of math is so vast. One thing that got me more interested in mathematics was its history.

Here are some of my favorite moments/discoveries in the history of mathematics. One of the most important mathematical works is Euclid's Elements. Euclid essentially took all of Greek geometry at his time and made it into a book. He had his axioms/postulates from where he started and then came up with many geometrical theorems and properties. Then of course we have the discovery of calculus by Newton AND Liebniz in the 1700s (or 1600s I think). After the discovery of calculus, the study of physics progressed rapidly. And then we have the mathematical discovery machine in the great Leonhard Euler. He was born a genius and had John Bernoulli as a mentor growing up. Euler made contributions in all areas of mathematics: complex analysis, calculus, geometry, number theory. Even after his death he had some of his works still being published. Euler is my favorite mathematician. I also would like to mention Rene Descartes. From what I've read, much of our algebraic notation we use today is from him. One last mathematician I would like to mention is Evariste Galios. I will not spoil everything for those who have not heard of him. Look him up, his story is astounding.