Book Review: e: The Story of a Number

I read e: The Story of a Number by Eli Maor.  This book was a very fast read.  The book was laid out in a very easy to read format, with each chapter talking about the history of some aspect of math.  At the end of each chapter there was a short section on some of the derivations that were talked about in the chapter.  The back of the book had additional appendices to show more derivations and show how some other parts of the book were concluded.

I liked the layout of the book.  Just talking about the history of a discovery and some of the most essential math in the actual chapters of the book was a good move by the author.  This helped focus the chapter on the essentials of the history.  It also allowed for those that have less mathematical background in general or on a specific subject to be able to finish a chapter and understand what the key messages were.  Adding the additional parts at the ends of chapters and the appendices were also good, because those that have more background in the subject or want to know more could go and look at those sections and find the derivations.  Personally, there were some that I looked at, and others that I found less interest in and just skipped over because I was not interested in the topic.

As far as the content goes, I found the book to be VERY interesting.  I chose this book by the recommendation of Duncan Vos and because I knew that being a science major as well, learning more about the number e would be more helpful than many other things that I could have chosen to read.  The history behind the number e is very long.  It has many twists and turns that finally get to the ending value of 2.7182818284. The naming of the number also took many years, it was initially called just the inverse of the logarithmic function, but Leonard Euler (of course) was the one that gave it the letter e.

The book was a definite good read, but I would not suggest it to everyone. The material is very interesting to those that like to learn the history/origins of numbers and ideas, but if that is not your favorite thing to read about, I would shy away from this book. A major flaw of the book comes in that aspect, in that, the author talks about feuds between families and the discovery of calculus, but doesn't do a great job tying all of the aspects of the book together well. He gets all the major discoveries that lead up to the number, but again, unless you have background knowledge in the area of e or enjoy having a history lesson, the book is not for you.

Doing Math: Magic Birthday Square

After class time on April 8th, I found one particular problem of Ramanujan both interesting and completely unfair.  I know all of the people in class that day felt the same way.  We were going through many of his accomplishments when it was shortly brought up that Ramanujan had a magic square that he had made out of his birth-date.  It made me wonder... Can any birth-date be made into a perfect square?  So of course I gave it my own go.  Luckily with today's technology, doing this is much easier than it had to have been in the past.  With the simple power of Microsoft Excel's "SUM" tool I could easily know if my perfect square would work.

I first started with my birthday, October 23, 1991 (10-23-1991), which adds up to be 143 (10+23+19+91).  So far the table is pretty empty, but it looks like the following,

10	23	19	91
-	-	-	-
-	-	-	-
-	-	-	-

I was trying to figure out what to do next.  I began with a simple guess.  I knew that all the squares (including the top left 2-by-2 square) had to be 143, so I took 19, added 1 to it and took 91 and subtracted one, leaving me with 90 and 20.  Now came the time for a decision, where to place the numbers.  I looked at the numbers and thought to try putting the larger number with the smaller number and vice verse.

10	23	19	91
90	20	-	-
-	-	-	-
-	-	-	-

So now I knew that the top left corner square was 143.  So again I went with what I thought to be a good educated guess.  I looked at the top right corner now.  I knew that the numbers need to add to be 33 so I first tried to do 32 and 1, with the same strategy as before, I thought to put the bigger number underneath the smaller number from the first row, so my magic square then looked like the following,

10	23	19	91
90	20	32	1
-	-	-	-
-	-	-	-

I was beginning to feel like I was getting somewhere.  I now had to take one more guess and the rest of the numbers should fall into place if they were to be right in this configuration, otherwise, I would have to start again.  I knew that the first vertical line was so far at 100 and I needed to get to 143, I also knew that the second vertical line was at 43 and needed to get to 143.  I took a wild guess and said why not try 2 and 98 for the two numbers for the second vertical line, I chose that it would be 2 then 98 in the vertical sense because I knew from the top that 23+19 was only 42, so it needed a lot more to get to be 143, so I was at,

10	23	19	91
90	20	32	1
-	2	-	-
-	98	-	-

I now knew 3 of the 4 squares from the square that makes up the middle two top numbers and middle two bottom numbers.  I did some simply subtraction (143-(23+19+98)) and found that the last number needed to be 3 in order to complete that square. I filled that one in next.

10	23	19	91
90	20	32	1
-	2	-	-
-	98	3	-

Since the vertical was now almost complete, it was only crucial to fill in the number above 3.  So I found what 143-32-19-3 is 89.  This now completes the third vertical row.

10	23	19	91
90	20	32	1
-	2	89	-
-	98	3	-

Next I will figure out the middle right square.  Since there is already a 32, 1, and 89, then 143 subtracted by those leaves 21.  So that gets filled in there and with that entry, there is the bottom right most square that can be filled in, subtracting 143 by 89, 21, and 3 leaves 30.  So we will fill those in below.

10	23	19	91
90	20	32	1
-	2	89	21
-	98	3	30

Using the same method, the two squares on the left can be filled in.  The middle left most square is then filled in with 31 since 143-(90+20+2) and then with that filled in the bottom left most square is filled in with 12 since 12=143-(31+2+98).  So the final square turns out to be as follows

10	23	19	91
90	20	32	1
31	2	89	21
12	98	3	30

After being elated that I found a magic square for my own birthday, I began to check myself.  I realized at that moment that my magic square was not completely correct.  I looked at the diagonals and the bottom and top middle squares and realized that these 4 sets did not add to the magic 143 number.  I tried for quite some more time, but did it to no avail.  I then questioned the thought that my birthday had a magic square.  I tried more numbers and worked through the problem the same way.  The first guesses turned out to be the closest I could get out of my 30 tries.  I then ended my trails at a magic birthday square believing that my birthday (along with the possibilities of many others) did not have a magic square associated with it.

Doing Math: March Madness

March Madness has many math elements that go into it.  All sorts of statistics go into each game in order to decide the percentage of winning for each team.  If the chances of winning were 50% for both teams for every game, then the chances of getting the bracket 100% correct would be,

$2^{63}=9.22337\cdot10^{18}$

Now, since it has never happened in history that a 16-seed team has beaten a 1-seed team in the first round (a whopping 0-96 record), then the chances get a little better if we factor those games out,

$2^{59}=5.76461\cdot10^{17}$

To put that large number in perspective, if we assume there are 7,222,626,382 (found from http://www.worldometers.info/world-population/ at 10:38 AM on March 28th, 2014), then each person would have to fill out 79,813,176.23 brackets each to fulfill every possible outcome of March Madness.  Again another large number, so lets put that number into perspective.  If every person were to fill one bracket completely every second, then it would take 2.53086 years to created every possible bracket outcome for this year's March Madness.

We can now show the chances of getting a perfect bracket if each team has a 50-50 shot to win each game.

$.5^{63}=1.08420\cdot10^{-19}$

$=1.08420\cdot10^{-17}\%$

Again, we could take out the 1-seed versus 16-seed games again, so the chances get a bit better,

$.5^{59}=1.73472\cdot10^{-18}$

$=1.73472\cdot10^{-16}\%$

Again, the chances are VERY slim of getting a perfect bracket.  These numbers can again be adjusted though since every game has statistics that go into them and each team has a certain percent chance of winning any single game.  If we factor in the odds.  If we never pick an upset, then the chances get much better.  The biggest toss-up of a game is the 8-seed versus the 9-seed at 47% of the time the 8-seed winning.  From there, the higher seeded team wins more consistently from in an almost linear fashion from the 7-seed to the 1-seed.  In the second round, 7-seeds win 60%, 6-seeds win 67%, 5-seeds win 67%, 4-seeds win 79%, 3-seeds win 85%, 2-seeds win 96%, and as mentioned above, 1-seeds win 100% of the time (from http://statistics.about.com/od/Applications/a/March-Madness-Statistics.htm).  Using these statistics, we can make a better estimate of what the chances would be if picking the higher percentage win,

$1.00\cdot.96\cdot.85\cdot.79\cdot.67\cdot.67\cdot.6\cdot.53=.092022$

$= 9.2022\%$

After only the round of 64, picking the seeds with the best probability still only leaves 9.2022% for each region of having a perfect bracket.  Factoring in that there are four different regions makes the percentage,

$.092002^4=.0071709$

$.71709\%$

Therefore, after only one round, and picking the higher seeded teams every time, the percentage of having a perfect bracket is only .71709%.  In the next round, we are only left with the 1-seed versus 9-seed, 2-seed versus 7-seed, 3-seed versus 6-seed, and 4-seed versus 5-seed in each of the regions.  In the third round, the 1-seeds win 91% against the 9-seed, the 2-seeds win 75% against the 7-seed, the 3-seeds win 57% against the 6-seeds, and the 4-seeds win 54% against the 5-seeds (from http://www.betfirm.com/seeds-national-championship-odds/).  That means that the chances of getting the next round correct is,

$.91\cdot.75\cdot.57\cdot.54=.2100735$

$=21.00735\%$

Now, that is only for one region, again if all four regions are factored in, then

$.2100735^4=.001948$

$=.1948\%$

As you can see, as the rounds progress, the chances get a little bit better that the region can be gotten perfect.  The next round would then be the 1-seed versus the 4-seed and the 2-seed versus the 3-seed.  The 1-seed wins 58% against the 4-seed (from http://bluedevilnation.net/2014/03/path-final-four/) and the 2-seed wins 59% against the 3-seed (from http://www.sportingcharts.com/articles/nba/2014-ncaa-march-madness-cheat-sheet.aspx).

$.58\cdot.59=.3422$

$=34.22\%$

This is for each individual region, for all four regions it would be

$.3422^4=.01371$

$=1.371\%$

Finally, the last round before the final four would be the 1-seed versus the 2-seed.  Picking the 1-seed has a 55% chance of being correct.  Therefore,

$.55^4=.09151$

$=9.150625\%$

Since the last three games are then 1-seed versus 1-seed, then the chances are 50-50 for each game, thus

$.5^3=.125$

$=12.5\%$

To figure out the whole percentage of getting the bracket perfect, you must multiply all the percentages together.  Therefore,

$.0071709*.0019475*.091506*.013713*.125=2.1905\cdot10^{-9}$

$=2.1905\cdot10^{-7}\%$

$2.1905\cdot10^{-7}\%-1.73472\cdot10^{-16}\%=2.1905\cdot10^{-7}\%$

Therefore, with going by the percentages, your chances are much better.  Your chances are about 10,000,000,000 times better than if each game was a 50-50 shot.

If this sort of stuff interests you more, there are many good sites to look at in order to get more information.  I have put where I put my references, but other places for information you can look at,

http://fivethirtyeight.com/features/nate-silvers-ncaa-basketball-predictions/

http://bracketodds.cs.illinois.edu/index.html

http://outsidethehashes.com/?p=471

http://www.sportingcharts.com/articles/nba/how-does-seeding-affect-success-in-the-ncaa-mens-basketball-tournament.aspx

http://www.esquire.com/blogs/news/2014-march-madness-bracket-stats

Book Review: The Math Book by Clifford Pickover

The Math Book by Clifford Pickover was a good read.  The book goes through history and talks about the relevant mathematical findings.  Pickover starts from c. 150 million years B.C. and works his way up to 2007.  Pickover, in his introduction, states that he realizes that some of the discoveries not mentioned may be more important than the ones that he states, but this is his own opinion on the most significant findings and the ones he enjoyed learning about the most.

The book has a good structure.  Each new page is a new "section" and is completely independent from the previous or next page.  Each page also has an accompanying picture that, in some ways, tries to help the reader in their understanding of the subject.  Pickover, in each entry, tries to give some background about the subject and who discovered it and also tries to explain some what the discovery did or is doing for math now.  Each section is short and easy to read, which makes the book a fast read and not too time intensive to read through.

The book did some things well.  The book did a good job going over many topics (249 in total) relatively fast and briefly.  This book gives the reader a good sense of what has been done in math since c. 150 million years B.C.  The book also does a good job trying to link together many of the subjects talked about.  Each page has bolded words that refer the reader to other passages with similar context or to help the reader with an understanding on how the discovery was made (usually due to something earlier in the book).  At the end of each page, you can use the references to find other similar works if reading this book for a leisure read and wanting to gain more knowledge on a specific topic.  Pickover does do a good job of explaining who discovered each mathematical finding too.  He gives a very brief history on the person (i.e. who they are, their nationality, their race/religion, etc.) so that the reader can gain a respect for the breadth of math.  I believe that Pickover does this because he wants the reader to understand that math is discovered everywhere and by all types of people, yet sometimes he can bog down the reader with the details.  The most interesting part of this book, which was sometimes the most frustrating too, was the "paradoxes" and "problems" that Pickover places in the book.  I had an easy time understanding some of the "paradoxes" and/or "problems," but other times, like other sections, I just had no idea what was going on and so I would just skim over the reading and not understand it.  The "paradoxes" and "problems" that I did understand were very interesting though.  I found myself waiting to read the answer until giving myself sufficient time to try and find the answer (or have a guess) myself.  It made the book more fun to read and gave a break from the strictly math sections of the book.

The book also has many things that I would change.  There are frequently (more so toward the end of the book) pages and sections that I would leave out entirely.  For the standard student, some sections are either WAY over their head, way to rudimentary, or just plain boring.  The book would therefore go into way too much detail or just not enough at all.  I found that some sections I would finish the page description and leave with a "what?" or "I don't understand this at all" feel.  After talking to some other students reading the book, they felt the same way.  On the other hand, the pages that were just plain simple, I felt like the discovery was important, but the details not so much because they are self explanatory (i.e. a least squares line he goes into a two paragraph description of what it is).  Another thing that I would change about the book is the amount of background given for each section.  Sometimes the amount of background given for a particular section would overtake the math of the section and cause me to miss the overall reason for the section in the book.  The background, while I understand is sometimes necessary, would just be too in-depth.  A more brief background would have benefited me more and then if I was interested in the subject I would look up more.  The last thing that I did not like about the book was the amount of quotes given from other mathematicians about a particular work.  These added nothing to the sections.  Sometimes it seemed that Pickover would add these quotes just because he realized that a finding was important, but didn't understand if fully and so he would take up space on the page with these quotes.  I found that by the end of the book, almost any time something was quoted from some other mathematician I would completely skip the quote because it added nothing to my understanding and was a waste of my time to read it.

While it may seem that I had a lot of complaints about the book, I did highly enjoy reading it.  I found that as I was reading about things I would be able to link them to other things I have learned in math.  I also enjoyed learning about the history of math from another perspective.  If you can get past the critics I have on the book, I would suggest reading this book, as it was not a hard read and had some very interesting topics.

History of Math: Fibonacci Sequence

Fibonacci has done many things that have advanced mathematics.  Most of his accomplishments came around 1200 A.D. since he had lived from c.1170 to c.1250.  One of his most well known works is the Fibonacci Sequence.  This sequence involves adding the latest two terms in the sequence in order to achieve the new term, written as F(n+1)=F(n)+F(n-1).  This mathematical equation is not too hard to derive, especially given that the first two terms are defined to be 1 and 1 or 0 and 1 depending who you ask.  There are other applications that add to this mathematical equation as well.

Using the Fibonacci Sequence, the Golden Spiral can be made.  Using the Fibonacci numbers as guidelines, you can make squares that continually grow.  Then, connecting corner to corner with an arc, a spiral can be made.  The reason that this spiral has a special name, the Golden Spiral, is because in order to make this spiral, the dimensions are based off of the golden rectangle.

However, how would you answer the problem, how do the Fibonacci Sequence and Pascal's Triangle relate?  Give an example such as 89 to help with understanding?  After some research, you may find that the Fibonacci Sequence is made up of the additions of Pascal's Triangle's angles.  Each line, always drawn parallel to the last, will make up the next number in the Fibonacci Sequence.  So with Pascal's Triangle and the Fibonacci Sequence in hand, errors can be checked for on both.  A good example is the number 89.  89 is the 11th number in the Fibonacci Sequence.  This number is attained by adding 55 and 34 from the Fibonacci Sequence.  Using Pascal's Triangle, 89 is attained by adding 1, 9, 28, 35, 15, and 1 (if you want to see more check here http://upload.wikimedia.org/wikipedia/commons/b/bf/PascalTriangleFibanacci.svg).

Nature of Mathematics: Connections between Algebra and Geometry

Algebra and Geometry both are derived from our understanding of numbers.  Without one, the other would never have existed.  The two mathematical studies evolved concurrently, because advances in Geometry would merit advances in Algebra and the same was true the opposite direction.

Geometry was the beginnings of Algebra.  Through Geometry, mathematicians could visualize different problems that they had been working on.  A good example is learning about different algebraic equations through the usage of squares and other geometric objects.

After many years, advances in Algebra started to lead the way to advances in Geometry.  Through different equations and algebraic problems, different geometric figures could be constructed or at least found to at least exist.

Algebra and Geometry do not cause the discovery of the biggest ideas in either category today.  Algebra also did not cause the discovery of the largest finding in Geometry.  Through the continued practice of Geometric figures, non-Euclidean geometry was found by Gauss, Boylai, and Lobachevsky, all independently of one another.  They found a contradictions to Playfair's Postulate.

While Algebra did not cause the biggest discovery in Geometry, I believe that Geometry did cause the discovery of the biggest Algebra discovery.  By looking at triangles and other geometric figures (mostly squares), the discovery of Pythagorean's Theorem was made.

History of Fractals

A fractal is an infinitely extending mathematical set that perpetuates the the same image over and over.  Contrary to what many people believe, a fractal does not always have a regular pattern.  Many fractals have a very irregular or even fractured appearance to them.  Fractals are not necessarily a first degree shape either.  Circles, squares, and other 2D shapes if increased, then the area is increased by a factor of 2 because that is the size of its dimensions; however, a fractal does not expand like 2D or even 3D objects.  Fractals grow by a value that is not always an integer.

Allegedly, fractals have been known about since about the 17th century by the mathematician and philosopher Gottfried Leibniz.  It was known by the late 19th century that fractals were recursive and that they were continuous but not always differentiable discovered by Georg Cantor, Felix Klein, and Henri Poincare.  In 1904 Helge von Koch started making figures that represented the ideas of fractals (which can be viewed at http://en.wikipedia.org/wiki/File:Von_Koch_curve.gif).  Later that century, around 1918, at about the same time, two French mathematicians Pierre Fatou and Gaston Julia, came up with how fractals map the complex numbers.  By 1975, the name "fractal" was finally coined for these figures.  The name is derived from the Latin fractus meaning "broken" or "fractured."  The name fractal was coined by Benoit Mandelbrot.  Although naming these objects fractals, debate is still had over these objects.  Many people do not know how to formally define a fractal and what its name should encompass, but Mandelbrot was known for saying that fractals are "beautiful, damn hard, and increasingly useful. That's fractals!"  Now the general consensus on what a fractal is, is infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions.

Fractals have been defined as many things, not just geometric objects.  Among other things, fractals can also describe processes of time.  Fractals have been studied in images, structures and sounds, and also found in nature, technology, art, and law.  My area of interest comes from fractals in nature.  Fractals in nature show extended,  but finite, scale ranges.  These fractals are being studied heavily and are helping in many ways.  One such way is leaves, based on the fractals of leaves it can be determined how much carbon is in a given tree.  Wikipedia.org gives the following extensive list of fractals in nature...

• clouds
• river networks
• fault lines
• mountain ranges
• craters
• lightning bolts
• coastlines
• Mountain Goat Horns
• Animal coloration patterns
• Broccoli and Cauliflower
• Heart rates
• Heartbeat
• Earthquakes
• Snowflakes
• Crystals
• Blood vessels and pulmonary vessels
• Ocean waves
• DNA
• Soil pores
• Psychological subjective perception

For those that are interested in fractals and in viewing more fractals, Wikipedia.org is a good source.  Some fractals that I found pretty cool were the Julia set (http://en.wikipedia.org/wiki/File:Julia_set_%28indigo%29.png), Karperien Strange Attractor 200 (http://en.wikipedia.org/wiki/File:Karperien_Strange_Attractor_200.gif), Uniform Triangle Mass Center grade 5 fractal (http://en.wikipedia.org/wiki/File:Uniform_Triangle_Mass_Center_grade_5_fractal.gif), Lichtenberg figure (http://en.wikipedia.org/wiki/File:Lichtenberg_figure_in_block_of_Plexiglas.jpg), and the Mandelbrot set (http://en.wikipedia.org/wiki/File:Animation_of_the_growth_of_the_Mandelbrot_set_as_you_iterate_towards_infinity.gif).

My information was either previously known information or found from http://en.wikipedia.org/wiki/Fractal.