A Slice of Pi

  
   I find myself thinking about math more these days than ever before.  I have to admit that these past four weeks have been intense.  My brain has been inundated with math concepts to the point that I even ponder ideas having to do with math when I could be thinking of other things.
   After revisiting many formulas, I began wondering about the concept of Pi.  Where did it come from?  How long has the idea been around?  Who came up with it?  Was his last name "Pi?"  I'll answer that last question right away.  No, it wasn't.
   After doing some investigating, it seems that a mathematician named William Jones was the first to use the symbol for Pi, although he did not invent the concept of Pi.  According to an article on Math.com, "Ancient civilizations knew that there was a fixed ratio of circumference to diameter that was approximately equal to three.  The Greeks refined the process and Archimedes is credited with the first theoretical calculation of Pi." 
   I have been finding out that there are many people who have devoted their entire lives to the study of mathematics.  I wish I could have met them and been in the presence of such intelligence.  I also wish that one of them could sit next to me as I do my homework, but I digress. 

                                                                 William Jones
                                                   Photo courtesy of: en.wikipedia.org




   I found this very interesting article on some of those math-minded geniuses titled "5 brilliant mathematicians and their impact on the modern world."  It is very clear that the contributions of these five men; Isaac Newton, Carl Gauss, John Von Neumann, Alan Turing, Benoit Mandelbrot, helped to make our world what it is today.  We would not have the everyday lives that we currently have without them and without what their minds helped to develop.   Many of the gadgets that we have come to rely so heavily on; computers and cell phones to name just two, would not be possibilities without the math concepts that are foundational in their operations.

                                                                  Isaac Newton
                                                 Photo courtesy of: www.brighthub.com

   So back to this notion of Pi.  The number represented by the symbol Pi, or π, is 3.1415926... ...
and goes on...and on...and on...  Here is a list of the first 500 digits of Pi as taken from an article titled,"Memorize the Number Pi to 500 Places:" 


500 digits of pi, written as 50 digits in each row:

3.1415926535897932384626433832795028841971693993751
05820974944592307816406286208998628034825342117067
98214808651328230664709384460955058223172535940812
84811174502841027019385211055596446229489549303819
64428810975665933446128475648233786783165271201909
14564856692346034861045432664821339360726024914127
37245870066063155881748815209209628292540917153643
67892590360011330530548820466521384146951941511609
43305727036575959195309218611738193261179310511854
80744623799627495673518857527248912279381830119491


Pretty amazing, isn't it?  Believe it or not, there have been many many people who have spent more time than we can imagine memorizing thousands and thousands of digits of Pi.  There's even a world record category for it.
   For those who have no desire to try to memorize the endless list of numbers in Pi, we can at least appreciate the old "Area equals Pi r squared" from elementary school.  The next time you're trying to figure out the area of a circle, you'll be serving up a slice of Pi with that formula.  If I were you I'd stop at 3.1415.  A little Pi goes a long way.

 
 
 

A Whole New World

   Here I am on this adventure of taking two college math courses during the summer session and I find myself stunned at the amount of information that is coming my way.  I've already admitted that math is not my strongest subject, but what I've been studying this past week has really given me a lot to think about. 
   I've been studying and working problems on topics with names such as:  "Connecting Euler's Formulas for Planar Networks and for Polyhedra," "Working With Angles in Regular n-Gons," and "Determining the Interior Points of a Simple Closed Curve in the Plane," to name a few.  Those are topics from my Mathematics For Elementary Teachers II class.  This blog is actually designed as part of my studies for Mathematics For Elementary Teachers I class.  I mention both classes simply to say that while summer is normally the time for outdoor fun and swimming in the sunshine, the only thing swimming around here right now is my mind.  It's a good thing.  Don't get me wrong, but it is a new thing.  New things always take a little getting used to.
   I did give myself a break from the studies the other night and went outside to play on the trampoline in the dark with my daughter.  We laughed and jumped and sang songs like a couple of dorks.  After a long time we laid down on our backs and looked up.  I laid there looking at the stars and feeling the breeze of a cool summer night.  As I noticed the hexagonal shape of the net that surrounds the trampoline, I thought to myself, "I bet I could figure out the angles of each adjacent side of the net."  That's when I realized, this math stuff gets in your head.  I suppose that's the whole point.
   After spending several hours this past week learning about numeration systems of other cultures from both the past and the present, I have a new found appreciation for all things math but especially for our numbering system, which now seems so much "easier" than others.  It's only my second blog post and I am already referring to math as "easier?"  Huh.
  In all seriousness, we can be glad to have the system that we do for counting and combining numbers.  I learned about the Egyptian, Babylonian, Mayan, Indo-Arabic, and Roman numeration systems.  I even learned something called "The Russian Peasant Algorithm" for Multiplying."  It consists of making two columns of numbers beginning with the first number to be multiplied placed at the top of a column on the left and the second number to be multiplied placed at the top of a column to the right.
    From there, the numbers in the left column are halved and the numbers in the right column are doubled until the last remaining pair of numbers begins with a 1.  The pairs of numbers are set up in a table.  Rows of numbers beginning with an even number are crossed off of the list.  At the end, the sum of all numbers remaining in the right column are added together to find the answer to the original multiplication problem.  It is time-consuming, but really rather amazing.  You'll want to try it.  Go ahead.  Just do it.

Multiplying Using the Russian Peasant Algorithm

 

 

 

 http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml


   Each of these systems is equally impressive.  The Mayan and the Babylonian systems were pretty incredible and left me scratching my head.  It's one of those things that at first is like a puzzle that you have to try to wrap your mind around and then once you get it, you get it and you can appreciate it.  If you've ever mastered one of those metal "brain teaser" types of puzzles, you'll know exactly what I mean.  You know, the ones where you have to get one part of this metal puzzle separated from the other and it seems impossible...until you do it...and after that you can't not know how to do it?
   Mathematics is a language of its own, but it is one that has been spoken in every culture across all times.  The wording and the symbols may be different, but the ideas are the same.  As for me, I all of a sudden really really like the way that we "do" math here in America.  And with that, I have some math to do... ...

Here is a chart of the Egyptian symbols for the powers of 10:

https://www.google.com/search?q=Egyptian+Symbols+for+powers+of+10&client=firefox-a&hs=FJV&rls=org.mozilla:en-US:official&channel=fflb&source=lnms&tbm=isch&sa=X&ei=dXCoU4XgNtevyASshoD4Bg&ved=0CAgQ_AUoAQ&biw=1218&bih=638

Getting Started

   I have this opportunity to create a blog for a Math Class that I am taking this Summer.  I can't say that I'm an expert on blogs, but just like so many other aspects of being back in college, I'm willing to give it a try.  
   It's been many years since I took my last Math Class, and I find myself thinking back to High School.  I can picture the classrooms where I took Algebra and Geometry.  I can remember where many of my friends and classmates sat too.  (In a graduating class of 39 kids, it's really not that difficult to do.)
   I also remember the feeling of fear; fear of being called on to come up to the board and show an answer, fear of not having the correct answer, fear of being laughed at.  My dad used to try to help me with my Math and if any subject made me cry, it was Math.  Well, that or Phy Ed., but that story is for another time.  I can still feel that feeling of sitting at the kitchen table trying not to cry over Math problems that I could not wrap my mind around.
   Some things in life change and some stay the same.  This week we've been learning about sets and operations on sets.  Again, I find myself back at the kitchen table trying to wrap my mind around a Math topic and, well, trying not to cry.  I'm finding that there are symbols that are very new to me.  I realize that their meanings simply need to be memorized, much the same as the multiplication sign, the minus sign, the plus sign and others.
   Here is a list that I found that is helpful for understanding the meaning of these symbols:
  
Table courtesy of: http://enpub.fulton.asu.edu/powerzone/fuzzylogic/chapter%202/chapter2.html
   
   Sets are basically collections; numbers, objects, anything that can be put into a category.  Georg Cantor was a German Mathematician who came up with the idea of sets in Mathematics.