Wednesday, October 21, 2015

Review: The Math Book

For class we each read a book of our choosing.  I chose the Math Book by Clifford Pickover because it looked like an interesting quick read.  The front states "250 Milestones in the History of Mathematics" and that's exactly what it is.  It's written in chronological order all the way from circa 150 Million BCE to 2007, two years before it was published.  Every page is a different milestone with an accompanying picture.  

Not only does it discuss the famous mathematicians that we've all heard of like Pythagoras, Euler, and Fibonacci, but it also discusses people, discoveries, and inventions that I've never talked about in math class.  My favorite pages were ones that talked about different puzzles and games like Rope around the Earth Puzzle (1702), and the board game Go (548 BCE) .  So although not every page is a gigantic breakthrough that everyone has heard about, they're all important in some way, even just for entertainment.  Most of the pages discuss some history which I found really interesting because not often do we put these discoveries in context of the times.   One that I found interesting that ties in the puzzles and also history was the entry on Hex from 1942.  It discussed the game of hex and how to play and then also tells about how it was manufactured by Parker Brothers.  It's inventor, Piet Hein had to go into hiding in 1940 because of WWII.  It really makes you think about what could have been discovered and invented by some people if their society had allowed them to keep doing math.

Overall I really enjoyed the book.  It really shows how vast math is and showed how interesting it could be.  It's written at a level that is easy to read and understand.  I believe the general public would enjoy this book, but it might help if that have a slight interest in math to start with.  No deep knowledge of math is needed to read and understand this book.  For the first half of the book I read straight through in chronological order, but the second half I skipped around a bit, so if the reader decides to read in chronological order or not, either way is interesting.  My only complaint is that sometimes I wish the pages were a little more in depth instead of just introducing the topic and then moving on to the next one.  Perhaps if I had had more time to get through this book I could have looked up the ones I was really interested in online instead of immediately moving to the next one.  I definitely liked this book though and would recommend it to anyone with the slightest inclination towards math and its history.  It was an easy read and entertaining.

Monday, October 12, 2015

The Tale of the Cubic

Once upon a time there was no uniform way to solve all cubic functions--how sad.  It was a problem that puzzled mathematicians up until 1535.  What happened in 1535 you ask?  Well there was a math competition in Italy that a guy named Niccolo Fontana attended.  You might know him by the nickname Tartaglia which means "the stutterer."  Or you might not.  Anyway this guy was an engineer and amateur mathematician--because who doesn't want to do math in their free time?  So he came to this competition and SURPRISE he won it by solving a cubic function using his general formula!  Everyone was shocked because they had thought it to be impossible.  

Since it was such a coveted formula, Tartaglia wanted to keep it to himself, even hiding it by encoding it in a poem.  Here's a picture of the formula from this Vanderbilt website. Can you imagine fitting that whole thing secretly into a poem? 

So Tartaglia wanted all the glory for himself and kept it a secret until the smooth talking Gerolamo Cardano came along and got the formula from Tartaglia which he subsequently published in his book "Ars Magna" in 1545.  Needless to say Tartaglia was not pleased that Cardano broke his promise of keeping it a secret, but got his revenge by helping Cardano get arrested for heresy after Cardano made a horoscope for Jesus.  

When we talked about the cubic formula in class I attempted to use the formula to solve the following cubic.  The (-3,0), (2,0), and (5,0) are the solutions I found from graphing.  My incomplete attempt gave me imaginary numbers after taking the first square root.  I may have messed up a negative somewhere...  But take a look and then take a moment to appreciate graphing calculators, the factor theorem, and long division of polynomials that can find the solutions a whole lot faster!
It makes me wonder what kind of applications they had that needed the solution of a cubic function.  As I thought back on my schooling to try to remember an application we learned that used cubics I couldn't think of a single one.  I think we were always just given a graph and/or the equation and had to find the solutions from there without giving an adequate reason why we should even solve it.  So I went to google and found this page where several people came up with applications of cubics.  Most of them seem to be far more advanced than what we would learn about when first seeing cubic formulas.  I thought the most interesting one was the claim that the typeset letters are formed from cubic functions.  So thank you cubics for helping me type this blog!  Do you know of any other applications of cubic functions?  Any that people would have needed in 1535?

Hey look!  If you want more information about the characters in this tale click here for a more in depth story courtesy of Luke Mastin!