Sticking with my nerdy theme as of late, the book I have chosen this time around is called Zero. Zero was written by Charles Seife, a mathematician who also has his Doctorate in journalism. The book is named "The Biography of a Dangerous Idea" but how dangerous can one silly number be? The answers inside the book may surprise and shock you. You can find this book on Amazon here http://www.amazon.com/Zero-Biography-Dangerous-Charles-Seife/dp/0140296476.
This book documents the progression of the number zero through the ages. Now I know what you are thinking, how can you even begin to follow the progression of a single number. More importantly how can there even be a history of a number other than, boom the number was invented, end of story. The revelations in are quite intriguing for the budding mathematician and the historian alike. If this book was about any other number it would be rather dull and boring. A limp noodle of a book one might say. But the idea of nothing or zero, is quite different. One might argue that the thought of nothingness philosophically is quite terrifying to most people. The book goes into such topics as philosophy, history, religion and my favorite pure mathematics. Seife talks about how the concept of zero sparked such ingenuity's as Calculus and scared away such brilliant minds as Einstein. He argues that most religion is based on the concept of zero, for the thought of nothing after our lives are over, is terrifying to most people. Where as the eternal or infinite, zero's twin, is a much easier pill to swallow. Most of the book refers to the "dance" between zero and infinite and how they are two sides of the same coin. Alpha and Omega, Yin and Yang, the never ending battle that occurs between these titanic twin ideals. I would say that this book is more philosophical and historical than it is mathematical but you can not help but see the beauty in the way Seife presents the ideas within. I have to say that this is my favorite read thus far and being only 5 books in I guess that is not saying much. But this will be a hard one to top and being only 213 pages long and costing only $10 you would be silly not to pick this one up and give it a read.
Now I have to include this wonderful little proof that essentially allows me to prove anything in the universe using a little bit of logic, although if you look carefully I will make a mistake which of course in the end nullifies my proof. The question is can you spot it?
Let a and b each be equal to 1, since a and b are equal it follows that,
b^2 = ab and
a^2 = a^2
Now subtract the 1st equation from the 2nd to get
a^2 - b^2 = a^2 - ab
Factoring this we get
(a + b)(a - b) = a(a - b)
Dividing by (a - b) we get
a + b = a
Subtract a to get
b = 0
But I set b = 1 at the very start how can 1 = 0???
Using this trickery I can essentially prove anything in the universe I would like, the power of zero reigns supreme!!!! Alas I may have provided you with the solution to my little trick and given away my diabolical plan. I guess you can't win them all. But oh the fun we can have with math!
Here's a more subtle one (or, at least, more difficult to to formally express why it's wrong):
ReplyDelete0 = 0 + 0 + 0 + 0 + ...
0 = (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + ...
Arithmetic is associative, so shifting the parenthesis over is a legitimate operation:
0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ...
0 = 1 + 0 + 0 + 0 + ...
0 = 1
And, for variety, a different-but-similar approach to yours:
Starting from yours, a^2 - b^2 = a^2 - ab
b^2 - ab = 0 <== cancel out the a^2's
b^2 - ab + (a/2)^2 = (a/2)^2 <== add (a/2)^2
(b - a/2)^2 = (a/2)^2 <== factor the left
b - a/2 = a/2 <== sqrt both sides
b = a
0 = 1