Amir D. Aczel - Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem

Note:  This review was originally posted to my Epinions account.

Anyone who took math in high school would remember the Pythagorean theorem. It states that the sum of the squares of the two sides of a right triangle equals the square of the hypotenuse. (More basically, that a squared plus b squared equals c squared.) Then, along came Fermat, who stated that two is the only whole-number power greater than one to which this equation is possible, at least with whole numbers as a and b. This means that you could have A cubed plus B cubed equaling C cubed. Supposedly, Fermat had proof, but he never wrote it down.

In the 300 years since Fermat wrote that down, many great minds have tried to figure out how to prove (or disprove) what became known as Fermat’s Last Theorem. This book shows many of the major players and how they went about trying to get the answer, which was eventually solved. (Yes, it was an extremely difficult problem.)

I thought the book was a little short. It was only 147 pages, which made for an easy read. While the book covered the subject matter pretty well, it didn’t go into a lot of detail. There were many mathematical theorems and proofs that built up to the proof for Fermat’s Last Theorem; someone that doesn’t know a lot about higher math will probably be lost. I was able to follow the book, but there were still a few things that I didn’t know much about.

Instead of focusing on the mathematical detail, the book is more of a historical account of what happened. For instance, Fermat lived for another 26 years, I believe, after writing out his famous equation. In that time, he never bothered to write out his proof. It’s believed that his proof was much simpler than the one we have now, mostly because it used a lot of math that Fermat didn’t have available. However, we’ll never know if Fermat actually had a proof or if he just wrote out this equation on a whim.

(On a side note, there was an episode of Star Trek: The Next Generation in which Captain Picard mentions Fermat’s Last Theorem, stating that it hadn’t yet been solved. That episode aired a few years prior to the finding of the solution.)

One thing that I noticed, and I might be imagining this, is that it seemed like there was a lot of repetition. I know that I wasn’t repeating any pages, but there were some passages that seemed familiar, as if I had just read them a few pages back. Maybe I had accidentally gone back a few pages, but I doubt it.

I’d give this book four stars, but I wouldn’t recommend it to everyone. You will need some understanding of math. It may be somewhat difficult for the average person. If you’re into math, this would be a good book to read. 

Fermat's Last Theorem on Wolfram Mathworld

A Short-Form Proof of Fermat's Last Theorem

Malba Tahan - The Man Who Counted: A Collection of Mathematical Adventures

Note:  This is a review I originally posted to my Epinions account.

I actually found this book by accident. After reading another math-related book, I decided to look in the library for other similar books. I couldn't find what I was looking for, but I came across this book, the story of Beremiz Samir, a man who has exceptional math skills. The story is told by Hanak Tade Maia and starts when the two meet. Beremiz demonstrates to Hanak the unusual abilities that he has and the two become traveling companions.

Much of the story takes place in Baghdad. Beremiz solves many problems for many people. The first one that he solves is for three brothers who have inherited 35 camels. The problem is that one of them is to receive half of the camels, which is impossible since 35 is not divisible by two. Beremiz comes up with an ingenious solution. From there, the problems get more complicated, but Beremiz handles them with ease. As the problems he solves become more difficult, his fame increases, as well. He is eventually tested by seven wise men and passes all of the challenges put before him. In so doing, he gets what he wants most.

I don't want to give away too much because that would take away from the book. You don't have to be a math nut to enjoy it. All of the solutions that Beremiz came up with were explained simply enough for anyone to understand. While I think that the story itself is fiction, there are some interesting mathematical curiosities. There's one section on getting any whole number out of four fours. For instance, 1 is 44 divided by 44. 2 is four fourths plus four fourths, or 1 plus 1.

This book was translated by Leslie Clark and Alastair Reed and illustrated by Patricia Reid Baquero. Aside from the cover, the illustrations were at the beginning of each chapter and in black and white. They were simple, but worked well with the story. I actually looked forward to each new illustration.

There was something about the story that made me not want to stop reading. I like math, but I don't think that was it. It's mostly to see how good this guy really is. You have to wonder if Beremiz will ever be presented with a problem that he can't solve. That's just it, though. While Beremiz's skills are very rare, the book illustrates how useful math is.

I'd recommend this book to anyone. There was a religious aspect that might turn some people off, but I don't think that it was offensive to anyone. It shouldn't be a problem to the vast majority of the people. Most of the religious references weren't too much. The chapters were short, which made for an easy read. I almost read the entire thing in the library, but I had to be somewhere, so I checked it out and brought it home. Five stars.

WIkipedia page

Publxihser Web Site

Edward H. Julius - Rapid Math: Tricks and Tips, 30 Days to Number Power (book review)

Note:  This is a review that's reposted from my Epinions account with a few minor modifications.

I’ve always been impressed by people that could write thousands of words on a topic when I could only get a few hundred. Many of these people knew a lot about the subject and had a lot to offer. I could only do this when I was reviewing something, such as a game, that required a lot of detail. Math is a subject that I know a lot about, so I’m about to reveal how and why some of the tricks work.

While going through the bookcase recently, I came across this book. My brother bought it many, many years ago. He seemed to like it, but hasn’t used it much since he left for college. The book promises greater calculating speed and over 2,000 practice problems. I figured that I’d take a look through it. Presumably, it’s meant to be done over the course of a month, since the subtitle is "30 days to number power". If you made it all the way to high-school math, the book won’t seem that  impressive.

Each day has two tricks. It starts off with multiplying and dividing with zeros. For example, if you have to multiply 50 by 30, remove the zeroes and multiply 5 by 3 to get 15, then put the two zeroes back to get 1500. That’s an entire trick. The next one is about multiplying and dividing with decimals. It’s the same concept with different powers of ten. You should have mastered this before leaving elementary school. Of course, both of these are on the first day. I don’t imagine that it would be a good idea to throw anything too difficult your way so soon.

Many of the ‘tricks’ could be consolidated. For instance, one trick is how to multiply two numbers that differ by 2. Almost a week later, you learn how to multiply two numbers that differ by 4. Both of these tricks rely on the same principle. Lets say you have two numbers, 31 and 29. According to the book, you multiply by the average of the two numbers and subtract 1, thus getting 899. If you have 32 and 28, you multiply the two numbers and subtract 4, thus getting 896. What the book doesn’t explain is that this works for any two numbers. (For the sake of convenience, it's generally only used for whole numbers that differ by an even number.) The rule is that (x+y)(x-y)=x?-y?. It’s just that the larger the difference between the two numbers, the less convenient it becomes. If you look closely, you’ll notice that this rule comes up several times throughout the book.

Look also at the trick for multiplying by 12. The author says to multiply by ten, then to double the amount so that 38 times 12 should be 380+76, which amounts to 456. The trick for multiplying by 11 is similar. To multiply by 11, take the number, split the digits and put the sum of the digits in the middle, carrying if you have to. 59 times 11 is 509+140, or 649. I have news for you: this is how you usually do math. The author is just pointing out two different cases that are easy to do in your head.

Multiplying a two-digit number by 101 is a similar case, where you just repeat the two-digit number. 48 times 101 becomes 4848. Multiplying by 99 is more of a trick which most people wouldn’t necessarily be able to figure out on their own. Instead of multiplying by 99, you multiply by 100 and subtract 1. 48 times 99 is done by multiplying 48 by 100 to get 4800. You then subtract 48 from 4800 to get 4752.

The author also points out that you can reassemble factors to make multiplying easier. Instead of 2 times 14, you can multiply 2 times 2 times 7, or 4 times 7. This is essentially the same thing as dripping zeroes. 30x50 is like 3x10x5x10, or 3x5x10x10. (It also works great until you hit prime numbers. If you have to multiply 13 by 7, you’ll have to find another way to do it.)

The author presents it in a way that is easy for someone to understand each trick, but the reader might not necessarily understand the underlying principle, and that’s the problem. The trick from the paragraph before last could be used with other numbers or in combination with other tricks. For instance, 48 times 98 would be like 48 times 100 minus twice 48, or 4702. The real lesson to be learned is to notice proximity to an easier number to multiply by.

I would say that if you paid attention in math up until high school, you’ll find about 30% of this book to be stuff that you could have figured out on your own. (I’d say that a person of average mathematical abilities will find at least a few tricks that they already have figured out on their own.) Another 20% will be stuff that will serve no practical purpose. For instance, the book has parlor tricks, which the author admits are nothing more than mathematical curiosities that are meant to amuse people at parties. One will allow you to tell the day of the week for any date in the 20th century.

There are also a few things, like adding large sets of numbers, which will probably still require pencil and paper of most people. On the whole, I’d say that among the 60 tricks in the book, very few of them are of any benefit to me. Many are of great use, but I already know much of the information contained herein. I think that it would have been better to write a book on why these tricks work.

And now for something completely different

You have entered 131,155 Bills worth $346,299
Bills with hits: 9,205 Total hits: 10,419
Hit rate: 7.02% Slugging Percentage: 7.94% (total hits/total bills)
George Score: 1,268.43
Your rank (based on George Score) is #199
(out of 49,660 current users with a George Score. [99.6 Percentile])
Your State Rank in Florida is: 19 out of 7,924 [99.8]
Your initial entries with hits have traveled a total of 4,879,028 miles.
They have averaged 474.9 miles per hit and 194.41 days between each hit.

For those of you that think that math has no real-world application, I present to you a case of statistical sampling. Very often, I will enter 100 or more bills at a time. Once in a while, I’ll have entered fewer bills than I should have. This means one of two things: either I was shorted a bill or I missed one.

Since I deal with banks and casinos, I’m dealing with people that are a bit paranoid about money and thus not likely to short me a bill. When I’m missing a bill like that, it means that I’ve skipped over one. You may ask how I can find one missing bill among 100, 200, 300 or more. Do I have to go through each bill?

I used to do that until I realized that I didn’t have to. That’s where statistical sampling comes in. Let’s say that I have 250 bills. I think I’ve entered all of them only to discover that I’ve actually entered 249. I count off 10 bills at a time and check the top bill against the recently entered bills.

The top bill on my pile should be the most recently entered bill. If I count off ten bills, the next should be the eleventh most recently entered. When I find one that doesn’t match, I’ve effectively narrowed it down to a range of ten bills; my missing bill should be in there. (If it’s not, I’ve probably miscounted.)

I know this may well be boring to most, but it is an example of how paying attention in math class can help.