Sunday, June 8, 2014

Weekly 5: Book Review

     The book I am reviewing is Euler: Master of us All, by William Dunham, published in 1999 by the Mathematical Association of America (incorporated). I've read several math books in the past but this book is by far my favorite. Probably because it's about my favorite mathematician, Leonhard Euler. Now, there were eight chapters; each chapter talks about a topic: algebra, number theory complex numbers, and so on. In each each chapter, the author gives some historical background on the topic, then what Euler accomplished with that topic, then some concluding thoughts. For my blog, I will essentially review each chapter. But there also was a historical background of Euler right before chapter 1, and that is where we will start.
    The history of Euler is a story of courage, doubt, and the triumph of human spirit. Euler's parents were heavily involved in the clergy. So it looked like that was where Euler was headed. But as he was taking his theology classes, he couldn't help but to study mathematics. He actually entered college at the age of 14. Even as a child, Euler's genius was very apparent. At the university of Basel, he met Johann Bernoulli, a professor. Each Sunday Euler met with him to discuss Euler's questions on mathematics and physics. Later, Euler got a position at St. Petersburg in Russia, as a physics professor. The head of the math department was held my Danioull Bernoulli, with which Euler became friends with. He later took Danioull''s position as the head of the math department. But, because of political turmoil and wars, Euler went to the Berlin academy in Germany. There he was able to write mathematical works all day. Later he went back to St. Petersburg because the king of Germany (Frank the Great) was jealous and didn't like Euler. In Euler's later years, he was essentially blind. Though he was still able to publish math works with the help of scribes.

Euler may have been a genius, but my god did he work hard. Being a family man and having to tend to the government, I'm surprised he had time to do so much math. He inspires me to work hard and become a math professor. With his intelligence, he didn't have to work so hard and he would still be successful; he could have been rich. But instead, he concentrated his focus on math, something he loves to do. He is truly and inspiration for anyone. That's why I study math, because I love it. I don't really care about money, I never did. I'm just very thankful there are jobs out there where you can study math and make money. Now let's discuss the actual chapters.

Chapter 1: Euler and Number Theory
     Euler made a pretty significant contribution to number theory, especially on perfect numbers.
Victor Klee and Stan Wagon (professors in the late 1900's) said that perfect numbers "is perhaps the oldest unfinished project of mathematics." Even with Euler's contributions, that's a pretty bold statement.
One major contribution Euler made was off one of Euclid's theorems:
If 2^k-1 is prime and if N = 2^{k-2}(2^k-1), then N is perfect. 
Euclid proved that one. But, Euler proved the next one
If N is an even perfect number, then N = 2^{k-2}(2^k-1), where 2^k-1 is prime.
The author gives proofs to both of these theorems. The proofs are not too difficult; you could understand them with a couple of years of college math. One of my favorite theorems Euler proved was that the sum of reciprocals of odd perfect numbers is finite, the proof of which is very intuitive. Euler also helped out the study of perfect numbers by considering each number's whole number, instead of just their proper factors.

Chapter 1 was very interesting, it made me keep wanting to read on. Understanding the proofs take some time, but it's worth your while. Though sometimes the author will refer to a theorem or statement he made, so you have to flip back to remember what that statement was; things do get a bit muddy this way. But, some proofs in this chapter can be long, so I guess that was his best option.

Chapter 2: Euler and Logarithms 

This chapter explores some of Euler largest contributions, that of logarithms. One of Euler's books: Introductio in analysin infinitorum was published in 1748. The author said it was one of the most influential math books of all time. It's essentially a pre-calculus text. I wouldn't mind taking taking a look at it. I wonder how students would fare if we gave them Euler's book instead of our usual pre-calculus book, just kidding. At the beginning the author discussed early methods of finding logs. From what I read: to compute logs, we first used square rooting, then series, and then carried on from there. Some of those methods took very long to explain; it took me like a half hour to understand that dreadful square rooting method. I won't mention it here because I would like you to keep reading my blog and not shut off your computer.
     Euler also worked with exponential functions. He found that if a^z = y then log_a y = z; any pre-calculus student must know this if they want any shot in passing his/her class.
Euler also found that
log_b (y) / log_b (x) = (log_a (y) / log_a (b)) / (log_a (x) / log_a (b)) = log_a (y) / log_a (x)
which is a way to find numbers, other than in base 10. Euler also found a series expansion for a^x, eventually finding the number e. One of my favorite parts was Euler's proof that the Harmonic series diverges.

The beginning of the chapter explains early methods of finding logs, so it was kind of a snore fest. It seems like Euler laid the foundation for the knowledge needed to study calculus, which was very important for future generations of mathematicians. Euler may have not discovered logs, but he definitely popularized them and found many ways to use them.

Chapter 3: Euler and Infinite Series

     Before Euler's time, infinite series were already pretty popular. Jakob Bernoulli loved infinite series, he found the sum of (k^2) / (2^k), which he found to be 6. He also found the sum of (k^3) / (2^k), which came out to be 26. But he had no idea of the Basel problem: the sum of (1) / (k^2). When Euler took a stab at it he found it to be Pi^2 / 6. His proof is not all that rigorous, he makes a lot of assumptions on certain infinite sums and products. Later on, with Issac Newton's help, Euler found the sum of (1) / (k^4), (1) / (k^6), and so on. There were doubters who thought Euler played too fast and loose with the Basel problem proof. So Euler gave alternate solutions, which were a bit more confusing than his original. Though one interesting fact was that Euler could not find the sum of (1) / (k^p) for odd p.

This was one of my favorite chapters. It was classics Euler: taking the natural logs of expressions, turning terms into infinite series and manipulating variables. It was all about infinite series and their sums. Euler was a master at finding these sums. Euler was so quick to recognize the sum for any know series. He was utterly a master at manipulating expressions. It was like watching Calvin Johnson run a route and catch a football.

Chapter 4: Euler and Analytic Number Theory

This chapter had a few things I had no idea about. It was known that odd primes are either in the form 4k+1, or 4k-1. Also, it was known that there are infinitely many primes; Euclid gives a very clear and simple proof of this in this chapter. The infitude of 4k-1 primes proof is similar, but the infitude of $4k+1$ primes proof was a lot more complicated, which was weird. You think it would be a similar proof as 4k-1. Also, I never knew that 4k+1 primes can be decomposed into the sum of unique squares, I think Fermat found this, but Euler proved it. For instance, 137 = 16 + 121 = 4^2 + 11^2. But, 4k-1 primes did not share this property.
Euler proved that the sum of reciprocals of primes is infinite. The proof is very long but very interesting. Andre Weil said the proof may be the birth of analytic number theory.

I like this chapter because because it discussed things I never knew about. Though it gets very confusing when the author discusses some of Euler's proofs. They are long and not intuitive. Though the author discusses some pretty interesting properties Euler found on the harmonic series. One of which is that
1+1/2^2+1/3^2+... = (2*3*5*7*11*...) / (1*2*4*6*10*...)
where the numerator is the product of all primes and the denominator is the product of all primes minus 1. This property blows my mind, but I still don't quite understand why it's true.

Chapter 5: Euler and complex Variables

From Cardano's cubic formula to Bombelli interpreting the results, mathematicians were still confused with where the other roots came from in the solutions from Cardano's formula. Not Euler though. This required the use of imaginary numbers. Even Liebniz was scared of the square root of -1. It got me to think "is there something mathematicians are scared of today?" Not sure. In Euler's book Elements of Algebra, he said there are imaginary numbers because they only exist in our imagination. Sounds like nursery rhyme or something. Euler didn't mind using imaginary numbers. He found the logs, exponent, and sines and cosines of imaginary numbers. He also found that e^x = cos x + i*sin x, one of his most famous formulas. Essentially, Euler popularized imaginary numbers and showed us that there is nothing to be afraid of. One interesting account was where Johann Bernoulli argued with Liebniz over what ln(-x) was. Euler found that 
ln(-x) = ln((-1)x) = ln x + ln(-1), a pretty amazing discovery. Though I was confused on how Euler knew he could do ln(ab) = lna + lnb where a,b may not be positive. Oh well, I'm sure he was correct.
Euler found that ln (a+bi) = ln c + i*(theta + 2*Pi*k)$ where c=a^2+b^2 and sin \theta = b / c. He also found that i^i = e^{-Pi / 2}*e^{+/- 2*Pi*k}$. Two very interesting discoveries.

Euler popularized imaginary numbers, which is so important to mathematics now-a-days. To me, this was his largest contribution. This chapter has many long confusing algebraic proofs, but they are worth understanding. This chapter covered everything I knew about imaginary numbers and much more. This chapter was probably the most informative, learning wise.

Chapter 6: Euler and Algebra

Euler knew how to solve a quartic: you have to depress it into a cubic, then use Cardano's formula. Euler also somehow knew that the solution was of the form sqrt{p}+sqrt{q}+sqrt{r} where p,q,r are complex numbers. How he figured this out, I have no idea, but that's just Euler. A lot of this chapter discusses Euler solving the cubic and quatric, and Attempting to solve the quintic. He couldn't find a formula for it (as we already know, there is none). Also, Euler tried to solve the fundamental theorem of Algebra: that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem brings about the fact that essentially, every polynomial can be factored as a product of linear or non-reducible quadratic factors. Euler failed in solving the fundamental theorem of Algebra, but Carl Fredrick Gauss found it. The author does not explain the full proof because it is very long and requires a lot more knowledge than a few years of math courses.

This chapter, like it's title, contains a lot of tedious algebra. So it probably is not the most exciting chapter. Euler did not prove many large algebraic theorems, be he laid the groundwork for others to do so. Also, we know about Evareste Galios, but you should looks up Niels Abel. At first I thought they were the same guy.

Chapter 7: Euler and Geometry

Euler did not do much in geometry. He did prove that the 3 centers of a triangle (orthocenter, centriod, circumcenter) all lie on a straight line, which is known as the Euler line. A lot of the chapter consists of geometrical proofs that requires tons of algebra. This was probably the most boring chapter, but included for completeness.

Chapter 8: Euler and Combinatorics

Euler dabbled a bit with this subject. Before his time, basic combinatorical theory was known. The book Ars Conjectandi was a text on probability theory published in 1713 and written by Jakob Bernoulli, so a good amount of combinatorics existed before Euler made his mark.

Euler's most notable work in this subject was on partitions. For any whole number, a partition is the number of ways you can add smaller numbers to get that certain number (1+3=2+2=1+1+2= ... = 4). A special case is the number of partitions with different numbers. So for 4, there are 2: 1+3 and 4 itself. Another special case is the number of partitions that contain just odd whole numbers. For 4, there are again, only 2: 1+1+1+1 and 1+3, interesting. As you can tell, I'm getting to a theorem Euler proved: The number of different ways a given number can be expressed as the sum of different whole numbers is the same as the number of ways in which that same number can be expressed as the sum of odd numbers, whether the same or different. 
That was truly a remarkable theorem. I understood the proof but I had no idea how he could have thought of it, it was a work of genius. Hopefully you will read about it.


Overall this book was great. With a few years of college math you can understand basically all the proofs. Also, some proofs proved to be a bit difficult and not so rigorous (classic Euler), so the author gave alternate proofs of certain theorems and he took extra time to explain certain parts of the proofs. This was very helpful. Like I explained, this book is not just on Euler. The author mentions the works of many other great mathematicians, so you get a pretty broad view of some mathematical history in general. I hope you enjoyed my review and even more so, I hope you read the book.

1 comment:

  1. Integration of the book notes was a nice way to give good coverage to Euler's math. And you're completely correct on nothing makes you feel more like a sloth than studying Euler. 5C's +.