The first book I reviewed was Euler: The Master of Us All William Dunham. I liked how the author presented the book so I decided to read (skim) another one of his books, Journey Through genius. The book is a nice read, it gives a broad view of the some of the greatest mathematicians, and their works, who ever lived. The book reads in chronological order, so you get a taste of each generation of mathematicians. This means the book starts out at around 300 B.C where was still young. Therefore, the first five chapters talk a lot about old school geometry, which I'm not really a fan of so I skimmed through those pretty fast. Though, the first chapters talks a little bit about transcendental numbers (numbers that are not solutions to any polynomial equation with integer coefficients); so I read that and it was pretty interesting. Then chapter 6 discusses Cardano and his cubic formula. It's a good chapter, but the first book I read discussed Cardano pretty extensively so I didn't spend to much time on chapter 6.
No book about mathematical, or scientific, geniuses would be complete without Issac Newton (chapter 7). So in that chapter you get to learn more about him and his works. The author gives a good explanation of Newton's binomial theorem, which is so widely used in many areas of mathematics; so it's worth it to read and understand his explanation.
Then the author gives two good chapters on Leonhard Euler himself. I pretty much knew everything he had to say in them; but, if you haven't read about Euler, the two chapters should convince you to learn more about him (and read Euler: The Master of Us All).
Now comes the good stuff, chapter 11: The Non-Denumerability of the Continuum. One of my favorite subjects in math is infinity. This chapter discusses Georg Cantor and his great work on infinite sets. If you don't know much on cardinality, you will love this chapter. The concepts of countability and different levels of infinity is so mind blowing, I just love Cantor's work. Though, many mathematicians of his time thought he was crazy for coming up with this stuff. I just want to give them a good punch in stomach.
Then we have chapter 12, which is essentially a continuation of chapter 11, which means it's awesome. This chapter has Cantor's proof that the cardinality of any set A is always smaller than the cardinality of the power set of A. This theorem is powerful because it works for infinite sets as well. Though, the actual proof in the book was long and a bit difficult to understand. Georg Cantor was like Euler in a way, he helped other mathematicians not be so scared of infinity, as Euler did with imaginary numbers.
Overall, this is a fine read for anyone interested in mathematics, and science overall. William Dunham touches upon pretty much all my favorite (and maybe yours) mathematicians. He gives a number of proofs, so there is plenty to learn in this book; it's not just a leisure read. I would actually recommend giving this book to any freshman math or science major; it would inspire them to keep working hard in their field. Personally, I would read any book by William Dunham, he's a good author.
Monday, June 23, 2014
Sunday, June 22, 2014
Weekly 7: David Hilbert's 9th Problem
For my final weekly post, we are going to have some fun. I will be trying to interpret the meaning of David Hilbert's 9th problem the best I can. But, before I state the actual problem, I would like to build up the terminology and appropriate knowledge required to understand it. When I type in a different color, that just signifies my thoughts and opinions on the statement at hand. So first, observe the following table I took from Wikipedia. Here, f(n) = n^2 - 5 and the column after f(n) gives the prime factorizations of f(n).
Let p, q be distinct odd primes. Then:
(i) If p = 1 (mod 4) (p = 5, 13, 17, 29,...) or q ≡ 1 (mod 4) (q = 5, 13, 17, 29,...),
Find the most general law of the reciprocity theorem in any algebraic number field.
PETE L. CLARK. QUADRATIC RECIPROCITY I, http://math.uga.edu/~pete/4400qrlaw.pdf
http://en.wikipedia.org/wiki/Quadratic_reciprocity#Other_statements
Whats interesting about this table is that for the prime factorizations, other than the numbers 2,5, the prime numbers that appear as factors end in either a 1 or 9. There is a cool way to state this: the primes q for which there exists an n such that n^2 = 5 (mod q) are precisely 2, 5, and those primes q such that q = 1 (mod 5) (so q = 11, 31, 41 ...) or q = 4 (mod 5) (q = 19, 29, 89, ...). The law of quadratic reciprocity gives something similar like the example above of prime divisors of f(n) = n^2 − c for any integer c. So how about we state that law, with my thoughts in blue. Law of quadratic reciprocity:
(i) If p = 1 (mod 4) (p = 5, 13, 17, 29,...) or q ≡ 1 (mod 4) (q = 5, 13, 17, 29,...),
p is a square mod q (so p = n^2 (mod q), which is equivalent to saying n^2 = p (mod q)) if and only if q is a square mod p (so q = n^2 (mod p), or n^2 = q (mod p)).
(ii) If p = q = 3 (mod 4) (so p = q = 3, 7, 11, 19, 23, ...), p is a square mod q if and only if q is not a square mod p.
The statement of the theorem is almost directly from my first reference at the bottom. This theorem seems pretty confusing. So how about we explain it by using an example; naturally, we will use the example I stated at the beginning of the blog. In that example, p = 5, which means p = 1 (mod 4) so we use the (i) statement of the reciprocity law. Also, it was given in the example that n^2 = 5 (mod q) so we know p (which is 5) is a square mod q. Then by the law of quadratic reciprocity part (i), we can see that q is a square mod 5, which means q = n^2 (mod 5). Now what does this mean? It means that q is a prime where q = 1^2 (mod 5), or q = 2^2 (mod 5). This means q = 1 (mod 5) (so q = 11, 31, 41, ...) or q = 4 (mod 5) (so q = 19, 29, 89, ...) and there you go, we used the reciprocity law to get the same results as in our example.
But, your thinking, HOLD ON KYLE, what about q = 0 (mod 5), q = 9 (mod 5), q = 16 (mod 5), q = 25 (mod 5), and so on. Well first, if q = 0 (mod 5) then q = 0,5,10,15, ...; where none of those numbers will ever be prime because they are multiples of 5, so we can throw that case out. But, for q = 9 (mod 5), we have q = 9 + 5c = 1 + 5 + 5c = 1 + 5(1+c) so q = 1 (mod 5), but that would mean q = 9 (mod 5) is redundant. I could go on with other cases, but how about not. Recall in MTH 310 when we had congruence classes. I won't go to far in depth, because every math major has to take the class anyway, but it's known that if a = b (mod 5), then a is congruent to either: 0 (mod 5), 1 (mod 5), 2 (mod 5), 3 (mod 5), or 4 (mod 5), there is no need to go on because each case will just revert back to the 5 cases stated above, ie. redundancy. The 5 cases above are known as the congruence classes of Z (mod 5). But, now you are saying: Kyle, what if we hit a case in which q = n^2 (mod 5) where it reverts back to cases 2 (mod 5), or 3 (mod 5). That won't happen though. If you want to know why, refer to my daily 13. Now back to my explanation of the Law of quadratic reciprocity.
So we went through statement (i), so then what about (ii)? Well if you understood my explanation of (i), then surely you can understand (ii). So there you go, you have a very rough understanding of the law of reciprocity. There are actually many ways to state this law: Euler, Lagrange, and Gauss had some I found them more confusing than the way stated above. Gauss is the one who actually proved the law. Now let's switch gears and talk about something a bit different.
So from Math 310, we all remember rings: sets with 2 binary operations that satisfy basic properties, like the set of rational numbers with addition and multiplication. Then a field was a ring with a few more properties: multiplication was commutative, had an identity, and each element had an inverse. Again, the rational numbers are an example of a field. Now let A be a field and let B be a field that contains A and has the same operations as A. Then B would be a field extension of A. So then with the field Q (set of rationals), the set L = {a + b*sqrt(5), where a,b are rationals} would be a field extension of Q. Also, the term algebraic number field is any finite field extension of Q.
So gathered with all this information, here is David Hilbert's 9th problem:
So this problem is like the law of reciprocity, except that we are working in a more general sense: not just rational numbers, but any field extension of the raitionals. Sadly, this problem has been partially. Luckily the guy supposedly only solved the law of reciprocity for abelian extensions (another confusing concept) of the rationals. So the non-abelian case is up for grabs. Now that you understand the problem you can go solve it, Good Luck!
References
http://en.wikipedia.org/wiki/Hilbert's_problems
http://en.wikipedia.org/wiki/Quadratic_reciprocity
Daily 13: Some Modular Mathematics
This daily assignment will serve as a reference to my weekly 7 assignment. The problem is essentially this:
For some integer q,n, if q = n^2 (mod 5) then either q = 0 (mod 5), q = 1 (mod 5), or
q = 4 (mod 5).
Proof: Let q and n be some integers and assume q = n^2 (mod 5). To prove the statement above, we will use cases. Since n is an integer, we know that either n = 0 (mod 5), n = 1 (mod 5), n = 2 (mod 5), n = 3 (mod 5), or n = 4 (mod 5). Out of all these cases, we will show that the only possibilities are that q = 0 (mod 5), or q = 1 (mod 5), or q = 4 (mod 5).
Case 1: n = 0 (mod 5). This means n^2 = 0^2 (mod 5) = 0 (mod 5); I used the well known theorem: if
a = b (mod n) then a^2 = b^2 (mod n), it's a strait forward proof. Since n^2 = 0 (mod 5), there exists an integer a such that n^2 = 0 + 5a = 5a. Also, since q = n^2 (mod 5), there exists an integer b such that
q = n^2 + 5b and plugging in n^2 = 5a we get q = 5a + 5b = 5(a+b) which means q = 0 (mod 5).
Case 2: n = 1 (mod 5). This means n^2 = 1^2 (mod 5) = 1 (mod 5). Since n^2 = 1 (mod 5), there exists an integer a such that n^2 = 1 + 5a. Also, since q = n^2 (mod 5), there exists an integer b such that
q = n^2 + 5b and plugging in n^2 = 1 + 5a we get q = 1 + 5a + 5b = 1 + 5(a+b) which means
q = 1 (mod 5).
Case 3: n = 2 (mod 5). This means n^2 = 2^2 (mod 5) = 4 (mod 5). Since n^2 = 4 (mod 5), there exists an integer a such that n^2 = 4 + 5a. Also, since q = n^2 (mod 5), there exists an integer b such that
q = n^2 + 5b and plugging in n^2 = 4 + 5a we get q = 4 + 5a + 5b = 4 + 5(a+b) which means
q = 4 (mod 5).
Case 4: n = 3 (mod 5). This means n^2 = 3^2 (mod 5) = 9 (mod 5) = 4 (mod 5). Then this case follows exactly like case 3, so we can conclude q = 4 (mod 5).
Case 5: n = 4 (mod 5). This means n^2 = 4^2 (mod 5) = 16 (mod 5) = 1 (mod 5). Then this case follows exactly like case 2, so we can conclude q = 1 (mod 5).
Therefore, considering all the cases, we can see that the only possibilities are either q = 0 (mod 5), q = 1 (mod 5), or q = 4 (mod 5). So the theorem has been proved.
Note: in the context of the weekly 7 assignment, q is prime. This means q cannot be congruent to 0 (mod 5) because then q would be a multiple of 5, which can't happen.
For some integer q,n, if q = n^2 (mod 5) then either q = 0 (mod 5), q = 1 (mod 5), or
q = 4 (mod 5).
Proof: Let q and n be some integers and assume q = n^2 (mod 5). To prove the statement above, we will use cases. Since n is an integer, we know that either n = 0 (mod 5), n = 1 (mod 5), n = 2 (mod 5), n = 3 (mod 5), or n = 4 (mod 5). Out of all these cases, we will show that the only possibilities are that q = 0 (mod 5), or q = 1 (mod 5), or q = 4 (mod 5).
Case 1: n = 0 (mod 5). This means n^2 = 0^2 (mod 5) = 0 (mod 5); I used the well known theorem: if
a = b (mod n) then a^2 = b^2 (mod n), it's a strait forward proof. Since n^2 = 0 (mod 5), there exists an integer a such that n^2 = 0 + 5a = 5a. Also, since q = n^2 (mod 5), there exists an integer b such that
q = n^2 + 5b and plugging in n^2 = 5a we get q = 5a + 5b = 5(a+b) which means q = 0 (mod 5).
Case 2: n = 1 (mod 5). This means n^2 = 1^2 (mod 5) = 1 (mod 5). Since n^2 = 1 (mod 5), there exists an integer a such that n^2 = 1 + 5a. Also, since q = n^2 (mod 5), there exists an integer b such that
q = n^2 + 5b and plugging in n^2 = 1 + 5a we get q = 1 + 5a + 5b = 1 + 5(a+b) which means
q = 1 (mod 5).
Case 3: n = 2 (mod 5). This means n^2 = 2^2 (mod 5) = 4 (mod 5). Since n^2 = 4 (mod 5), there exists an integer a such that n^2 = 4 + 5a. Also, since q = n^2 (mod 5), there exists an integer b such that
q = n^2 + 5b and plugging in n^2 = 4 + 5a we get q = 4 + 5a + 5b = 4 + 5(a+b) which means
q = 4 (mod 5).
Case 4: n = 3 (mod 5). This means n^2 = 3^2 (mod 5) = 9 (mod 5) = 4 (mod 5). Then this case follows exactly like case 3, so we can conclude q = 4 (mod 5).
Case 5: n = 4 (mod 5). This means n^2 = 4^2 (mod 5) = 16 (mod 5) = 1 (mod 5). Then this case follows exactly like case 2, so we can conclude q = 1 (mod 5).
Therefore, considering all the cases, we can see that the only possibilities are either q = 0 (mod 5), q = 1 (mod 5), or q = 4 (mod 5). So the theorem has been proved.
Note: in the context of the weekly 7 assignment, q is prime. This means q cannot be congruent to 0 (mod 5) because then q would be a multiple of 5, which can't happen.
Sunday, June 15, 2014
Weekly 6: Marie-Sophie Germain
For my weekly 6 assignment, I will give a biography on Marie-Sophie Germain, along with some of my thoughts. Sophie was born on April 1, 1776 in Paris, France. Sophie's parent's house was actually a meeting place for those interested in liberal reforms. That would be interesting to have an upbringing like; it probably influenced Sophie greatly as a kid. In her teen years, Sophie was able to teach herself Latin and Greek. She also read Newton and Euler at night while under blankets as her parents were sleeping. They took away her fire, her light and her clothes in an attempt to get her away from books. Wow, if my parents did that, I probably would never have even pursued math. Anyways, her parents did lessen their opposition to her studying the sciences. What I found interesting was that her father actually supported her financially throughout her life, even though she never really had a well paying job. So maybe we (as in me) shouldn't be so quick to judge her parents.
At the end of some of Lagrange's (we should all know who he is) lecture course on analysis, using the pseudonym M. LeBlanc, Sophie submitted a paper that even made Lagrange look for its author. When Lagrange found out Sophie was a woman, he still respected her work and would eventually become her sponsor and mathematical counselor. Sophie collaborated with many mathematicians, but the most notable is Karl Friedrich Gauss. Between 1804 and 1809, she wrote several letters to him, again taking M. LeBlanc as her name. Gauss gave her tons of praise for her number theory, which is amazing because he was a highly intelligent schmuck. When Gauss did find out about her true identity he gave her even more praise, for learning science even with society's harsh gender roles at that time. One of Germain's most famous papers was her work on Fermat's last theorem in where she broke new ground and used divisibility as an attempt to prove Fermat's Last theorem.
Then came the Institut de France prize competition which brought about the following challenge:
formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence.
Most mathematicians didn't even try to solve the problem. But, Germain spent the next decade attempting to derive a theory of elasticity, collaborating with some of the most famous mathematicians and physicists of her time. Sadly, she did not win this time. Her hypothesis was not formed from the principles of physics, nor did she have any training in analysis or calculus (which was important in solving the problem). Finally, Germain's third attempt was deemed worthy of the prize: one kilogram of gold. Though, to public disappointment, she did not receive the prize. She thought the Judges did not fully appreciate her work, which probably was true. I bet if a man submitted her work, he would have one the prize, so unfair. In an attempt to extend her research, Sophie submitted a paper in 1825 to a commission of the Institut de France, whose members included Poisson, Gaspard de Prony, and Laplace. Her work suffered from a number of deficiencies (which probably could have been avoided if she had the proper training, which was inaccessible to her), but rather than reporting them to the author, the commission ignored the paper. It was recovered from de Prony's papers and published in 1880. Wow that just makes me mad that they just ignored her. It sounds like her research was pretty significant since de Pony's kept it. Sadly, Germain got breast cancer in 1829, but still she completed papers on number theory and on the curvature of surfaces (1831). Even on her death certificate in 1831, she was not even listed as a mathematician or scientist, bullcrap.
Sophie Germain was a very strong person. She never caught a break and her work was insulted left and right. I think that for any class, or book, that discusses woman's rights and the history overall of woman, Germain's name should be mentioned. From woman who campaigned for their rights, who voiced their opinion even in the face of adversity, Germain is different. Back then, society thought of woman as lesser than men, but Germain proved she was not lesser. She PROVED she was intelligent, brave, and had so much to offer. She was a shining example of the hardships woman had to go through, but prevailing through them. I know way more about her than that de Prony guy; I've never even heard of him (hahahaha). If she was actually given some proper training, and had a bit more support, who knows what more she could have done. As a mathematician, scientist, woman, or anyone really, Sophie Germain is truly someone to admire.
Marie-Sophie Germain, (JOC/EFR, October 1998 School of Mathematics and Statistics, University of St. Andrews, Scotland), http://www-groups.dcs.st-and.ac.uk/history/Biographies/Germain.html
References
Marie-Sophie Germain, (JOC/EFR, October 1998 School of Mathematics and Statistics, University of St. Andrews, Scotland), http://www-groups.dcs.st-and.ac.uk/history/Biographies/Germain.html
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:
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
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
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.
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.
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}$. Two very interesting discoveries.
Euler found that ln (a+bi) = ln c + i*(theta + 2*Pi*k)$ 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.
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.
Conclusion
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.
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.
Conclusion
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.
Friday, May 30, 2014
Weekly 4: History of Fibonacci
For this post, let's explore the history of Fibonacci. Leonardo Pisano (Fibonacci) was born on 1170 to Guglielmo Bonacci, a merchant of some kind. Fibonacci was educated in Bejaia, Algeria in North Africa (where his father worked). Fibonacci actually was enrolled in Bejaia's school of accounting. From all of Fibonacci and his fathers travels, Fibonacci learned the Hindu-Arabic number system, the system we use today. He was one of the first people to introduce the number system to Europe. His book on how to do operations in the new number system was called the Liber abbaci. This book convinced many mathematicians, and others, in Europe to start using the new number system. Actually, much of the book had to do with accounting mathematics: price of goods, how to calculate profit, currency conversion, etc... This number system seems very natural to us. In class it was very difficult to perform operations using roman numerals. I tried to do it and I pretty much was fed up after the first problem. I couldn't imagine doing every day operations in those cruel roman numerals. It's amazing that Fibonacci could even understand more than one number system. But, for me the truly astonishing thing Fibonacci did was to actually recognize that the Hindu-Arabic number system was easier to use. As we have seen, any new number system is hard to get used to, but to recognize one is easier than another seems so difficult. Who knows, maybe our grand children will be learning a different number system than the one we learned (hope not).
Leonardo Pisano Fibonacci (JOC/EFR, October 1998 School of Mathematics and Statistics, University of St. Andrews, Scotland) http://www-groups.dcs.st-and.ac.uk/history/Biographies/Fibonacci.html
Fibonacci presented many problems in his Liber abbaci, including the one about bunnies we studied in class. Out of this problem came the Fibonacci sequence:
As we have seen in our math 495 class, the Fibonacci sequence has a lot to do with the golden ratio. For instance, if you take any two successive numbers in the Fibonacci sequence, the ratio gets closer and closer to the golden ratio. The golden ration also has ties to finance, biology, architecture, and many other areas. There are so many interesting things in nature that deal with the golden ratio, but this post is about Fibonacci. If you wan't to learn more about nature and the golden ration, you should read this article
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc...
As we have seen in our math 495 class, the Fibonacci sequence has a lot to do with the golden ratio. For instance, if you take any two successive numbers in the Fibonacci sequence, the ratio gets closer and closer to the golden ratio. The golden ration also has ties to finance, biology, architecture, and many other areas. There are so many interesting things in nature that deal with the golden ratio, but this post is about Fibonacci. If you wan't to learn more about nature and the golden ration, you should read this article
,
Fibonacci's book also discussed perfect numbers, rational approximations of square roots of numbers, and sums of arithmetic and geometric series. This may be his most famous book, but another one of his works was Liber quadratorum. Here he discussed number theory including Pythagorean triples, square numbers, and interesting number theory results such as
there is no x, y such that x^2 + y^2 and x^2 - y^2 are both squares.
Since Fibonacci lived before printing, it was difficult to keep some of his original works. Some of his papers were actually lost, including his book on commercial arithmetic Di minor guisa.
Frederick II (the holy roman emperor at the time) actually was aware of Fibonacci because the Emperor's scholars corresponded with him after his return to Pisa in 1200 A.D. One of those scholars, Johannes of Palermo, gave some problems as challenges to Fibonacci. Fibonacci solved three of these problems and gave the solutions in his book Flos which he sent to Frederick II. I never knew mathematicians were so competitive back in the day. I imagine there was trash talking going on all over the intellectual community back then. Actually, when I think about it, we do have math competitions today in colleges and high schools; though I have yet to hear trash talk going on in my math classes. We'll save the trash talk for Richard Sherman.
Fibonacci was a very important figure in mathematics and history in general. The Hindu-Arabic number system he brought to Europe was so important in the study of mathematics, science, business, and pretty much any discipline that used numbers (which was like all of them). The Fibonacci sequence is probably the most popular of all sequences because of how much of it is related to science and nature. I hope you enjoyed this non-rigorous post on Fibonacci. I usually like to post proofs and other interesting math theorems, but I'm so thankful for Fibonacci that I thought a brief biography would be nice.
References
Nature, The Golden Ration, and Fibonacci too (Copyright © 2011 MathsIsFun.com)
Saturday, May 24, 2014
Weekly 3: Al Kharaji's formulas
One of my favorite things about mathematics are series. That's why when we started talking about Al Kharaji and his series formulas I couldn't help but explore and prove them. The first one we talked about was an interesting closed form for the sum of squares: (note: I'm using sum( ) instead of the summation symbol)
sum(k^2) = sum(k) + sum(k(k-1)).
To see why this is true, observe that k^2 can be manipulated in the following way:
k^2 = kk = k(k+1-1) = k(1+(k-1)) = k+k(k-1),
which means
sum(k^2) = sum( k+k(k-1)).
Now the question is, can you distribute the sum ( )? Yes of course; observe that
sum(k+k(k-1)) = (1 + 1(1-1)) + (2 + 2(2-1)) + (3 + 3(3-1)) + ...
and from each expression in the parenthesis, we can group all the first terms together and then all the second terms together to get
sum(k+k(k-1)) = (1 + 2 + 3 + ...) + (1(1-1) + 2(2-1) + 3(3-1) + ...)
where we can see that
sum(k+k(k-1)) = sum(k) + sum(k(k-1)).
Then since sum(k^2) = sum(k+k(k-1)), we have
sum (k^2) = sum(k) + sum(k(k-1)).
Note that I merely explained why this is true, I didn't really prove it for the most part. Now another one of Al Kharaji's claims was
sum(k^3) = (sum(k))^2.
This one I will prove using mathematical induction (one of my favorite proof methods). To make the proof easier to read, I typed it up and put it on a pdf file. You can't really put pdf files on blogger.com so I inserted them as pictures. To see them, just hold the "control" key and push the "+" key a few times to zoom in. Then to zoom out, hold the "control" key and push the "-" key, sorry about that.
I probably missed on some minor subtitles, but the main part of the proof should be fine. Now proving this wasn't so bad, it was fun. Make sure to observe the "note" at the end of the proof. I had to actually use the sum of integers formula (lemma 1) to prove sum(k^3) = (sum(k))^2 (lemma 1 actually serves as my daily 5 assignment). I imagine that to prove the sum of squares or sum of cubes formulas, you could use mathematical induction and the proofs would be straightforward; but, let's save those for a different post. I hope we explore more series in the future. If we do, expect more proofs in my blogs. :)
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