Fermat’s Enigma, Chapters Two and Three
After reading chapters 2 and 3, list and describe three ideas from each chapter that interested you.
Then, read other people's entries and comment on four other people's entries.
(Your response can be in the form of a comment or a question which can lead others into a discussion).
Finally, go back to your entries, and respond to at least person's comment to you.
(Your response can be in the form of a comment or a question which can lead others into a discussion).
Finally, go back to your entries, and respond to at least person's comment to you.
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ReplyDeleteChapter 2 ideas:
ReplyDelete1) Pi can never be written down exactly because the decimal goes on forever without a pattern.
2) The use of zero (Ex: 52=5(10)+2(1))
3) Friendly and amicable numbers
Chapter 3 ideas:
1) Method of infinite descent
2) Bridges of Konigsberg
3) imaginary numbers/axis.
Ch 2:
Delete1) Pi is an irrational number that cannot be solved exactly because it never repeats. Why is that?
2) How was the idea of the use of zeros developed. (Ex: 52=5(10)+2(1))
3) What is the reason for friendly and amicable numbers?
A man named Yasumasa Kanada from University of Tokyo calculatied Pi to six billion decimal places.
DeleteChapter 2:
ReplyDelete- For what reasons do these groups dedicated to mathematics in the past, such as the Pythagorean Brotherhood and Parisian mathematicians, value secrecy in their work?
- Was the game of points (seventeenth century) where the idea of gambling and probability began?
- Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number.
Chapter 3:
- Algorithm is used to predict the phases of the moon.
- (a+b^n)/n = x was said to be proof for the existence of God. We know that that is false, but is there any other use for that formula?
- Hindus discovered negative numbers. Greeks discovered irrational numbers.
What made irrational numbers so controversial?
DeleteThe Pythagorean Brotherhood sounds like a cult.
DeleteFRIENDLY NUMBERS LIKE 220 AND 284
DeleteI looked this up and I read that irrational numbers were controversial because Pythagoras said they MUST exist. The idea was stolen by Pythagoras from an ancient civilization. Pythagoras did not give a real proof to why these numbers HAD TO exist.
DeleteIts crazy how they used algorithms to predict the phases of the moon
DeleteChapter 2
ReplyDelete1) During the seventeenth century, math was not viewed as an important subject.
2) Fermat was an academic amateur, math was his hobby.
3) Friendly numbers are somewhat similar to perfect numbers. Friendly numbers have a relationship among the sum of its divisors.
For example:
220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 45, 110; sum is 284.
284: 1, 2, 4, 71, 142; sum is 220.
Chapter 3
1) Fermat's Last Theorem was unresolved for over 300 years.
2) Mathameticians can believe a statement is true, when in reality it could have always been false.
3) Euler developed an algorithmic method to calculate an accurate position of the moon. He later gave his method to the British Admirality and received a prize of 300 euros.
^
DeleteHAHAHAH AMATEUR...FERMANT WAS A LOSER, jk jk.
DeleteIt's honestly amazing how much Fermant contributed to mathematics even though it was just his hobby.
DeleteThis comment has been removed by the author.
Delete"math was his hobby" reminds me of someone who studied the first 100 numbers of Pi.
Delete*cough* Andrew *cough*
Math not important?!?!? Take me there
DeleteChapter 2:
ReplyDelete- Likelihood of sharing the same birthday is over 50% in the same field, COOL. Could this be like how my little brother and I share the same birthday but we are not twins? Freakkkyyyy
- The Library in Alexandria sounds SO COOL. Too bad Julius Ceasar had to ruin it for everyone and burn it down. Thousands of books to waste. The horror!
- Ha. E.T. Bell wrote that "civilization would probably come to an end before Fermant's Last Theorem could be solved." SIKE
Chapter 3
- Catherine the Great. A woman in charge! Get it girl.
- Euler still continued studying math like crazy with a vision handicap. What a guy.
- Hilbert's Hotel is so interesting. An infinite size hotel with all of the rooms occupied? Infinity + 1 = infinity yo.
- No!!! Hypatia's man went against her and started oppressing the intellectuals! My guess was he felt inferior to her and couldn't stand that. What a horrid death. RIP.
"civilization would probably come to an end before Fermat's Last Theroem could be solved." LOLOL not today
DeleteI never knew "the likelihood of sharing the same birthday [was] over 50%" and I didn't know you and your brother shared the same birthday! That's cool!
DeleteI thought that birthday thing was so cool! But I've only met one other person with the same birthday as mine. I realize there are so many people in the world but I wonder, are some birthdays more common than others?
DeleteChapter 2
ReplyDelete-One of the most probability problems that are contrary to intuition is the likelihood of sharing birthdays.
-Fermat and Pascal were the ones who discovered essential rules that govern all games such as gambling and betting.
-Fermat changed the power in Pythagora's equation (x^2+y^2=z^2) from a 2 to a 3. However, his changes had no whole number solutions.
Chapter 3
-Andrew Wiles dared to prove Fermat's Last Theorem since he was a child!
-Euler's greatest achievements was developing the algorithmic method.
-Back then, women were discouraged from studying mathematics. However, one woman, Theano went against the discrimination and became one of Pythatgora's disciples.
Math even governs everything, whether we know it or not.
DeleteTheano must have been a pretty dominant woman for studying mathematics during a time of constant discrimination.
DeleteGo Theano!!!!
DeleteIt's crazy that something like changing the power of the Pythagorean theorem from a 2 to a 3 created a seemingly unprovable theorem
DeletePascal was a good gambler? Teach me your ways!
Deletethats so amazing how they used mathematics for the basis of gambling. lol.
DeleteChapter 2
ReplyDelete-There was a time when mathmaticians wouldn't exchange information but a monk, Father Mersenne, would share all of his ideas even papers sent to him by others in confidence.
-Fermant greatly impacted three areas of mathematics: percentage of probability theory, calculus, and the theory of numbers. Dang what am I doing with my life?
-Okay this is my favorite thing I have read so far: people in the middle ages believed that the friendly numbers, 220 and 284, would promote love. An Arab mathematician ate a fruit with 220 carved on it and gave a fruit with 284 carved on it to a lover as a form of "mathematical aphrodisiac." How romantic LOL.
Chapter 3
-The great mathmetician Euler became blind and continued doing math for the next seventeen years and he was more productive than ever. Wow I have trouble doing math with 20/20 vision, that's skill.
-This chapter also talks about the topic of infinity. It describes that if you start with infinity numbers and remove a large bulk of it you would still get an infinite amount of numbers but the first is a larger infinity than the second.
-I love that this chapter talked about the awesome women mathmeticians such as Hypatia. Even males would come to her for advice on math problems.
GIRL POWER. Loved that about this chapter too. GO HYPATIA!
DeleteMathematicians were/are so secretive about their work
DeleteDo you think it's wise to keep theorems and proof to yourself, or would you share them?
DeleteI love romance, and when you mentioned how he gave the "284" carved apple to his lover, it made me want to go back and find that part. Probably my ONLY favorite part of the book.
DeleteChapter 2
ReplyDelete-Irrational numbers were seen as horrific because they were neither whole numbers nor a a fraction. The most famous irrational number is Pi.
-The Arithmetica described the theory of numbers using a series of problems and solutions. It inspired Fermat to study mathematics.
-Fermat claimed that he had a proof for every single one of his observations.These proofs were proved one by one except for Fermat's Last Theorem.
Chapter 3
-"Often the object of the proof is clear, but the route is shrouded in fog, and the mathematician stumbles through a calculation." It is feared that each step taken may be in the wrong direction or that no route exists.
-Bombelli created an imaginary number to answer the question, "What is the square root of negative one?" Imaginary numbers are restricted to their respective number lines.
-Female mathematicians were highly discriminated over the centuries and were discouraged from studying mathematics. Although some made an impact by forging their names in the annals of mathematics.
Not a huge fan of irrational numbers.
DeleteI don't favor irrational numbers either
DeleteIt's crazy to think that Fermat's Last theorem could have possibly had no explanation at all
DeleteI think the idea that a theorem you spend much of your life on can possibly be impossible to solve, and have no proof, is scary, because it seems like wasted time.
DeleteIts great that intellectual women managed to break through the barriers of society to be able to forge their names into the history of mathematics.
DeleteI find it baffling how he just decided that the answer to his problem was imaginary. How does someone come up with that and everyone just agrees it is correct?
DeleteI still don't understand how imaginary numbers were widely accepted by society and how they abide by their own rules. Ex. they have their own number line
Deletewhat was the major causes for women to be discriminated within the mathematics field?? :o
DeleteFor Chapter 2:
ReplyDelete-When Pythagoras was presented with proof for the existence of irrational numbers by Hippasus he did not act accordingly. He could not destroy Hippasus’ argument with logic so instead he “resorted to force rather than admit he was wrong." Hippasus was sentenced to death by drowning. This became known as his most shameful act.
-The first two friendly numbers found were 220 and 284. The sum of the divisors for each equal the other number. These numbers were believed to be a symbol of friendship and love. They were often worn on charms. One person usually held an item with one number and the other person held the other. This is similar to how people wear friendship bracelets or how couples get each other items with their initials in it. It is even mentioned in Genesis where Jacob gave 220 goats to Esau. Fermat later discovered another pair, followed by Descartes, Leonhard Euler, and Nicolo Paganini.
-Arthur Porges’ Deals with the Devil explores the hardship of "solving" Fermat’s Last Theorem in its short story “The Devil and Simon Flagg.” In this short story the devil implores Simon to ask him a question and if he gets it right within a day then he can take his soul. If not, then he must give Simon $100,000. Simon asks the devil if Fermat’s Last Theorem is true. Its funny that later (after much studying and learning as much math as he can) the devil comes back and admits defeat. “…Bah! Do you know,” the Devil confided, “not even the best mathematicians on other planets-all far ahead of yours- have solved it? Why, there’s a chap on Saturn- he looks something like a mushroom on stilts- who solves partial differential equations mentally; and even he’s given up.”
For Chapter 3:
-They’re usually prestigious families known for producing well known scientists, doctors, or artistically inclined offspring. It was surprising however to find out that there was such a “royal” family per say during the eighteenth-century that was known to be the “most mathematical of families.” The Bernoullis created “eight of Europe’s most outstanding minds within only three generations.” It is great to hear that they encouraged the importance of mathematics.
-“It is unthinkable for mathematicians not, in theory at least, to be able to answer every single question, and this necessity is called completeness.” This necessity led to the discovery of negative, irrational, and imaginary numbers. I think this necessity of wanting to explain/understand something questionable is something we all can relate to at one point in our lives.
-Pythagoras is known as the “feminist philosopher” because he actively encouraged women scholars. It wasn’t until the fourth century AD that a women mathematician founded her own influential school.
If I would have lived during Pythagoras's time, I would have greatly appreciated his sense of equality when it came to "actively encourag[ing] women scholars."
DeleteI like how Pythagoras encouraged women scholars and not just men
DeleteThe idea of completeness is really interesting, and I believe it is the reason we strive to learn and grow in knowledge. I agree that everyone has probably gone through a point in their life and it gives us the math we have and the technology we have.
DeleteI wonder why 6 was not one of the first amicable numbers discovered. You would think that they would have started at 1 and worked their way up.
DeleteChapter 2:
ReplyDeleteWhat is the purpose of finding a "sociable" number?
Was Claude Gaspar Bachet de Meziriac's compilation of puzzles, titled Problems plaisans et delectables qui se font par Les nombres, the foundation to math magic tricks?
Thanks to Fermat's eldest son, Clement-Samuel, who appreciated the significance of his father's hobby, was determined that his discoveries should not be lost to the world.
Chapter 3:
Leonhard Euler is the founding father of the algorithmic method. The point of his algorithms was to tackle "impossible" problems
First woman known to have made an impact on the subject is Theano 16th century bc (1 out of 28 women in Pythagorean Brotherhood)
Andrew Wiles teen years he studied the work of Euler, Germain, Cauchy, Lame, and Krummer. He hoped he can learn from their mistakes.
Andrew Wiles ended up being stumped the same way Kummer was even by the time he was an undergraduate.
DeleteWould mathematics have been any different without Fermat's last theorem?
DeleteImagine the horror of the loss of knowledge that would have occurred if Clement-Samuel had not appreciated the significance of his father's hobby.
DeleteI wonder that too. What's the point of finding sociable numbers, or amicable numbers too. I mean they're interesting, but does it hold a purpose in the real world, or does it just stop just being awesome?
DeleteI don't think much of the mathematics that we have learned would be affected if they had not found a proof for Fermat's Last Theorem. Though, I think it is great and a major accomplishment that we do now have a proof!
DeleteCh. 2:
ReplyDelete- the counterintuitive probability of sharing birthdays
Fermat and Pascal discovered the first proofs in the probability theory. The likelihood of sharing a birthday say with someone on the same team as you consisting of two teams and a referee. The answer is just over 50%. The high probability due to the fact that in configuring this you would look at the pairs you can make between people.
-the most famous rational number
Pi (3.14) is an never ending number, it has been calculated to six billion decimal places by Yasumasa Kanada and even then it is not the exact number of pi. Pi can be calculated by an equation -> Pi=4(1/1-1/3+1/5-1/7+1/9-1/11+...)
-friendly numbers or amicable numbers
Friendly number are pairs of numbers where each number is the sum of the divisors of the other number. The numbers 220 and 284 are friendly numbers because 220 is divisible by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 which equal to 284. 284 is divisible by 1, 2, 4, 71, 142 which equals 220.
Ch. 3:
-the problem concerned with the Prussian city of Konigsberg
A city is divided by four separate sections and is linked by 7 bridges. Given the layout, the task is to cross each bridge only once but in attempting to one would find it impossible. Euler concludes that it would only be possible to make a trip crossing each bridge once if there was an even number of bridges or that two landmasses both have an odd number of bridges.
-imaginary numbers
What is the square root of -1? This can't be solved by 1 or -1 because their squared value would both equal 1. For this, Bombelli created i also known as an imaginary number.
-Hilbert's Hotel
This hypothetical hotel was meant to help explain the mystery of infinity. It suggests that all infinities are as large as each other and some are bigger than others.
Hilbert's Hotel is such a great way of explaining the varying sizes of infinities. Its such an abstract concept too.
DeleteHow could Bombelli just create an imaginary number?
DeleteI think its really cool and interesting that we can figure out the numbers in pi by solving for 1/1, 1/3, 1/5, 1/7, etc. Maybe that's how Andrew memorized it! Oooo!
DeleteI find it rather amazing as to how pi has no ending, even as Yasumasa Kanada reahed about 6 billion decimal points, it still continues further.
DeleteChapter 2:
ReplyDelete-The fact that math can sometimes be counterintuitive. This is interesting because it goes beyond our seemingly common logic and we are proven wrong through numbers. For example, the probability that 2/23 people would share the same birthday is over 50% which goes against what we would commonly assume.
-The discovery of irrational numbers. Mathematics was always dealt with in terms of whole numbers and fractions, so once mathematician Euclid saw that that wasn't working in certain situations he proved the existence of the irrational number.
-The theorem of primes. I never realized the impact or importance of prime numbers but it is interesting to see that mathematicians have been able to theorize that all primes fit into two equations. (4n+1 and 4n-1)
Chapter 3:
-Hilbert's Hotel. This hypothetical scenario represents the idea that some infinities are bigger than other infinities. This was interesting because it is hard to conceive the concept of infinity.
-Prime lifecycles. It is fascinating that mathematicians have pointed out a relationship between primes and a cicadas lifecycle (17 years). They reason that this was an adaptation in order to avoid a parasite. It is cool to see the connection between biology and mathematics.
-Women in mathematics. Although these women studied during a time of constant discrimination and prejudices they were still able to make a significant impact in mathematics.
It is fascinating that prime numbers have been acknowledged to have taken importance in the life cycles of beings such as the cicada.
Deletethe concept of smaller and larger infinities is very interesting, and David Hilbert creates Hilbert's Hotel, this hypothetical hotel helps effectively explain this concept
DeleteThankfully Pythagoras encouraged woman to continue studying mathematics
DeleteChapter 2:
ReplyDeleteWhy was Euclid's book "Elements" known as the most successful book?
How did the concepts of rational and irrational numbers come about?
What was the importance of Diophantus's Opus?
Chapter 3:
Where did Eulerget his inspiration for the development of the Algorithmic Method?
Where did the concept of imaginary numbers come from?
Why were women mathmaticians so rare in the past?
Chapter 2
ReplyDelete- Fermat preferred keeping his theorems to himself, and teasing other mathematicians to solve them. This bothered many mathematicians, but meant he could give a proof for his theorem and move on quickly to creating and proving another theorem.
-A man named Gombaud was gambling and couldn't finish, he asked Pascal to solve a probability problem (pascal wrote Fermat). Together this led to deeper exploration of more subtle and sophisticated questions related to probability.
- Fermat was deeply involved in founding calculus (ability to calculate the rate of change, or derivative, of one quantity with respect to another. Though for a while Isaac Newton thought he single handedly discovered calculus.
Chapter 3
- "A mathematician may believe that a statement is true, and spend years trying to prove that it is indeed true, when all along it is actually false. The mathematician has effectively been attempting to prove the impossible." I think this is a crazy chance you are taking to dedicate your life to something that is impossible, but sometimes you do have a proof and that is evident with Andrew Wiles.
-Euler earned a reputation for being able to solve any problem that was posed. He provided an accurate enough algorithm to provide a position for the navy and gave it to British Admiralty. Euler succeeded in making little progress on proving Fermat's Theorem, using imaginary numbers.
-the idea of completeness is the ability to answer every question. This idea led to the Hindus discovery of negative numbers, the Greeks discovery of irrational numbers, imaginary numbers, and complex numbers.
But wouldn't Fermat solve theorems faster with the help of other mathematicians?
DeleteThank you, Fermat (and Newton), for giving us the gift of cal-CU-lUUUUs!
DeleteChapter Two:
ReplyDelete-Fermat would state his theorem witomething I would do). He did this as to not waste time fully developing his methods (a mathematician avoiding math... what) and to avoid relentless nit-picking of other mathematicians (I get this)
-Euclid exploited reductio ad absurdum (proof by contradiction). The approach uses the idea of proving a theorem is true by first assuming it is false. This kind of reminds me of “innocent until proven guilty” but like “false until proven true”. This is how Euclid established the existence of irrational numbers-a number that is neither whole or a fraction, π being the most famous irrational number
-Fermat is really smug - “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain”. He thought he'd be able to prove it so he just claimed that he did in advance…... Fermat's Last Theorem was famous because of the difficulty of proving it
Chapter Three:
-Isaac Newton as well as the European powers believed in applying mathematics to the physical world to solve practical problems. Euler created an algorithm to determine future phases of the moon for navigation purposes (the only time I like math is when it's applied to the real world; I love word problems)
-There was a time when some numbers were not known; negative numbers, fractions, and irrational numbers were all discovered. This is true but trippy to think about.
-Pythagoras actively encouraged women scholars (28 sisters in the Pythagorean Brotherhood); known as the “feminist philosopher”. Some female mathematicians Theano, Hypatia (known for being a great problem solver), Maria Agnesi (treatises on the tangent to curves), Sophie Germain/Monsieur Antoine-August Le Blanc.
I think proof by contradiction is interesting. They are proving something is right by demonstrating it can't be wrong, eliminating any doubt in the proof. This makes mathematical proofs true without a doubt even after hundreds of years. I think there are few things that can withstand time like that which is pretty cool.
DeleteWhat would math be without negative numbers, fractions and irrational numbers? Probably a lot easier
DeleteI wonder how negative numbers correlate with imaginary numbers
DeleteWhat do you mean by European powers??
ReplyDeleteWho are you? Are you interested in the book as well?
DeleteChapter 2:
ReplyDelete1. The most famous irrational number is 3.14... Pi
2. The two men that discovered the essential rules for gambling
and betting
3. Pi can be calculated into 6 billion decimal number.
Chapter 3:
1. 220 and 284 are a symbol of love and friendship. The sum of their divisors equal the other number.
2. Women had it hard when it came to learning mathematics. They had to study mathematics in secret just to learn for their own sake of learning.
3. Since the square root of 1 and -1 both equal to 1. Bombelli then decided to create "imaginary numbers."
GREAT IDEAS SUJATA
DeleteBEST ONES YET!
^ I agree
DeleteCameron Basa
ReplyDeleteChapter 2:
-Fermat would state his theorem with something I would do. He did this as to not waste time fully developing his methods and to avoid relentless nit-picking of other mathematicians
-A man named Gombaud was gambling and couldn't finish, he asked Pascal to solve a probability problem. Together this led to deeper exploration of more subtle and sophisticated questions related to probability.
-The discovery of irrational numbers. Mathematics was always dealt with in terms of whole numbers and fractions, so once mathematician Euclid saw that that wasn't working in certain situations he proved the existence of the irrational number.
Chapter 3:
-Euler earned a reputation for being able to solve any problem that was posed. He provided an accurate enough algorithm to provide a position for the navy and gave it to British Admiralty. Euler succeeded in making little progress on proving Fermat's Theorem, using imaginary numbers.
-Andrew Wiles teen years he studied the work of Euler, Germain, Cauchy, Lame, and Krummer. He hoped he can learn from their mistakes.
-female mathematicians were highly discriminated over the centuries and were discouraged from studying mathematics
Cameron Basa
ReplyDeleteChapter 2:
-Fermat would state his theorem with something I would do. He did this as to not waste time fully developing his methods and to avoid relentless nit-picking of other mathematicians
-A man named Gombaud was gambling and couldn't finish, he asked Pascal to solve a probability problem. Together this led to deeper exploration of more subtle and sophisticated questions related to probability.
-The discovery of irrational numbers. Mathematics was always dealt with in terms of whole numbers and fractions, so once mathematician Euclid saw that that wasn't working in certain situations he proved the existence of the irrational number.
Chapter 3:
-Euler earned a reputation for being able to solve any problem that was posed. He provided an accurate enough algorithm to provide a position for the navy and gave it to British Admiralty. Euler succeeded in making little progress on proving Fermat's Theorem, using imaginary numbers.
-Andrew Wiles teen years he studied the work of Euler, Germain, Cauchy, Lame, and Krummer. He hoped he can learn from their mistakes.
-female mathematicians were highly discriminated over the centuries and were discouraged from studying mathematics