History of Math – Sophie Germain

Sophie Germain was an 18th – 19th century mathematician, and she was one of the first female mathematicians.  It was difficult for her but she eventually became a respected mathematician despite the sexism and classism she faced.  Her formative years were spent during the French Revolution were she used mathematics to escape from the fighting outside.  She was raised in an upper middle class family who were not supportive of her pursuit of mathematics.  Ms. Germain taught herself mathematics using the books from her father’s library.  Her parents disapproved of her studies and confiscated her candles and clothes to dissuade her from studying at night.  She hid candles and studied at night in secret.

When she turned 18 she ended up borrowing lecture notes from friends who attended college for mathematics in order to further her mathematical studies.  After she collected more mathematical knowledge from these notes and other sources she started creating her own proofs.  Once she had gained a bit of confidence in her proofs she started writing to J.L Lagrange under the name M. Leblanc in order to conceal her true gender.  Lagrange was very impressed with her work and was shocked when he found out that she was female.  He then started teaching her math once he saw her potential.

Later on in her life she was tutored by Gauss, she again used the name M. Leblanc in order to keep her gender a secret.  Gauss helped guide her research for three years before Gauss discovered her true gender, and when he did he was thrilled and quite impressed that a woman could be so skilled and interested in mathematics.  She also communicated with Legendre about her work on Fermat’s Last Theorem and she made a lot of headway into solving the problem.

Ms. Germain later ended up winning a prize for her paper “Memoir on the Vibrations of Elastic Plates” though she won it anonymously since she knew that they would judge her based on her gender instead of her math.  She did not win the prize until her third try since the judges could tell by her language that she was not well educated, so she kept fine tuning her proof until she eventually won the prize even though her proof still had a few holes in it.  It took many years for these holes to be corrected.

In her life she became a gifted mathematician despite the social stigmas that tried to stop her from getting very far in a scholarly field.

Work Cited

http://www.agnesscott.edu/lriddle/women/germain.htm

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Doing Math – Durer’s Polygons

Albrecht Durer was a 15th century artist and mathematician.  He started out his life as a gold smith who thought that adding mathematics to art would greatly improve it.  In the past goldsmith’s only had compass’s and straight edges in order to create complex polygons.  So Durer created his Four Books on Measurement which detailed how to make many different polygons with different numbers of sides.  Not all of his polygons are regular but he did at least come up with ways to make it close.  We will use his methods in order to work through constructing these polygons, though geogebra was used to create them the methods would work with pencil and paper.

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A regular quadrilateral is created by first using the compass and drawing a circle.  A horizontal line is then drawn through the center of the circle, the center is labeled A while the edges are labeled B and C.  We then draw a vertical line through A and perpendicular to line BC, we label these points D and E. We then connect D to B, B to E, E to C, and C to D.  We have now constructed a regular quadrilateral according to Durer’s method.

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We start the hexagon with a circle and a line drawn through the center.  We then create two more circles, with the same radius where the horizontal line intersects.  We place points where these two circles intersect the first one and then connect all of the points.

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To create a regular triangle (equilateral) we connect points CE, and F of the regular hexagon.

 

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In order to create and octagon we start with our regular quadrilateral and then create lines bisecting all of the sides of the quadrilateral.  Where these lines intersect the circle we place points and then we connect these points with those of the square and we create a regular octagon.

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To create a pentagon we first create horizontal and vertical lines through the center A.  We will then create a point E that is in the middle of A and B. We then form a circle with center E and a radius of ED. Where this circle intersects the horizontal will be point F.  We will take the length DF to be the length of the sides and then use this length to construct sides DG and DH We will use this same length for GI and HJ. We then connect points and Jthus completing our pentagon.

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Communicating Math – The Origins of Calculus

Calculus in the modern day has many uses and once a person has learned calculus it is hard to fathom how anything was built successfully without it.  Calculus in its simplest terms is the how quickly numbers change.  The two mathematicians credited with the creation of Calculus are Isaac Newton and Gottfried Leibniz, though it is argued that if Archimedes had lived a few more years that he would have created calculus a few centuries before either of them were even born.

Archimedes started studying how volume changes and how to calculate volume in a variety of different shapes.  Though without the use of calculus his proofs about volume are long and cumbersome.  But he did start to discover limits though he called it exhaustion and used it to approximate the area of circles by calculating the area of polygons, with roughly the same diameter, each with more edges than the previous iteration.

In the 17th century the idea of a derivative, the rate of change of a function, came into being. Newton and Leibniz are the main two mathematicians credited with creating calculus. While Newton was the first to come up with calculus, he used it to solve problems in physics, Leibniz was the first to publicize Calculus in its own right.  Newton did not publish any papers formalizing calculus instead he just used it in physics problems as can be seen in his correspondence.  To figure out the total area he used ratios of changes and showing that each change was infinitesimally small but the number of changes were also infinitesimal.

Leibniz on the other hand started his development of calculus by taking a graph and breaking it up into many small rectangles and then finding the area of these rectangles.  Leibniz made many of his discoveries by experimenting with the notation of calculus.  He is also who we can thank for the notation behind calculus.

Calculus has been centuries in the making and its development has had contributions from many intelligent mathematicians.  In its simplest terms it is the calculations behind the rate of change.

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Book Report of Visions of infinity by Ian Stewart

Visions of Infinity

by:  Ian Stewart

            Visions of Infinity by Ian Stewart is a book about how the field of mathematics is advanced not just about the field of mathematics.  This book was much more about how mathematicians go about discovering their ideas about mathematics then about the proofs themselves.  The book gives the reader insight into how mathematicians think and how they approach problems.  It also teaches us about the struggle between the field of mathematics as a whole wanting more recognition, and the fact that most mathematicians would rather live without publicity. Mathematicians went a long time without any sort of large accolades being given to the study of mathematics. Unlike other fields which had the Nobel Prize for their scientists to strive for.  The book also covers many different theorems and conjectures in mathematics, though it focuses on how the mathematicians focused on the problem.  The book also goes over the seven Millennial Problems which are seven problems in mathematics that were so difficult that a $1,000,000 reward was offered for a correct solution.  To date only one of these problems the Poincare Conjecture has been solved.

            The book starts off with explaining how surprised the BBC was when they presented a TV show about mathematics and people actually watched it with interest.  This is because most individuals were, and still are to the most part, that mathematics had stopped growing a long time before.  The problem with the program was that it proposed a solution to Fermat’s Theorem which had been proposed 300 years before, so many individuals wrongly assumed that mathematics hadn’t been furthered in 300 years.  This could be because while other fields that relied on mathematics received prizes for their works received prestigious prizes, such as the Nobel Prize, and mathematics most prestigious reward was the Fields Medal which is largely unknown about outside of the mathematical community.

            Mathematicians are also very cautious about giving out accolades.  When Grigori Perelman first proposed his solution to the Poincare conjecture it took eight years for him to be offered the $1,000,000 reward for his solution to the Millennial Problem. He then declined the prize since the recognition and the reward was not important to him after waiting for so long.  No mathematician wanted to throw their weight behind Perelman for fear about him being incorrect.  This shows how cautious mathematicians can be and how many mathematicians will try and refuse the spotlight if given the chance.  Many mathematicians also want their opinions to be accepted on merit and not based upon public opinion, which is problematic because the solution needs to be publicized for other mathematicians to know to look at it.  The private nature of most mathematicians prevents the field of mathematics.

            The book Visions of Infinity did a very good job about explaining how mathematicians approach problems and why it takes so long for them to receive recognition for their work.  It also shows accolades are necessary for public recognition of the advances of mathematics.  The book also shows how the nature of mathematicians private lives can contradict what needs to be done in order to achieve greater recognition for the field of mathematics.  Overall this book was worth the read for anyone curious about how mathematicians think or about the field of mathematics as a whole. 

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Why a Proper Number System is Important in Mathematics

A proper numbering system in mathematics is a very important part of being able to do math.  We might be able to do algebra using letters, but without the numbers these letters are meaningless.  The numbers themselves might start off as arbitrary but they do gain value once it is figured out how they work.  A simple number system can make predicting how the world will react to different things possible.  Trying to do complex math with roman numerals sounds to cumbersome to even attempt.  A simple number system can even make learning math easier, while a complex numbering system would make it difficult to grasp the more complex parts of mathematics.

In mathematics numbers act as placeholders in an arbitrary way.  They are arbitrary in the sense that at one point someone said this squiggle line is worth more than that squiggle line.  This holds true for the numbers 0-9 and then the numbers have a pattern that they follow.  Without these placeholders doing math would be impossible it would lead to statements like finding the derivative of that big number multiplied by that slightly smaller number.

Having placeholders in math is important, but it also matters how easy the placeholders are to work with.  Using a cumbersome number system like roman numerals would make doing math significantly more difficult.  Trying to do even simple division with roman numerals would take an incredibly long time.

It has been shown that places with simpler number systems tend to have better test scores than countries with a complicated numbering system.  Students in China have an easier time learning math because instead of having eleven their number translates closer to one-ten which would make learning how to count in the teens easier.

So while the shape of the numbers might not be important, how easy they are to use and the fact they exist is very important.  The easier the numbering system itself is the easier it will be to learn and the more complex the math can be that uses the numbering system.

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Tesselation

Tesselation made using Geogebra

Tesselation

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Fibbonacci

            Leonardo Pisano Bigollo, or better known as Fibonacci, was a mathematician that lived between 1170 – 1245 in Italy, though his exact date of death is unknown it was between 1240 and 1250.  Most of what we know about Fibonacci and his theories comes from the book he wrote “Liber Abaci” or the book on calculation.  He is most well known for the Fibonacci Sequence which is an infinite string of numbers where any given number is the sum of the two previous numbers (i.e 1, 1, 2, 3, 5, 8, …).  Even though he is most well known for this sequence he did many other things that are very important for the world of math today.  He was the one to introduce decimal numbers to Europe.  He is also the reason why people stopped using Roman numerals for their calculations.

            Fibonacci was originally a wealthy trader which is why he traveled to faraway places like North Africa.  As he traveled he learned many different mathematical techniques which were not yet known to the Europeans which he shared with them once he returned from his travels.  He ended up bringing Hindu-Arabic numerals to the Europeans.  Which while they weren’t our modern day number system they were much easier to use than Roman numerals. 

            The Fibonacci sequence came from Fibonacci proposing how rabbits would multiply given ideal conditions.  Though the sequence was known to Indian mathematicians far earlier than that, but it was Fibonacci’s book that made it widespread.  The Fibonacci numbers do lead to what mathematicians refer to as the golden ratio.  This ratio is a Fibonacci number divided by the previous number in the sequence, and leads to approximately 1.61538.  The Fibonacci sequence has also popped up in nature.  It can be seen in places like the spirals of seashells, and in how plant leaves are arranged.

            Without Fibonacci’s input math would not be nearly as advanced as it is today.  People might still be trying to do math using Roman numerals which are far more cumbersome than the script we use today.  The famous Fibonacci Sequence has been found to pop up in various forms through things that at first glance seem unrelated to the Fibonacci sequence.  So thanks to Fibonacci and his travels European mathematicians learned many things that they would not have been privy to.  This consolidation of techniques from around the world helped mathematicians from different areas of the world have the same knowledge base.  This helped other mathematicians make even more advances in mathematics.

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