Pierre de Fermat Joey Arnone Pierre de Fermat's life Pierre de Fermat was born in 1607 in Beaumont-de-Lomagne, France. It was believed that his birth year was 1601 - however, it was discovered that it was actually his half-brother, also named Pierre, that was born in this year. Fermat grew up with three other siblings, having two sisters and one brother. His father was both a successful merchant and an appointed second consul to his hometown, and his mother was a parliamentary noblesse de robe, in which she held a high rank in state office. Clearly, Fermat was born with a substantial amount of money. This proved much to his advantage, as he was given the opportunities to attend expensive and prestigious schools. As an adolescent, he attended the Les Cordeliers convent (as he was a Catholic) for secondary, or high school. He was very interested in languages, and by adulthood became fluent in Greek, Latin, Italian, Spanish and Occitan (a Romance language that is spoken in Southern France). A portrait of Pierre de Fermat He is known as one of the worlds most influential mathematicians, and for this reason it is slightly ironic that he studied law at the University of Orleans. He began his studies in 1623 and graduated a mere three years later as an 18 year old. In his young adulthood, Fermat took a strong liking to not only law, but also that of art and science. When Fermat moved to Bordeaux in the late 1620s, right after graduating from University of Orleans, he began making mathematical research of importance. He solved conjectures left by mathematicians from early decades and made significant advances in the fields of algebra and geometry through this. After Fermat's father passed away in 1630, he inherited a great deal of money, and used a portion of that money to obtain a senior position in the High Court of Toulouse. This was the profession that he carried out until the end of his life. He married his wife, Louise de Long, only a year later, and they had eight children together, five of whom survived into adulthood. Throughout his life, he collaborated with many different mathematicians and solved many different mathematical proofd, producing invaluable work, including the creation of probability theory with Blaise Pascal and the development of prime number theories that came as a result of collaboration with Marin Mersenne. It was around this time that he developed his most famous work: Fermat's Last Theorem. Soon after creating his last theorem, he received a large promotion in his profession as a lawyer. In 1638, he was granted a position as a lawyer in Toulouse's criminal court. Soon afterwards, he was also granted to work in the Grand Chamber, which was the highest honor in law work at the time, as he was able to deal with largely political and international affairs. During his lifetime as a mathematician, he continuously kept climbing the career ladder as a professional in the field of law. A decade after being selected for work in the Grand Chamber, he became Parlement's chief spokesman, acting as a speaker on behalf of official politicians in power. He worked mainly as a spokesman for Pierre Seguier, who was the chancellor of France. Through various obscure writings from his time appointed, it can actually be implied that his work in being a chief spokesman was not his best. In his later adulthood, a plague gripped the town of Toulouse, where Fermat resided, and he had been unlucky enough to contract it. However, he was able to pull through and survive. He continued making mathematical work that was considered influential up until his death. He died in 1665 in Castres, France. Pierre de Fermat's mathematical works In his years at the university, he would make conjectures based from uncompleted work by famed mathe- maticians, and one of the pieces he analyzed was Plane Loci by Apollonius. In going over this work, Fermat discovered that loci could be analyzed through usage of coordinates, in which case both algebra and geometry is needed. This work alone has proved invaluable to the field of analytic geometry. In the 1630s, Fermats life was preoccupied with many other matters besides mathematics - he became a lawyer/government official and also started a family during this time. However, he still worked on developing numerous theories, and it was at this point where he was at his most fruitious period. Fermats research began leading him to analyzing curves and equations related to curves. He extended the equations of parabolas ay = x2 and hyperbolas xy = a2 and was able to generalize these equations into the form a(n − 1)y = xn Curve estimates are then attained from using this formula, and they are now known as the parabolas and hyperbolas of Fermat. The equations (and curves obtained from these equations) that he generalized for both this and the Archimedian spiral became used for his discovery of what we now know as basic differentiation. Most of his life works (coordinate algebra and geometry) came to a summation on this theory, as it deals with locating maxima, minima, and inflection points, along with finding tangents to the curve. This work is known as finding "adequlities", and it was first published in his treatise "Methodus ad disquirendam maximam et minimam", or "Methods for distinguishing maximums and minimums", published in the 1630's. Soon after this, he made another important discovery. He was able to obtain a summation of the area bounded by a curve through his earlier equations and as of consequence discovered the early form of what we now know as integration. Given that Fermat delivered such important work dealing with derivation and integration decades before Newton and Leibniz, one could possibly believe that it was Fermat himself who founded calculus. 2 However, Fermat's examples only worked in specific cases, and no general solutions were ever worked out with major success. Nonetheless, Fermat's discoveries were highly influential in Newton's work that led to the founding of calculus, making Fermat possibly the biggest influence in its discovery. Newton himself wrote that his work for developing derivation was influenced by "Monsieur Fermat's method of drawing tangents." Fermat also contributed to physics at this time, as his work on the law of refraction is still seen as influential to this day. According to [Fermat's] principle, if a ray of light passes from a point A to another point B, being reflected and refracted (refracted; that is, bent, as in passing from air to water, or through a jelly of variable density) in any manner during the passa ge, the path which it must take can be calculated- all its twistings and turnings due to the refraction, and all its dodgings back and forth due to reflections- from the single requirement that the time taken to pass from A to B shall be an extreme. Fermat believed that light passes through denser material more quickly, which in essence is a justification of Snell's law, which repeats basically the same statement. He used his own theories on maxima and minima to help prove this - it later became known as Fermat's Principle. However, Fermats findings weren't entirely true, as some ideas conflicted with Aristotlean theory that nature (in this case, light) chooses the shortest path. The essence of what Fermat's Principle now states is that the path that a ray of light takes is the one in which it takes the least amount of time. Fermat began finding more success when he started his work on number theory. It was Fermat that discovered the idea that prime numbers are numbers that can only be divisible by 1 and themselves. He found an equation: 2k + 1 With this equation, Fermat objected, one can find many prime numbers - the numbers generated from this equation are now known as Fermat numbers. Even with his discoveries, he still had a few shortcomings in the mathematics he used. Euler proved later that one of the numbers generated from the Fermat number equation actually has factors other than 1 and itself, proving Fermats number theory not entirely true. The level of research that Fermat underwent to produce an equation like this, however, is astounding, as he studied not only the properties of prime numbers, but also perfect numbers (positive integers that are equal to the sum of their proper divisors), amicable numbers (two different numbers that are so alike that the sum of their proper divisors is equal to the other number) and Pell's equation x2 − ny2 = 1 Fermat learned from his mistakes while working on his prime number theory and used the knowledge he obtained for the development of another theory on numbers. He soon discovered that every single prime number that is divisible by 4 and leaves a remainder of 1 when this division occurs n = 1; mod4 can be expressed as a sum of two square numbers. p = x2 + y2 His findings became known as the Two Square Theorem. An even more influential theorem that he responsible for is his Little Theorem. In this, he states that if a and p are two numbers, and p is a prime number not divisible by a, then multiplying a by itself p-1 times and thereafter dividing by p will leave a remainder of 1. This work is still highly influential today, as it is used in codes for credit card transactions, among other things. Fermat to this day is seen as the father of modern number theory.