The Impact of Arthur Cayley on Modern Day Math

Arthur Cayley was born on August 16^th, 1821 in Richmond, Surrey, England to the parents Henry Caley and Maria Antonia Doughty. He also had a brother William Doughty. His father was of Russian decent and Arthur was bilingual from childhood. He spoke Russian his first eight years of his life during their stay in St. Petersburg, Russia, and later learned English once he moved to England. Arthur showed great skill in numerical calculations at school. After he moved to King's College School in 1835, his aptitude for advanced mathematics became apparent. His mathematics teacher advised that Arthur be encouraged to pursue his studies in this area rather than follow his father's wishes to enter the family business as merchants. His father said, " My son is not here to make math his career, but to carry on my business like it is expected of him."

At the unusually early age of seventeen, Cayley began going to school at Trinity College in Cambridge, England. While here, Cayley contributed many papers to his Math professor which were published and highly touted for his new age thinking and ingenuity. Cayley finished his undergraduate course by winning the place of Senior Wrangler, and the first Smith's prize. His next step was to take the M.A. degree, and win a Fellowship by competitive examination. He continued to reside at Cambridge for four years, during which time he took some students to mentor, but his main work was the preparation of twenty-eight memoirs to the Mathematical Journal.

Due to the lack of money involved in the Math profession, he had to take up a career as a lawyer. While working at a local law firm, he and a friend by the name of Slyvester, worked together on math problems while working at the firm. After spending foruteen years working at this law firm, he decided he could not bear working here anymore and decided to focus on solving math problems and other mathmetical ideas he had and wanted to express. " The law office where I worked was a dark and cold room. It was more of a room for a peasnat, let alone a lawyer." Over the course of time while at the law firm, he published over 250 papers expressing his mathmatical ideas.

He took a job as a Sadleirian professor of Pure Mathematics at Cambridge University, it was his goal to apply himself to the advancement of that science. He said it was his duty "To explain and to teach the common principles of math to his students". And it was here when he made the most substantial and notable math findings. It was Cayley who first introduced matrix multiplication in 1860. He saw through his hundreds of papers he wrote, that you could mulitply numbers while in a bracketed formation. He also was consequently was able to find the Cayley-Hamilton theorem, which was that every square matrix is a root of its own characteristic polynomial. He was the first to define the concept of a group in the modern way which was a set of groups with a binary operation satisfying certain laws. Back then, when mathematicians spoke of "groups", they had meant permutation groups. Arthur Cayley died January 26^th, 1896 in Cambridge England.

Cayley's impact on soceity is everlasting. Every time we do a matrix in math class, we are brining up the past that Cayley brought to life. We also use permutations very much in our everyday lives when we want to find out how many people may take place in an event by using permutations. He came up with many useful stratigies that we use in our every day lives. - Arthur Cayley " Math is not just a way of thinking, but it is an every day event which encompasses you and makes its prescence felt."