The Proof of Euler that Captures the Soul of Mathematics - Research Paper Example

‘A proof that captures the soul of mathematics’. Introduction Since the dawn of time, there has existed a bond between all creations of god, biological or chemical. With the advent of humans many with IQ’s far greater than our own have ventured, far and wide in search of the link that could answer the equation of all existence. What is the language that this equation will be written in? Galileo Galilei a great astronomer and mathematician once said Mathematics is the language with which God has written the universe. Mathematics in essence provides the basic learning code for all subjects of studies whether that is Finance, Marketing, Physics, Chemistry, Biology or robotics. The biggest credit to the Homo sapiens is their invention of mathematics; this invention has revolutionized every aspect of learning. Would it be possible to the code to human gene without mathematics? Or send Neil Armstrong to the Moon? Mathematics has indeed provided clues to phenomenon hidden to human eyes for centuries. Mathematics can give absolute and divine pleasure to a mathematician. This is not only because it is beautiful and capable of artistic divination as any other art but because unlike poetry, music etc it provides complete and absolute satisfaction, purity and above all freedom. It is capable of unrivaled perfection compared to any other art form. A mathematician does not do indulge in mathematics not only because it is useful but because it gives a reason of being. The insignificant human existence can find purpose and greatness beyond its boundaries. To understand this beauty we should take it as an expression of structure and pattern, like a painter paints a picture. Just like different colors combine to create a unique masters piece, similarly numbers and notations integrating to produce a beauty. Just look at the Rubik’s for instance, no human being with out the help of mathematic can imagine solving it. It has more solution than many times the stars of this galaxy but with mathematical algorithms we have accomplished to solves it in as less as 7.9 seconds. Leonhard Euler (1707 – 18 September) One of the leading physicist and mathematician of all time Euler was born in Basel Switzerland but spent most of his life in Russian and Germany (Scientists of Faith by Dan Graves) His childhood was greatly influenced by Johann Bernoulli one of Europe’s well known mathematician of his time(Remarkable Mathematicians: From Euler to von Neumann. Cambridge by Loan James). His father a pastor wanted Euler to continue in his footsteps but Johann Bernoulli intervened convincing his parents that the child had unmatched skills in mathematics and was thus intended for greater things than pastor ship. Thus after Euler completed his M.Phil from the University of Basel(Leonhard Euler: The First St. Petersburg Years by Ronald Calinger) he started taking Saturday sessions with Johann. To date he is known for his diversity of study and knowledge, he made astounding discoveries in diverse fields such as calculus and graph theory. His works are also well known in fields of astronomy, fluid dynamics, optics and astronomy. Euler’s first mathematical paper came out when Euler was only eighteen years old. It was a treatise on the masting of ships. Although he was still only seventeen, he competed in the annual contest held by the French academy of sciences. Mathematicians and top scientist from all over Europe were a part of that competition, still against all odds Euler managed to get second prize. Switzerland his native town was a landlocked country, and Euler had no first had experience with ships, luckily his genius was based not on experience but concrete basis of mechanics; thus his correctness couldn’t be questioned. Later however Euler went on to win the same competition 12 times. On recommendations of Daniel Bernoulli, Euler left for Russia to join the Imperial Academy of science (1727). He prospered in St. Petersburg becoming the head of mathematics department at the Academy. In 1741 fearing the turmoil in Russia with family he moved to Berlin joining the Berlin Academy of science. There he did much of his greatest works, publishing more than 380 articles. Another interesting work of Euler was in fact his letters to the princess of Prussia whom he tutored. These 200 letters were complied and published under the name of Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This compilation sold more copies than any of Euler’s mathematical works. This is indeed the power mathematics bestows upon a mathematician, just like Euler they give theories about far away galaxy and even time ; but they cant be question as their arguments are based to logic and reality. As Benjamin Pierce a world famous mathematician said about Euler’ Identity It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth. Euler’s Identity The most beautiful aspect of the identity is its simplicity; with pure artistry three simple mathematical operations are combined to form a complex paradox. Euler’s number is denoted by the symbol ‘e’. If we explain ‘e’ using a function it would be defined as the function ( An Introduction to the History of Mathematics by Howard Whitley Eves). If we draw it on a Cartesian plane, the value of the function is 1 when x=0. Another interesting constant used in this equation is (pi). This mathematical constant is the ratio of any circles circumference to its diameter. The numerical value of the is 3.142 if rounded of three decimal places that is; this is because pi is also an irrational number. This basically means that it can not be represented as a fraction, thus its decimal representation never ends nor repeats it self. Euler also uses an imaginary number in his equation denoted by. An imaginary number is the square root of any negative number, thus it’s the only number which gives negative answer when squared (). To understand the true beauty of Euler’ identity we must first of all try to comprehend what is an imaginary number. Only a few centuries ago imaginary number were though to be something of fiction and useless to mathematician (Albert A. Martinez, Negative Math). Imaginary number although might seem useless in the real life but in the fields that involve complex calculation may they be related to electrical engineering, quantum Physics, business mathematics or computer programming. If we use geometry to explain imaginary numbers lets imagine a line perpendicular to the straight number line. This number line is called the y-axis in geometry. If we move upwards from zero all the numbers visible are positive imaginary number and below it are negative imaginary numbers . The Proof How come when e is raised to the power of and imaginary number multiplied by and 1 is added the answer is 0? Let’s find out how. To understand Euler’s identity first we will have to take a look at Euler’s formula. To understand this equation lets start expanding each element. First of all using Taylor series we will expand. According to Taylor if we expand e using values of x this is how the equation will progress. Let the value of x be equal to 2. =7.389056099 the answer will continue but this is as far our calculator displays. Not substitute the value of x=2 in the equation. This will give you … as you further the denominators will become larger and the values therefore smaller until reaching zero (Beauty of Mathematics by Surien Aziz). Now let apply the Taylor series to Sin(x) and Cos(x); …. Substituting the value of in the equation by we get: If we simply the equation using different powers of i.e., and, this what we will get; Further simplification gives; Just look has it start making sense? Observe the Taylor series for Cos and Sin. A little simplification will result in this: . Insert the value of the function for x. Result will be: By now the beauty is emerging just like a diamond from a dense field of cold. As you know the value of and Thus we get! After a series of complex operations and procedure we are back to where we started. If we shit -1 to the left we will get back to Euler’s Identity. Conclusion The proof of Euler Identity twists and turns complex confusing, yet results in the most simplest and elegant equations. Just like a painting of Da Vinci, withholding a beauty hidden to the naive’s eyes. What you just saw was a glimpse of the beauty that mathematics withholds. It has its own way of integrating unrelated, illogical inputs and creating something that is truly unique and logical. To understand this beauty one doesn’t need the most advanced degrees in mathematics but sense to feel and sense it. Just like Euler’s equation other mathematical equations exist which have changed even the meaning of life for ever. Just as not every one can appreciate the beauty in abstract art; neither can every one sense the beauty of numbers. These numbers contain the secrets to all life and one day a mathematical proof might even answer why we Exist! Works cited Maor p.160 and Kasner & Newman p.103–104 Howard Whitley Eves (1969). An Introduction to the History of Mathematics. Holt, Rinehart & Winston Surein.Aziz. Beauty in Mathematics. Plus Magazine.2000 http://plus.maths.org.uk/issue51/features/aziz/2pdf/index.html/op.pdf Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. pp. 17 James, Ioan (2002). Remarkable Mathematicians: From Euler to von Neumann. Cambridge. pp. 2. ISBN 0-521-52094-0. Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 125. doi:10.1006/hmat.1996.0015. Read More