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Essays on leonhard euler
Essays on leonhard euler
Essays on leonhard euler
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The number e
Introduction
Leonhard Euler was a brilliant Swiss mathematician and physicist, living between 1707 and 1783. Euler had a phenomenal memory, so much so that he continued to contribute to the field of mathematics even after he went blind in 1766. He was the most productive mathematical writer of all time, publishing over 800 papers. Euler’s dedication towards the subject intrigued me and motivated me to choose a topic related to Euler himself. Amidst his many contributions, I came across e. After further research, I soon learned the multiple applications of the number, and its significance to math. I chose to study the topic of e because I wanted to learn the many ways e can be represented and how it impacts our lives, as well as to share my findings with my peers.
The aim of this exploration is to identify the different ways e can be defined, and its application to both mathematical topics and topics related to other areas of study.
What is the number e?
The number e is a famous irrational number. It is often referred to as Napier’s constant, but is named after Leonhard Euler. Its value is approximately:
2.7182818284590452353602874713527…
e carries great importance in mathematics and can be represented by a wide variety of equations. The number is also transcendental, which means that it cannot be the real root of any polynomial equation with integer coefficients.
Equations that define the number e
The number e is defined by: e=lim┬(n→∞)〖(1+1/n)^n 〗
This equation states that as the value of n increases, the value of (1+1/n)^n approaches e. A limit is the intended height of a function at a given value of n. For any epsilon greater than 0, there is such number n that after a certain value, the entire tale ...
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...athematical exploration would be very limited and strained. There wouldn’t be much information I could discuss. However, I was pleasantly surprised when I came across an array of subjects that held some type of relationship to the number e, and were often heavily dependent on the number. It helped me to realize that many of the topics we are taught throughout our academic career can be expanded upon by new ideas and theories.
For example, I had previously learned about exponential growth and decay during my Grade 9 and Grade 10 science courses. The relationship between the topic and the number e has increased the extent of my knowledge to be more specific pertaining to exponential growth and decay. I’ve learned how to accurately calculate exponential growth and decay, when previously I was only aware of the topic and how it was applicable to various situations.
In many applications, the natural base e is the most convenient base in an exponential equation. The value e is approximately 2.718281828. The natural base e works exactly like any other base. It is easy to think of e as a substitution for a in f (x) = ax. Its graph looks as so:
Infinity has long been an idea surrounded with mystery and confusion. Aristotle ridiculed the idea, Galileo threw aside in disgust, and Newton tried to step-side the issue completely. However, Georg Cantor changed what mathematicians thought about infinity in a series of radical ideas. While you really should read my full report if you want to learn about infinity, this paper is simply gets your toes wet in Cantor’s concepts.
Since its inception, science relied on predictability and order. The true beauty of science was its uncanny ability to find patterns and regularity in seemingly random systems. For centuries the human mind as easily grasped and mastered the concepts of linearity. Physics illustrated the magnificent order to which the natural world obeyed. If there is a God he is indeed mathematical. Until the 19th century Physics explained the processes of the natural world successfully, for the most part. There were still many facets of the universe that were an enigma to physicists. Mathematicians could indeed illustrate patterns in nature but there were many aspects of Mother Nature that remained a mystery to Physicists and Mathematicians alike. Mathematics is an integral part of physics. It provides an order and a guide to thinking; it shows the relationship between many physical phenomenons. The error in mathematics until that point was linearity. “Clouds are not spheres, mountains are not cones, bark is not smooth, nor does lightning travel in a straight line.” - Benoit Mandlebrot. Was it not beyond reason that a process, which is dictated by that regularity, could master a world that shows almost no predictability whatsoever? A new science and a new kind of mathematics were developed that could show the universe’s idiosyncrasies. This new amalgam of mathematics and physics takes the order of linearity and shows how it relates to the unpredictability of the world around us. It is called Chaos Theory.
Chapter two of The Universe and the Teacup deals with exponential numbers. More precisely, it deals with the difficulty humans have in processing very large and very small numbers. The term the book uses to describe this difficulty is "number numbness."
The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.
...eas that had been around for a long time but had also been thought to be different. He put together the concept of mass and the concept of energy and showed that they are actually the same thing when you think about them correctly. So his equation, E = mc2, theE is for energy and the m is for mass, and he showed that given a certain amount of mass you could calculate the amount of energy it contains. Or, alternatively, given an amount of energy, you can determine how much mass you can create from it. So mass and energy, he showed, are the ultimate convertible currencies. They are different carriers of some fundamental stuff that you can call energy, with mass simply being one manifestation of energy. But there are other manifestations: heat and light, radiation, and so forth. These are now recognized to all be different facets of one idea, one entity called energy.
The Fibonacci sequence was introduced as a problem involving population growth based on assumptions. Fibonacci got the idea from early Indian and Arabian Mathematics. He grew the theory and introduced it to the western world. The sequence is explained by starting at 1, 2 then adding the two t...
The great field of mathematics stretches back in history some 8 millennia to the age of primitive man, who learned to count to ten on his fingers. This led to the development of the decimal scale, the numeric scale of base ten (Hooper 4). Mathematics has grown greatly since those primitive times, in the present day there are literally thousands of laws, theorems, and equations which govern the use of ten simple symbols representing the ten base numbers. The field of mathematics is ever changing, and therefor, there is a great demand for mathematicians to keep improving our skills in utilizing the numeric system. Many great people, both past and present, have made great contributions to the field. Among the most famous are Archimedes, Euclid, Ptolemy, and Pythagoras, all of which are men. This seems to be a common trend in mathematics, for almost all classical mathematicians were male.
Perkowitz, Sidney. "E = Mc2 (equation)." Encyclopedia Britannica Online. Encyclopedia Britannica, n.d. Web. 26 Dec. 2013. .
imaginary numbers you wouldn’t be able to listen to the radio or talk on your cellular
» Part 1 Logarithms initially originated in an early form along with logarithm tables published by the Augustinian Monk Michael Stifel when he published ’Arithmetica integra’ in 1544. In the same publication, Stifel also became the first person to use the word ‘exponent’ and the first to indicate multiplication without the use of a symbol. In addition to mathematical findings, he also later anonymously published his prediction that at 8:00am on the 19th of October 1533, the world would end and it would be judgement day. However the Scottish astronomer, physicist, mathematician and astrologer John Napier is more famously known as the person who discovered them due to his work in 1614 called ‘Mirifici Logarithmorum Canonis Descriptio’.
Carl Friedrich Gauss was born April 30, 1777 in Brunswick, Germany to a stern father and a loving mother. At a young age, his mother sensed how intelligent her son was and insisted on sending him to school to develop even though his dad displayed much resistance to the idea. The first test of Gauss’ brilliance was at age ten in his arithmetic class when the teacher asked the students to find the sum of all whole numbers 1 to 100. In his mind, Gauss was able to connect that 1+100=101, 2+99=101, and so on, deducing that all 50 pairs of numbers would equal 101. By this logic all Gauss had to do was multiply 50 by 101 and get his answer of 5,050. Gauss was bound to the mathematics field when at the age of 14, Gauss met the Duke of Brunswick. The duke was so astounded by Gauss’ photographic memory that he financially supported him through his studies at Caroline College and other universities afterwards. A major feat that Gauss had while he was enrolled college helped him decide that he wanted to focus on studying mathematics as opposed to languages. Besides his life of math, Gauss also had six children, three with Johanna Osthoff and three with his first deceased wife’s best fri...
...ms that are present in today’s society would not be possible without Gauss’s effort on number theory.
Prime numbers have been of interest to mathematicians for centuries, and we owe much of our existing knowledge on the subject to thinkers who lived well before the Common Era––such as Euclid who demonstrated that there are infinitely many prime numbers around 300 BCE. Yet, for as long as primes have been an element of the mathematician’s lexicon, many questions about prime numbers remain unreso...
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.