RESEARCH AND OUTLINE EXAMPLE
-Euler’s number is a constant(like pi or i)
-Discovered in the early 18th century by Leonard Euler
-In the graph of e^x, the slope of the curve(derivative) at any point equals e^x and the area under the curve(integral) at any point equals e^x
-The number e is not only important in calculus, but also physics(shows up in equations for waves)
The number e (Euler’s number) was discovered by Leonard Euler centuries ago while he was trying to solve a problem that was brought attention to the math world about 50 years earlier by Jacob Bernoulli. Bernoulli’s problem was related to compound interest. The number e is an extra important constant similar to pi and i. One phenomenon of this number is that in the graph of e^x, the slope of the curve at any point equals e^x and the area under the curve(integral) at any point equals e^x.
-The number e can be represented by:
|-This limit can be expanded by Binomial Theorem to result in this:
-This limit converges
-e is a transcendental number(this means it is not the root of any algebraic equation with integer coefficients)
-When the function e^x is derived with respects to x, its result is the same as the function.
The main point of this article is to not just give the basics of the history of the finding of e, but focuses more on the mathematical properties of it. The reason why a lot of high schoolers don’t know a whole lot about the number e, is because most of the proving mathematically for this number can only be done with at least SOME calculus experience.
-Leonard Euler lived in the years 1707-1783 and was Swiss
-Jacob Bernoulli comes from the same family that brought forth “the Bernoulli effect”(causes lift in airplane wings)
-The number carries on forever and does not repeat
-The world record for the most decimal place is held by Rajveer Meena at 70,000 decimal places
This article starts out with a very brief, but in depth history of the two men who were the original discoverers of this number. The bulk of it though, is focused on compound interest. I’m not going to type the explanation of the mathematics because that would take years, but the main thing we need to realize is that this number was found thanks to compound interest.
-The number e is the base of the natural logarithms (invented by John Napier)
-e is irrational. Not the ratio of two integers
-”Cut up then multiply” property. Too hard to write, so just read the article about it
-e comes into compound interest, because the formula of compounding interest are very similar:
Compound interest: 1+rnn
-When 100% is chosen as the rate of interest, the formulas are the exact same
This article is good because it shows every way(I think) that e appears in the mathematical world. This will be a good place to remind myself of all of the properties of e. Well, at least mathematically.
-The farther along in math AND science you go, the more often you come across this number
-There are many useful applications that involve the number e. Look at the article for the whole and complete list
-The more times that you compound interest yearly, the closer you will get to the complete number e. But you will never truly reach that point
- Give a brief history of your topic:
In the 1700’s, Leonhard Euler discovered the letter e. Though he was the one to discover it, he wasn’t the one that brought the lack of the number e to the table. That was Jacob Bernoulli. He was researching compound interest when he found the need for the letter e. From then on, e was brought to life in the math community. It is related often to pi and i, because they are all “special” constants.
- Major people who play a role in your topic:
The two major people that play a role in the discovery of the letter e, is Leonhard Euler and Jacob Bernoulli. Euler was the real person to discover the number, but Bernoulli played a part as well 50 years before Euler’s discovery. John Napier also has a part of the discovery of the properties of e. He discovered that e is the base of natural logarithms.
- How is your topic significant now?
You know, it is hard to say that e is very significant now for the ordinary person in our society. In the normal math courses that high schoolers take, most people don’t even reach the concept of e, and if they do they are more taught how to avoid it and manipulate it, and not actually how to solve it. Also, the latest discovery of the letter e, could arguably be the original discovery. Not a lot has happened around it in the past couple of centuries.
Intro: Talk about the properties of e.
- How it is related to pi
- Everything that is unique to the number
- All of the traits of e
Body Paragraph(s) Point #1: History
- Jacob Bernoulli and his original question
- Leonhard Euler and his answer to Bernoulli’s question
Body Paragraph(s) Point #2: Where does the letter e come from(further explanation of Euler’s finding)
- Compound interest
- As we have a higher interest rate, the closer we get to Euler’s number
- Explain the mathematical reasoning
Closing: Why is it relevant to YOU
- Talk reasoning why not many people have heard of it contrary to pi, and i
- Compound interest pt. 2