NationStates Jolt Archive

Now, I know how much you all have enjoyed answering my questions on "e" and my waste of forum space. So now, I bring to you my draft of my essay on "e" in the hopes that the mathematical geniuses of this forum can pick it to bits.

On another note, I was hoping some of you who are proficient in Latin could help me with a sufficient title for my paper. Something like Euler or his home dogs would write. Maybe the Principals of E? Or the Mechanics and Foundation of E? Something along those lines.

Here it is:



E

With an approximate value of 2.718281828459045, an intricate history, and several fascinating properties, it is no wonder that e, the base of natural logarithms, is so renowned. Its use rivals that of even pi and some could argue it’s infinitely more important. It is a natural constant that appears in a plethora of mathematical problems. There are several ways to represent e other than the actual letter and each form represents e in a slightly different way. While e was first referenced to in the early 1600s, and possibly as far back as the time of the ancient Babylonians, its actual establishment and association with the letter e is credited to a man named Leonhard Euler. Since then it has been used extensively in calculus classes and has been the fascination of thousands.

The pioneer of the constant e is attributed to a Swiss mathematician, who worked at the Petersburg Academy and the Berlin Academy of Science, named Leonhard Euler. He is the most prolific mathematical writer of all time, publishing over 800 papers in his lifetime. He won the Paris Academy Prize twelve times and introduced the symbols e, i, and f(x) to mathematics. He also made major contributions in optics, mechanics, electricity, and magnetism. While many believe that e was named after him, it is very unlikely due to his modest nature and determination to attribute proper credit to the work of others. Others believe it to be the abbreviation of “exponential” which is also ridiculous. The most logical explanation is that Euler had been using vowels for mathematical notation and e was merely the next one in line.


The constant e itself is commonly represented in three ways. The first is e defined as a limit, or behavior.

http://en.wikipedia.org/math/15f5460b0d41750d9f3f23f47e0ba5fd.png

In this form, as n approaches positive infinity, the function grows closer and closer to the approximate value of e. From here we can gather that e is frequently an asymptote, a line that a graph approaches but will never reach.

The second way is presenting e as the sum of an infinite series.

http://en.wikipedia.org/math/5d787122d4e0a1a698aa112593b45778.png

Here, n! is the factorial of n. E.g., the factorial of 5 is http://en.wikipedia.org/math/a6c50400ec47c19de5a547a7aae40f59.png ; as the series approaches positive infinity so too does the value of the function grow closer to the value of the constant e.

Finally, the third way is representing e as a unique number x > 0.

http://en.wikipedia.org/math/9da4ec2806df4887a2ad6965b99434d3.png

This is to say that e is the unique number with the property that the area of the region bounded by the graph 1/x, the x-axis, x = 1, and x = e is equal to one. As the area of the region is equivalent to ln (e), the solution of ln (x) is x = e.

The history of e is rather interesting as well. The first references to it were published in 1618 in the table of an appendix of work on logarithms by John Napier, a Scottish mathematician and astrologer. The appendix did not contain the constant itself but simply a list of natural logs calculated from it. The first indication of e as a constant was made some time in 1683 by Jacob Bernoulli, a Swiss mathematician, trying to find the value of the expression http://img55.echo.cx/img55/5964/15f5460b0d41750d9f3f23f47e0ba5.png . Many other mathematicians such as Gregorius Saint-Vincent and Christiaan Huygens made significant findings regarding natural logarithms, the area under a rectangular hyperbola, and other mathematical issues that involved using e and its properties, but none of them ever directly approached e as an explanation or recognized it. It wasn’t until 1690 in a letter from Gottfried Wilhelm von Leibniz to Huygens that e was directly acknowledged, albeit notated as the letter b. Finally in 1731, in a letter to Christian Goldbach, Euler notates the constant with the letter e. In 1748, Euler publishes his Introductio in Analysin infinitorum which gives a thorough explanation of his ideas surrounding e. Through his work, Euler proved e to be irrational. In 1844 French mathematician, Joseph Liouville, proved that e did not satisfy any quadratic equation with integral coefficients, and in 1873 another French mathematician, this time Charles Hermite, proved that e was a transcendental number; a number that is not the root of any integer polynomial.

Today, e is used in problems ranging from exponential growth or decay, statistical bell curves, the shape of hanging cables, probability, counting, and even the distribution of prime numbers. It shows up frequently in calculus whenever logarithmic or exponential functions are used. Sebastien Wedeniwski has calculated e to 869,894,101 decimal places and at the initial public offering (IPO) of Google Inc. in 2004, the company announced its plan to raise $2,718,281,828. In other words, they planned to raise e-billion dollars. Donald Knuth, a famous computer scientist, even let the version of his book METAFONT approach e, e.g. 2, 2.7, 2.71, 2.718, etc.

As is easily seen, e is one of the most influential and important constants in the world of mathematics. From its history and all those mathematicians before Euler who failed to recognize it, to its creator who published countless papers despite loss of his eyesight, to its intricate properties and many ways of representation, and finally to its place in modern day mathematics and as a symbol that contends with pi, e is, simply put, a wonder of mathematics. E is 2.718281828459045.

E is the fifth letter in the alphabet. It is a vowel. It is put on the end of many words. It makes other letters have a long sound rather than the short one.

NARF!!

Narf?

Now, I know how much you all have enjoyed answering my questions on "e" and my waste of forum space. So now, I bring to you my draft of my essay on "e" in the hopes that the mathematical geniuses of this forum can pick it to bits.

On another note, I was hoping some of you who are proficient in Latin could help me with a sufficient title for my paper. Something like Euler or his home dogs would write. Maybe the Principals of E? Or the Mechanics and Foundation of E? Something along those lines.

Here it is:



E

With an approximate value of 2.718281828459045, an intricate history, and several fascinating properties, it is no wonder that e, the base of natural logarithms, is so renowned. Its use rivals that of even pi and some could argue it’s infinitely more important. It is a natural constant that appears in a plethora of mathematical problems. There are several ways to represent e other than the actual letter and each form represents e in a slightly different way. While e was first referenced to in the early 1600s, and possibly as far back as the time of the ancient Babylonians, its actual establishment and association with the letter e is credited to a man named Leonhard Euler. Since then it has been used extensively in calculus classes and has been the fascination of thousands.

The pioneer of the constant e is attributed to a Swiss mathematician, who worked at the Petersburg Academy and the Berlin Academy of Science, named Leonhard Euler. He is the most prolific mathematical writer of all time, publishing over 800 papers in his lifetime. He won the Paris Academy Prize twelve times and introduced the symbols e, i, and f(x) to mathematics. He also made major contributions in optics, mechanics, electricity, and magnetism. While many believe that e was named after him, it is very unlikely due to his modest nature and determination to attribute proper credit to the work of others. Others believe it to be the abbreviation of “exponential” which is also ridiculous. The most logical explanation is that Euler had been using vowels for mathematical notation and e was merely the next one in line.


The constant e itself is commonly represented in three ways. The first is e defined as a limit, or behavior.

http://en.wikipedia.org/math/15f5460b0d41750d9f3f23f47e0ba5fd.png

In this form, as n approaches positive infinity, the function grows closer and closer to the approximate value of e. From here we can gather that e is frequently an asymptote, a line that a graph approaches but will never reach.

The second way is presenting e as the sum of an infinite series.

http://en.wikipedia.org/math/5d787122d4e0a1a698aa112593b45778.png

Here, n! is the factorial of n. E.g., the factorial of 5 is http://en.wikipedia.org/math/a6c50400ec47c19de5a547a7aae40f59.png ; as the series approaches positive infinity so too does the value of the function grow closer to the value of the constant e.

Finally, the third way is representing e as a unique number x > 0.

http://en.wikipedia.org/math/9da4ec2806df4887a2ad6965b99434d3.png

This is to say that e is the unique number with the property that the area of the region bounded by the graph 1/x, the x-axis, x = 1, and x = e is equal to one. As the area of the region is equivalent to ln (e), the solution of ln (x) is x = e.

The history of e is rather interesting as well. The first references to it were published in 1618 in the table of an appendix of work on logarithms by John Napier, a Scottish mathematician and astrologer. The appendix did not contain the constant itself but simply a list of natural logs calculated from it. The first indication of e as a constant was made some time in 1683 by Jacob Bernoulli, a Swiss mathematician, trying to find the value of the expression http://img55.echo.cx/img55/5964/15f5460b0d41750d9f3f23f47e0ba5.png . Many other mathematicians such as Gregorius Saint-Vincent and Christiaan Huygens made significant findings regarding natural logarithms, the area under a rectangular hyperbola, and other mathematical issues that involved using e and its properties, but none of them ever directly approached e as an explanation or recognized it. It wasn’t until 1690 in a letter from Gottfried Wilhelm von Leibniz to Huygens that e was directly acknowledged, albeit notated as the letter b. Finally in 1731, in a letter to Christian Goldbach, Euler notates the constant with the letter e. In 1748, Euler publishes his Introductio in Analysin infinitorum which gives a thorough explanation of his ideas surrounding e. Through his work, Euler proved e to be irrational. In 1844 French mathematician, Joseph Liouville, proved that e did not satisfy any quadratic equation with integral coefficients, and in 1873 another French mathematician, this time Charles Hermite, proved that e was a transcendental number; a number that is not the root of any integer polynomial.

Today, e is used in problems ranging from exponential growth or decay, statistical bell curves, the shape of hanging cables, probability, counting, and even the distribution of prime numbers. It shows up frequently in calculus whenever logarithmic or exponential functions are used. Sebastien Wedeniwski has calculated e to 869,894,101 decimal places and at the initial public offering (IPO) of Google Inc. in 2004, the company announced its plan to raise $2,718,281,828. In other words, they planned to raise e-billion dollars. Donald Knuth, a famous computer scientist, even let the version of his book METAFONT approach e, e.g. 2, 2.7, 2.71, 2.718, etc.

As is easily seen, e is one of the most influential and important constants in the world of mathematics. From its history and all those mathematicians before Euler who failed to recognize it, to its creator who published countless papers despite loss of his eyesight, to its intricate properties and many ways of representation, and finally to its place in modern day mathematics and as a symbol that contends with pi, e is, simply put, a wonder of mathematics. E is 2.718281828459045.


Greek to me. The "e" I know and love is a commonly used vowel. You had better be going to a damn good school though.

*wonders*

People that do not know what "e" is.. how old are you ?
If you are over 16 or so.. do you consider it disturbing that you have no idea whatsoever what one of the most fundamental numbers in math and science is ?

Narf?

Sorry, got Pinky & the Brain in my head and I don't know what to say about this topic...

*wonders*

People that do not know what "e" is.. how old are you ?
If you are over 16 or so.. do you consider it disturbing that you have no idea whatsoever what one of the most fundamental numbers in math and science is ?

I wonder as well. However, I doubt most of the stuff I learn in math will ever see the light of day once I leave college.

I prefer phi, myself. It has nowhere near as many uses, as far as I know, but it's prettier!

And pi is cooler than e because pi is just cooler than e.

e is an amazing value

DARN! I thought it was going to be Dale Earnhardt

*wonders*

People that do not know what "e" is.. how old are you ?
If you are over 16 or so.. do you consider it disturbing that you have no idea whatsoever what one of the most fundamental numbers in maths and science is ?

Not all countries have the same mathematics curriculum. In Sweden, for instance, you don't learn about e until upper secondary school (~ at an age of 17-18), and you only do it if the scholastic programme you've chosen incorporates maths C. The courses of maths go through A-E (with an optional F, which mainly consists of combinatorics and other, more advanced forms of maths), of which only A is compulsory for all programmes, although many do choose to do B. I did A-E, and haven't had any use for anything past maths B in real life.

So, not knowing of "e" is not that big of a deal.

e^(i*π)+1=0

*wonders*

People that do not know what "e" is.. how old are you ?
If you are over 16 or so.. do you consider it disturbing that you have no idea whatsoever what one of the most fundamental numbers in math and science is ?

Well, I didn't really learn much at all about e until calculus, which I'm in now..most people don't take calc in high school anymore.

e^(i*π)+1=0

Yet another wonder of irrational numbers.

Not all countries have the same mathematics curriculum. In Sweden, for instance, you don't learn about e until upper secondary school (~ at an age of 17-18), and you only do it if the scholastic programme you've chosen incorporates maths C.

Then how on earth does one learn powers, integrals and differentials ? Just not tell people what they are doing ?
Well.. actually that could work..

So, not knowing of "e" is not that big of a deal.

The practical use of knowing the earth orbits the sun and there are other planets in our solar system is also nonexistant in daily life for most people (including me). Yet it is still something I think everyone should know...

e is the natural log. Damn guy who discovered it :mad:

Dont educate the masses I liked being the only one to know it.

NOw tell me the exact value of

e*i*pi

Then how on earth does one learn powers, integrals and differentials ? Just not tell people what they are doing ?
Well.. actually that could work..

Well it does, but you don't have to know of e to understand powers. Integrals and differentials, maybe, but how many people outside of scientific circles really need to know that?

The practical use of knowing the earth orbits the sun and there are other planets in our solar system is also nonexistant in daily life for most people (including me). Yet it is still something I think everyone should know...

I think you ask too much of people.

NOw tell me the exact value of e*i*pi
Approximately 8.54i.

Now, I know how much you all have enjoyed answering my questions on "e" and my waste of forum space. So now, I bring to you my draft of my essay on "e" in the hopes that the mathematical geniuses of this forum can pick it to bits.

On another note, I was hoping some of you who are proficient in Latin could help me with a sufficient title for my paper. Something like Euler or his home dogs would write. Maybe the Principals of E? Or the Mechanics and Foundation of E? Something along those lines.

Here it is:



E

With an approximate value of 2.718281828459045, an intricate history, and several fascinating properties, it is no wonder that e, the base of natural logarithms, is so renowned. Its use rivals that of even pi and some could argue it’s infinitely more important. It is a natural constant that appears in a plethora of mathematical problems. There are several ways to represent e other than the actual letter and each form represents e in a slightly different way. While e was first referenced to in the early 1600s, and possibly as far back as the time of the ancient Babylonians, its actual establishment and association with the letter e is credited to a man named Leonhard Euler. Since then it has been used extensively in calculus classes and has been the fascination of thousands.

The pioneer of the constant e is attributed to a Swiss mathematician, who worked at the Petersburg Academy and the Berlin Academy of Science, named Leonhard Euler. He is the most prolific mathematical writer of all time, publishing over 800 papers in his lifetime. He won the Paris Academy Prize twelve times and introduced the symbols e, i, and f(x) to mathematics. He also made major contributions in optics, mechanics, electricity, and magnetism. While many believe that e was named after him, it is very unlikely due to his modest nature and determination to attribute proper credit to the work of others. Others believe it to be the abbreviation of “exponential” which is also ridiculous. The most logical explanation is that Euler had been using vowels for mathematical notation and e was merely the next one in line.


The constant e itself is commonly represented in three ways. The first is e defined as a limit, or behavior.

http://en.wikipedia.org/math/15f5460b0d41750d9f3f23f47e0ba5fd.png

In this form, as n approaches positive infinity, the function grows closer and closer to the approximate value of e. From here we can gather that e is frequently an asymptote, a line that a graph approaches but will never reach.

The second way is presenting e as the sum of an infinite series.

http://en.wikipedia.org/math/5d787122d4e0a1a698aa112593b45778.png

Here, n! is the factorial of n. E.g., the factorial of 5 is http://en.wikipedia.org/math/a6c50400ec47c19de5a547a7aae40f59.png ; as the series approaches positive infinity so too does the value of the function grow closer to the value of the constant e.

Finally, the third way is representing e as a unique number x > 0.

http://en.wikipedia.org/math/9da4ec2806df4887a2ad6965b99434d3.png

This is to say that e is the unique number with the property that the area of the region bounded by the graph 1/x, the x-axis, x = 1, and x = e is equal to one. As the area of the region is equivalent to ln (e), the solution of ln (x) is x = e.

The history of e is rather interesting as well. The first references to it were published in 1618 in the table of an appendix of work on logarithms by John Napier, a Scottish mathematician and astrologer. The appendix did not contain the constant itself but simply a list of natural logs calculated from it. The first indication of e as a constant was made some time in 1683 by Jacob Bernoulli, a Swiss mathematician, trying to find the value of the expression http://img55.echo.cx/img55/5964/15f5460b0d41750d9f3f23f47e0ba5.png . Many other mathematicians such as Gregorius Saint-Vincent and Christiaan Huygens made significant findings regarding natural logarithms, the area under a rectangular hyperbola, and other mathematical issues that involved using e and its properties, but none of them ever directly approached e as an explanation or recognized it. It wasn’t until 1690 in a letter from Gottfried Wilhelm von Leibniz to Huygens that e was directly acknowledged, albeit notated as the letter b. Finally in 1731, in a letter to Christian Goldbach, Euler notates the constant with the letter e. In 1748, Euler publishes his Introductio in Analysin infinitorum which gives a thorough explanation of his ideas surrounding e. Through his work, Euler proved e to be irrational. In 1844 French mathematician, Joseph Liouville, proved that e did not satisfy any quadratic equation with integral coefficients, and in 1873 another French mathematician, this time Charles Hermite, proved that e was a transcendental number; a number that is not the root of any integer polynomial.

Today, e is used in problems ranging from exponential growth or decay, statistical bell curves, the shape of hanging cables, probability, counting, and even the distribution of prime numbers. It shows up frequently in calculus whenever logarithmic or exponential functions are used. Sebastien Wedeniwski has calculated e to 869,894,101 decimal places and at the initial public offering (IPO) of Google Inc. in 2004, the company announced its plan to raise $2,718,281,828. In other words, they planned to raise e-billion dollars. Donald Knuth, a famous computer scientist, even let the version of his book METAFONT approach e, e.g. 2, 2.7, 2.71, 2.718, etc.

As is easily seen, e is one of the most influential and important constants in the world of mathematics. From its history and all those mathematicians before Euler who failed to recognize it, to its creator who published countless papers despite loss of his eyesight, to its intricate properties and many ways of representation, and finally to its place in modern day mathematics and as a symbol that contends with pi, e is, simply put, a wonder of mathematics. E is 2.718281828459045.


We're using it in Honors Algebra 2, but I couldn't figure out why. This is explanative. ^_^

We're using it in Honors Algebra 2, but I couldn't figure out why. This is explanative. ^_^

Good to know I helped someone!!

Title suggestion. De rerum erum - on the matter of e. If anyone can come up with a better genitive for 'e' then please help, my latin is not of the best. Failing that, e by gum [only works if you're English, preferably Northern].

Well it does, but you don't have to know of e to understand powers.

To truly understand them you do. 5^2 is easy, as is 5^(1/2), but 5^2.7 ?
But agreed, you can work with them without knowing.

Integrals and differentials, maybe,
Definately. Something as simple as differentiating 5^x requires knowledge of the ln function. In most physics e recurs over and over again.

but how many people outside of scientific circles really need to know that?

Everyone who will ever read a graph involving exponential growth or has some interest in how things work.

I think you ask too much of people.
Actually I ask too little in my opinion...
*depressing*

Title suggestion. De rerum erum - on the matter of e. If anyone can come up with a better genitive for 'e' then please help, my latin is not of the best. Failing that, e by gum [only works if you're English, preferably Northern].

If that's correct, then it shall be my title!

And here was me thinking I'd come into a social commentary of the use of ecstasy in modern society, or something like that :p

Silly me :rolleyes:

And here was me thinking I'd come into a social commentary of the use of ecstasy in modern society, or something like that :p

Silly me :rolleyes:

I was expecting a paper on the charge of the electron. I also draw bars through my h. A cookie for the person to guess which field I am in.

I was expecting a paper on the charge of the electron. I also draw bars through my h. A cookie for the person to guess which field I am in.

The reduced Planck's constant? Well, it's physics at least. Subatomic.

Seeing as no one has picked my essay apart and feasted upon its organs, I'll take that as a thumbs-up for completion.

Now, is the Latin correct?

"De rerum erum"

Seeing as no one has picked my essay apart and feasted upon its organs, I'll take that as a thumbs-up for completion.

Now, is the Latin correct?

"De rerum erum"

Are you going for the ablative of "res"? Because "rerum" is the genitive plural.

"De" is followed by the ablative case. The ablative of "res" is "re" in the singular and "rebus" in the plural.

So, maybe "De re erum" or "De rebus erum".

"De re erum" : On the Matter of E?

"De rebus erum". : On the Matters of E?

Is that the correct translation?

"De re erum" : On the Matter of E?

"De rebus erum". : On the Matters of E?

Is that the correct translation?

It should be.

I like "On the Matters of E", I'll go with that one. Thanks.

I was expecting a paper on the charge of the electron. I also draw bars through my h. A cookie for the person to guess which field I am in.

Particle or some field of Theoretical Physics?

I prefer phi, myself. It has nowhere near as many uses, as far as I know, but it's prettier!

And pi is cooler than e because pi is just cooler than e.

Pi is cooler than e cos you can eat pi. duh

e is something you want to see when doing logarithms :)

Also, remember this mnemonic device, Americans...

(2.7-Andrew Jackson-Andrew Jackson), because Andrew Jackson was elected in 1828.

A heads up: do not cite wikipedia! Ever! Although it is certainly fun and useful for quickly obtaining data, wikipedia has not and may never obtain the level of accuracy and academic confidence necessary for citation in any researched paper. Surely it cannot be hard to find some college history of mathematics course and cite from that?

*applauds*

As a Math major, I may end up writing a paper on the same subject. Except I'm not taking any English courses anymore, and I'm only in my second semester :P.

The exact value of e*pi*i is e*pi*i.

If you wanted something else, then I guess you could take the limit definition, multiply it by i, then multiply it by the solution the upper solution of:

sum((-1)^n*x^(2n+1)/((2n+1)!),n,0,infinity) = 0

(Function notation: sum(expression, variable, lower limit, upper limit) )

And I also like phi. It is a beautiful number. (1+sqrt5)/2. Does it get any better than such an exact value?

Another probable title is "Euler Was a @#$%ing Genius"

You neglected to mention the important relationship between f(x)=e^x and the basic trigonometic functions.

You do know that e^x is a periodic function on the domain of complex numbers, right?

Yes but is there a "Monte Carlo Method" for choosing e? (I mean, come on- that's good comedy right there.)