Mathematical significance of the number 2.7?

[X-Rated30]

Any math whizzes on the board?

I was reading an article on the GM Volt saying it got like 230mpg. Unfortunately, it is going to cost $40,000. A Prius is only like $20,000 so I was wondering if the thing would ever pay for itself. So I took the price difference of $20,000, and divided it by what I guessed might be the average price of a gallon of gas over the next couple of years, $2.70. Thats how many gallong of gas you can get with $20K. Then I multiplied it by the mileage I get, 20mpg, and you would have to drive 148,148.148 miles to pay for the difference in the cars. (actually more, but that's beside the point.)

I noticed the 148,148.148 and thought it was odd, so I decided to see if anything unusual happened if I used 10,000 and divided by 2.7. I came up with another repeating number, 3703.70370370370370370... I thought it was odd again, so I played with it a little. I multiplied it by 3 and got 11111.1111111... Again odd. I got back to the original 3703.70370... and divided it by three and got 1234.56790123456790...

Now I am thinking I am on to something, like 2.7 is really the number of the beast or the true value of Pi.

3703.70370370370370370 divided by 4 = 925.925925925925...

3703.70370370370370370 times 4 = 14814.8148148148...

3703.70370370370370370 divided by 5 = 740.740740740740...

3703.70370370370370370 times 9 = 25925.9259259259...

So what is significant about 2.7 that you get all the repeating numbers and the 11111.11111111 and 1234.56790123456790?:willy_nilly:

 

[old377guy]

this is kinda fun and kinda wasting a bunch of time on my calculator................

 

[Bobcat]

and another hundred thousand to find your dignity after being seen in either car !:sifone:

 

[X-Rated30]

and another hundred thousand to find your dignity after being seen in either car !:sifone:

true.

 

[Dude! Sweet!]

and another hundred thousand to find your dignity after being seen in either car !:sifone:

I saw some fat broad in a Smart Car this morning as I was walking into my office and acatually started laughing, out loud, uncontrollably... at her. I'm actually cracking up again just thinking about it.

Ah thank you, ironically named "Smart" car...

 

[MarylandMark]

Imagine Sunkin & Fundy stuffed up in to a Smart car!

 

[X-Rated30]

No answers from the math whizzes. Just try any number - you get a repeating number eventually.:ack2:

 

[cigdaze]

No significance.

It's close, however, to the the Euler number, "e," which is the number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x=0 is exactly 1. The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

e=2.71828182845904523536….

 

[LaughingCat]

Cig,

Your post and signature work well together.

 

[Ratickle]

Cig,

Your post and signature work well together.

Sometimes he's just scary...........:leaving:

 

[Buoy]

I saw some fat broad in a Smart Car this morning as I was walking into my office and acatually started laughing, out loud, uncontrollably... at her. I'm actually cracking up again just thinking about it.

Ah thank you, ironically named "Smart" car...

It's funny we had that conversation yesterday. I want to find a Smart car in a parking lot some where and just go tip it over - kinda like cow-tipping.
I had one of those things pass me on the freeway the other day, I was doing 70, and just thought, if something happens to that car, it's gonna be about as thick as a penny when my 2500 Av either crushes it into the truck in front of him, or runs it over...
I'd hate to be responsible for the death of someone.

No significance.

It's close, however, to the the Euler number, "e," which is the number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x=0 is exactly 1. The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

e=2.71828182845904523536….

Nick, thanks for letting me in on that secret that you just make this Bullchit up because you know that no one will refute it (because they can't even wrap their head around it):rofl::willy_nilly:

 

[Ratickle]

Nick, thanks for letting me in on that secret that you just make this Bullchit up because you know that no one will refute it (because they can't even wrap their head around it):rofl::willy_nilly:

Yeah, this is the real one.... (Cut and paste):sifone:

The Euler numbers, also called the secant numbers or zig numbers, are defined for by

where is the hyperbolic secant and sec is the secant. Euler numbers give the number of odd alternating permutations and are related to Genocchi numbers. The base e of the natural logarithm is sometimes known as Euler's number.

A different sort of Euler number, the Euler number of a finite complex , is defined by


This Euler number is a topological invariant.

To confuse matters further, the Euler characteristic is sometimes also called the "Euler number," and numbers produced by the prime-generating polynomial are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47).

Some values of the (secant) Euler numbers are

In the area of number theory, the Euler numbers are a sequence En of integers defined by the following Taylor series expansion:


where cosh t is the hyperbolic cosine. The Euler numbers appear as a special value of the Euler polynomials.

The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A028296 in OEIS) have alternating signs. Some values are:

E0 = 1
E2 = −1
E4 = 5
E6 = −61
E8 = 1,385
E10 = −50,521
E12 = 2,702,765
E14 = −199,360,981
E16 = 19,391,512,145
E18 = −2,404,879,675,441
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/or change all signs to positive. This encyclopedia adheres to the convention adopted above.

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics; see alternating permutation.


[edit] Asymptotic approximation
The Euler numbers grow quite rapidly for large indices as they have the following lower bound



(Sloane's A000364).

The slightly different convention defined by

(16)
(17)

is frequently used. These are, for example, the Euler numbers computed by the Mathematica function EulerE[n]. This definition has the particularly simple series definition

(18)

and is equivalent to

(19)

where is an Euler polynomial.

The number of decimal digits in for , 2, 4, ... are 1, 1, 1, 2, 4, 5, 7, 9, 11, 13, 15, 17, ... (Sloane's A047893). The number of decimal digits in for , 1, ... are 1, 5, 139, 2372, 33699, ... (Sloane's A103235).

The first few prime Euler numbers occur for , 6, 38, 454, 510, ... (Sloane's A103234) up to a search limit of (Weisstein, Mar. 21, 2009). These correspond to 5, 61, 23489580527043108252017828576198947741, ... (Sloane's A092823). was proven to be prime by D. Broadhurst in 2002.

The Euler numbers have the asymptotic series

(20)

A more efficient asymptotic series is given by

(21)

(P. Luschny, pers. comm., 2007).

Expanding for even gives the identity

(22)

where the coefficient is interpreted as (Ely 1882; Fort 1948; Trott 2004, p. 69) and is a tangent number.

Stern (1875) showed that

(23)

iff . This result had been previously stated by Sylvester in 1861, but without proof.

Shanks (1968) defines a generalization of the Euler numbers by

(24)

Here,

(25)

and is times the coefficient of in the series expansion of . A similar expression holds for , but strangely not for with . The following table gives the first few values of for , 1, ....

Sloane
1 A000364 1, 1, 5, 61, ...
2 A000281 1, 3, 57, 2763, ...
3 A000436 1, 8, 352, 38528, ...
4 A000490 1, 16, 1280, 249856, ...
5 A000187 2, 30, 3522, 1066590, ...
6 A000192 2, 46, 7970, 3487246, ...
7 A064068 1, 64, 15872, 9493504, ...
8 A064069 2, 96, 29184, 22634496, ...
9 A064070 2, 126, 49410, 48649086, ...
10 A064071 2, 158, 79042, 96448478, ...

 

[Airpacker]

No significance.

It's close, however, to the the Euler number, "e," which is the number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x=0 is exactly 1. The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

e=2.71828182845904523536….

Friggin rocket scientist :26:

 

[MikeyFIN]

I saw some fat broad in a Smart Car this morning as I was walking into my office and acatually started laughing, out loud, uncontrollably... at her. I'm actually cracking up again just thinking about it.

Ah thank you, ironically named "Smart" car...

I hate those...We were doing a 60 miles trip with one and all we had was my bag of clothes with me and the driver at ca 240lbs and me at 160 and that crap couldn´t take a 4 grade hill without downshifting TWICE @ WOT.

No matter how you calculate GM ( and the 2 others) should go back to build real cars, rearwheeldriven, safe and cheap and somewhat appealing too.

I´d make the company do "world cars" with an Opel/Saab front wheel drive basis.
Sporty and midsize cars with a global market too ( Ask Holden to help).
And then Finally the Full Size cars rear or allwheeldriven for the upscale market.
And make them honestly a bit retro and actually more like Drivers cars.

Thursday I saw a 69 GTO Judge convertible in Helsinki and I know the car from the past..now it had the proper orange paint and the RAIV engine sounded healthy.
I am glad we see those still being used overhere and let me tell you NOBODY here would buy a GM volt....and that makes GM REVOLT.

 

[LaughingCat]

I officially have brain matter dripping from my ear.

:willy_nilly:

 

[cigdaze]

I officially have brain matter dripping from my ear.

:willy_nilly:

:rofl: :willy_nilly:

 

[X-Rated30]

Yeah, this is the real one.... (Cut and paste):sifone:

The Euler numbers, also called the secant numbers or zig numbers, are defined for by

where is the hyperbolic secant and sec is the secant. Euler numbers give the number of odd alternating permutations and are related to Genocchi numbers. The base e of the natural logarithm is sometimes known as Euler's number.

A different sort of Euler number, the Euler number of a finite complex , is defined by


This Euler number is a topological invariant.

To confuse matters further, the Euler characteristic is sometimes also called the "Euler number," and numbers produced by the prime-generating polynomial are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47).

Some values of the (secant) Euler numbers are

In the area of number theory, the Euler numbers are a sequence En of integers defined by the following Taylor series expansion:


where cosh t is the hyperbolic cosine. The Euler numbers appear as a special value of the Euler polynomials.

The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A028296 in OEIS) have alternating signs. Some values are:

E0 = 1
E2 = −1
E4 = 5
E6 = −61
E8 = 1,385
E10 = −50,521
E12 = 2,702,765
E14 = −199,360,981
E16 = 19,391,512,145
E18 = −2,404,879,675,441
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/or change all signs to positive. This encyclopedia adheres to the convention adopted above.

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics; see alternating permutation.


[edit] Asymptotic approximation
The Euler numbers grow quite rapidly for large indices as they have the following lower bound



(Sloane's A000364).

The slightly different convention defined by

(16)
(17)

is frequently used. These are, for example, the Euler numbers computed by the Mathematica function EulerE[n]. This definition has the particularly simple series definition

(18)

and is equivalent to

(19)

where is an Euler polynomial.

The number of decimal digits in for , 2, 4, ... are 1, 1, 1, 2, 4, 5, 7, 9, 11, 13, 15, 17, ... (Sloane's A047893). The number of decimal digits in for , 1, ... are 1, 5, 139, 2372, 33699, ... (Sloane's A103235).

The first few prime Euler numbers occur for , 6, 38, 454, 510, ... (Sloane's A103234) up to a search limit of (Weisstein, Mar. 21, 2009). These correspond to 5, 61, 23489580527043108252017828576198947741, ... (Sloane's A092823). was proven to be prime by D. Broadhurst in 2002.

The Euler numbers have the asymptotic series

(20)

A more efficient asymptotic series is given by

(21)

(P. Luschny, pers. comm., 2007).

Expanding for even gives the identity

(22)

where the coefficient is interpreted as (Ely 1882; Fort 1948; Trott 2004, p. 69) and is a tangent number.

Stern (1875) showed that

(23)

iff . This result had been previously stated by Sylvester in 1861, but without proof.

Shanks (1968) defines a generalization of the Euler numbers by

(24)

Here,

(25)

and is times the coefficient of in the series expansion of . A similar expression holds for , but strangely not for with . The following table gives the first few values of for , 1, ....

Sloane
1 A000364 1, 1, 5, 61, ...
2 A000281 1, 3, 57, 2763, ...
3 A000436 1, 8, 352, 38528, ...
4 A000490 1, 16, 1280, 249856, ...
5 A000187 2, 30, 3522, 1066590, ...
6 A000192 2, 46, 7970, 3487246, ...
7 A064068 1, 64, 15872, 9493504, ...
8 A064069 2, 96, 29184, 22634496, ...
9 A064070 2, 126, 49410, 48649086, ...
10 A064071 2, 158, 79042, 96448478, ...

Well, 2.7 is cooler.:26:

 

[BBB725]

You just happen to get a number that when dividing 1 by the number got a repeating decimals just like 3.3 or 3.7.

1/2.7=.370370

1/3.3=.303030

1/3.7=.270270270

 

[DollaBill]

I used to hate math in school but over the past 10 years I've actually warmed to it quite a bit. Getting interested in quantum physics lately

 

[Chris]

I was in a truck stop restaurant on Saturday listeninbg to a truck driver talking to a waitress with a strong eastern Eurpoean accent discuss her Intro to Algebra studies for her GED. The truck driver's take on the subject was pretty entertaining.

"Well how the hell much is "X" anyway? Is that ten?"