Finding the Deduction for Euler Number

[live4physics]

Hi,

Can anyone show me what´s the deduction for e Euler number ?

Thank you

 

[mathman]

Hi,

Can anyone show me what´s the deduction for e Euler number ?

Thank you

What do you mean by "deduction"?

 

[Gear300]

One way the number came about (I don't remember if it originates from this or not) was through analysis of the derivative of logarithmic/exponential functions. Evaluating the derivative of such functions involved taking the limit of an expression, which through analysis came out to be e (you could say it approaches an irrational number e).

 

[symbolipoint]

There is a YouTube video about "e" from Khan Academy, dealing with derivative of the exponential function. It might be instructive for you.

This base of the natural logarithms can also be inductively developed through compounded interest, increasing the number of compoundings per year period, and then imagining or seeing where this goes as the number of compoundings per year becomes infinite. This I have seen developed in an old Intermediate Algebra book.

 

[Integral]

e is the number which satisfies this integral

$$\int^e_1 \frac {dx} x =1$$

In other words the area under the curve of 1/x from 1 to e equals 1.

 

[HallsofIvy]

Another way of phrasing it is that e is the number such that
$$\lim_{x\to 0}\frac{e^x- 1}{x}= 1$$

 

[robert Ihnot]

Once you find that e^x is its own deratative, you can get things from the Taylor series:

$$e^x =1+x+x^2/2! +x^3/3! +++=\sum_{i=0}^{i=\infty} \frac{x^i}{i!}$$

 

[Bohrok]

Also,
$$\lim_{x\rightarrow\infty}\left(1 + \frac{1}{x}\right)^x = e$$

 

[robert Ihnot]

Also,
$$\lim_{x\rightarrow\infty}\left(1 + \frac{1}{x}\right)^x = e$$

Using this expression, you an also obtain the infinite series from the expansion:

(1+1/n)^n = 1+n/n + (n)(n-1)/n^2 *2! +n(n-1)(n-2)/n^3(3!) +++

So that taking the limit term by term becomes 1+1/1! +1/2! + 1/3! ++++as n goes to infinity.

 

[Tac-Tics]

Also,
$$\lim_{x\rightarrow\infty}\left(1 + \frac{1}{x}\right)^x = e$$

This equation without the limit is the equation for compound interest.

Say you found a bank that promised to double your money every year, compounding it daily. You start off by putting $1 in your account. After a year, you end up with pretty close to $2.71 = "e dollars".

Magically, e pops up everywhere in math. It's probably more prominent than pi.

 

[robert Ihnot]

Tac-Ticks: This equation without the limit is the equation for compound interest.

What he means is that the amount is compounded instantly, instead of every day or every month, etc.

For example if the nominal rate is 5%, then compounding instantly would give e^.05 =1.05127

Where as if it was compunded every day (1+.05/365)^365 = 1.05126, or only about a dollar difference on $100,000.

Bankers are often, or used to be, inclinded to use 360 days for the year. It makes the calculation easier--at least before computers.

 

[Tac-Tics]

What he means is that the amount is compounded instantly, instead of every day or every month, etc.

I know where it comes from, but it's nice to know the name for it. The equation by itself doesn't really help you understand where it comes from.

 

[symbolipoint]

I know where it comes from, but it's nice to know the name for it. The equation by itself doesn't really help you understand where it comes from.

More commonly called "compounded continuously" or "continuous compounding".