Calculating Natural Number e: History & Methods

In summary: With a calculator, we find that e is between 2.716 and 2.718.In summary, the natural number e is a mathematical constant that is approximately equal to 2.718281828, and has many interesting properties such as being the limit of the sequence (1 + 1/n)^n as n approaches infinity, being the only number for which the rate of change of the function e^x is equal to e^x itself, and being the sum of the infinite series 1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! + ... . Its numerical value can be approximated using various methods such as the Taylor series
  • #1
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Would you please tell me how to calculate natural number e ? How that number came into being ?
Thanks
 
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  • #3
So I just realized that all this doesn't get me to the numerical value of e, just one of the nice properties of it. So if anyone wants the limit derivation for e^x, let this be your guide:

[tex]\lim_{h\rightarrow 0} \frac{n^{x+h} - n^x}{h} \Rightarrow \frac{n^xn^h - n^x}{h} \Rightarrow \frac{n^x(n^h-1)}{h}[/tex]


The limit of the product is the product of the limits:

[tex] \lim_{h\rightarrow 0} \frac{n^h}{h} \lim_{h\rightarrow 0} n^x [/tex]

L'hospitals simplifies the left to [itex] n^hln(n)[/itex]

[tex] \lim_{h\rightarrow 0} n^hln(n) \lim_{h\rightarrow 0} n^x = n^0ln(n)n^x[/tex]

Result: [tex] n^x' = n^x(ln(n))[/tex]

For some n, n^x' = n^x, and that is only when n = e.
 
  • #4
The definition is with the limit of that sequence,but the calculation is done with the series of [itex] e^{x} [/itex] for x=1.


Daniel.
 
  • #5
Or you could use the limit [tex]e = \lim_{x\rightarrow\infty} (1 + \frac{1}{x})^{x}[/tex]

Jameson
 
  • #6
As to "How that number came into being ?": it has the very nice property that the rate of change of the function ex is just ex itself- no other function has that property. That's why it was recognized as being important. I wouldn't begin to speculate as to how any number "came into being"!
 
  • #7
yet another way would be to integrate the hyperbola, and then take the inverse of that function:

[tex]\int \frac{1}{x} dx = ln(x)[/tex]

[tex] ln^{-1}(x) = e^x [/tex]

this integration could be done numerically, and a table of [itex]ln[/itex] values be built (ala Napier and friends) and then a column could be made of [itex]e^x[/itex] by indexing backward into the table.

tables of logs were actually very popular at one time (and led to the slide rule) because multiplication and division could be reduced to addition and subtraction (think algebra of exponents). this greatly benefited fields like artillery and naval navigation since complicated calculations could be done quickly by indexing into a precomputed table (same for trig functions).
 
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  • #8
of course all functions of form ce^x have that property as well, and these are the only ones.
 
  • #9
The way I was introduced to e was to try to find the exponentional which was its own derivative. So say we have a^x, we know that a^0 is 1, so try to find the value of a such that the gradient of a^x is 1 at the point (0,1). From this you get the definition of a, or as we know it e: Lim (1+(1/n))^n as n tends to infinity. I think this is how it was approached, I think Eli Maor has a book on the history of e if you're interested, it's on Amazon. Cheers, Joe
 
  • #10
You could always use:

e = 1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! + ...

which is from the Taylor series for [itex]e^x[/itex].
 
  • #11
James R: You could always use:

e = 1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! + ...

which is from the Taylor series for e^x.


This is a really effective way, that is if the student seeks a good approximation. As my professor once said, "e gets to its limit very fast."

After n terms, starting with 0, the error is less than 1/(n!) for n =1 or greater. This can be shown. Consider:

1/n! >1/(n+1)! + 1/(n+2)! +1/{(n+3)! ++++

Since 1>1/(n+1) +1/{(n+1)(n+2)} + 1/{(n+1)(n+2)(n+3)} ++++

Since for n=1, we have 1>=1/2 + 1/(2*3) + 1/(3*4) +1/(4*5) +1/(5*6) ++++++ =(number of terms)/(number of terms +1)> the series above.

So that even at n=1, we can conclude 2<e<3. At n=3 we find that 2+2/3<e<2+5/6. At n=7, we have 2+3620/5041<e<2+3621/5041.
 
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1. What is the natural number e and why is it important in mathematics?

The natural number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828. It is important in mathematics because it is a fundamental constant that appears in many formulas and mathematical models, including compound interest, growth and decay, and normal distributions.

2. How was the natural number e discovered?

The natural number e was first discovered and studied by Swiss mathematician Leonhard Euler in the 18th century. He initially called it the "base of natural logarithms" and studied its properties in relation to logarithms and exponential functions. Later, it was named after him as Euler's number.

3. What are the different methods used to calculate the natural number e?

There are several methods used to calculate the natural number e, including infinite series, continued fractions, and limits. One of the most common methods is the infinite series, which is expressed as the sum of 1/n! from n=0 to infinity. Another method is using continued fractions, where e is the limit of the sequence of convergents. Lastly, e can also be calculated using limits, such as the limit of (1+1/n)^n as n approaches infinity.

4. What are some real-life applications of the natural number e?

The natural number e has many real-life applications in various fields, including finance, physics, and statistics. In finance, it is used in compound interest calculations, which is the basis for many financial investments. In physics, e appears in equations related to exponential growth and decay, such as radioactive decay and population growth. In statistics, e is used in probability distributions, such as the normal distribution.

5. Can the natural number e be calculated to an exact value?

No, the natural number e cannot be calculated to an exact value because it is an irrational number, meaning it has an infinite number of non-repeating decimal places. Its decimal representation is never ending and non-repeating, making it impossible to calculate to an exact value. However, it can be approximated to any desired degree of accuracy using the various methods mentioned earlier.

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