I wanna learn how mathematicians found the number e=2.7 I do not mean the way(1/1 + 1/1*2 ....),but why they found that?? How they imagine that this number will be so usefull?? What was their difficulty and they discover this number?? Thank y!!!
1. Well, within economics, if you go from discretely calculated compound interest to calculations of continuously calculated compound interest, the number "e" will appear as the most natural number base for such calculations. 2. Another way in which "e" naturally appears are in simple growth models: If the rate of increase is strictly proportional to the actual population level at that time, again "e" will occur as a naturally preferred base for the exponential growth. 1. and 2. are obviously conceptually related, but comes from two different fields of reality, so to speak.
e was used for a long time before anyone calculated its value: http://en.wikipedia.org/wiki/E_(mathematical_constant)
[tex] \int_1^e \frac {dx} x = 1 [/tex] This relationship is a key definition for my understanding of e.
Why? To find the area under the multiplicative inverse function (i.e, standard hyperbola) was a MAJOR undertaking in the 17th century. That is, the gradually developed understanding of the (natural) logarithm function was a critical step in the history of the development of modern mathematics.
In his "First course in calculus", Lang has a nice argument to introduce ##e##. It's a bit informal, but he makes it rigorous eventually. Basically, he introduces the exponential function ##f(x) = a^x##. He then proves that [tex]f^\prime(x) = f^\prime(0) a^x[/tex] Then he shows (by using graphs), that there must exist some number ##e## such that if ##a=e##, then ##f^\prime(0) = 1##. Thus holds that [tex]\frac{d}{dx}e^x = e^x[/tex] In my opinion, this is the absolute best way to think of ##e## since this is its fundamental property. I know historically it has been introduced with interest rates and stuff. But the importance nowadays of ##e## seems to be its use in derivatives.
I agree with 1MileCrash here. It's obviously a very important property, but this doesn't really make a good definition.
Actually, you are the one needing to argue for the "silliness" of Integral's comment. As yet, you haven't
That seems rather silly only after the fact when you know that Integral's integral evaluates to ##\ln(e) = 1##. Before the fact, I see Integral's integral as rather fundamental.
If e was discovered to answer the question "what is the maximum of the real valued function x^(1/x)", sure, maybe that had some important application and was an important question in this hypothetical universe, but that does not make it a good definition. "The number a such that d/dx a^x = a^x" says the same thing denotatively as "the number a such that when 1/x is integrated from 1 to a, the result is 1" except the former is more concise, and requires less to understand. Connotatively, saying that e is "a such that d/dx a^x = a^x" immediately tells me where e is going to be involved, and its significance. Clearly, e is going to make an appearance in situations where growth or decay is proportional to current value. It isn't even a leap, or a derivation, it is plainly stated. Saying "the number a such that when 1/x is integrated from 1 to a, the result is 1" does not allow me to immediately see where e would appear. Showing me that, I could not see its significance in application immediately, I would only know it as an interesting number with a relationship to 1/x.
"After the fact" requires knowledge of e in the first place, so "before the fact" is how any reasonable person should consider that definition.
Nope. If you go back to 17th century maths, the question of what number would make the the area under the hyperbolic curve equal to 1 was a fairly important question.
Basically, here you presuppose the conception of the derivative. Finding the area under a curve does not, and questions about how to evaluate it will occur independent of fruitful mathematical definitions of derivatives and integrals.
What exactly are you objecting to? That we should consider Integral's definition without knowing that the integral of 1/x evaluates to ln(x)? I would presume that Integral's definition of e does not require me to know this property about e (that it is the base of the logarithmic function whose derivative is 1/x). Intergral's definition is about area, it has nothing to do with 1/x's antiderivative as a function.
Area is not a more basic concept than a rate of change. I used the derivative notation to symbolize the rate of change just as the integral symbol has been used to symbolize area. Neither requires a fruitful mathematical definition of derivatives or integrals.
"Intergral's definition is about area, it has nothing to do with 1/x's antiderivative as a function. " QUITE so. That's why it can be regarded as more fundamental, rather than as silly.
Actually a more interesting discussion would be the universality of e: [tex] \sum_{k=0}^{\infty} \frac{1}{k!} = \lim_{n\rightarrow\infty} \left(1+\frac{1}{n}\right)^n [/tex]
Aah, happy memories! Reminds me of my student days, when I and a couple of co-students became determined to prove that identity directly. We were proud of ourselves when we managed to do so, not the least because we found it rather troublesome to achieve.