Hi,
I have a fundamental question about math as a whole.
I can see why the constant $\pi$ would get into so many field in math (through radians, circles, trig... etc.) but why does $e$, and HOW does the constant $e$ get into so many fields of math.
I don't know differential equations, but i am learning physics, and as an answer to a differential equation we get e to the power of some paramters.
How did e jump from its base in logs, and the differential of logs, to the whole fields of math.
For instance 1/log_e n is the probability of n being prime for a very large n.

Is there an answer, and maybe i'm just missing it..?


Thanks,

Yatir

Firstly, be very wary of saying "the probability that n is prime." For fixed n, either n is prime or n isn't prime. The number 230 isn't prime some of the time and non-prime at other times. What you really mean is that the proportion of primes to non-primes approaches this number.

Anyway probably the best way to explain what's going on is to tell you that your teachers cheated. They gave you e first, then showed that it had a whole load of useful characteristics, which appear to come from nowhere. Actually, it's probably better to think of e as the number which satisfies d(e^x )/dx = e^x . You can obtain all the properties of logarithms from this equation, and from there it's obvious that the canonical base for a logarithm must be e. Any other base would cause lots of natural logarithms hanging around. So we don't start by saying "e=2.71828..." but instead say "what is the number which is natural for logarithms?" and eventually find the answer.

Another important consequence of the differential formula is that e^x has a particularly simple Taylor expansion which is convergent for all real numbers. This explains why many innocent looking expressions converge to e, like (1-1/n)ⁿ .

If you apply the same train of thought to pi, you'll start to ask yourself, "Why do we define radians the way we do? Why do we use radians as the canonical choice of angle?" Trig involves pi lots but radians are a weird choice if you just decide the total angle in a circle - why 2pi? The correct way to approach this is instead to ask "what number should we use in order to make the Taylor expansion work easily? Is there a canonical expression?" You then find that 2pi is the obvious choice.

Hope that gives you an idea.

-Dave

Dave, I see where you're getting to...
But, if i recall correctly, e was first found by trying to diffirintiate logs, and then the base of natural logs was decided do be e because it caused the differential to be much simpler.
After choosing, e as the base of the natural logarithms, i can see why, d(e^x )/dx=e^x . But you can not find all of this without first find e when try to differintiate logs and find a constant e=(n+1/n)ⁿ . You can than use the binom of newton to find that e=1+1/2!+1/3!...

What i mean is that e was found when trying to find what is d(log_a x)/dx and than found that it involves a constant, e. Now, by choosing e as a base of logs, the differntitation is a lot simpler...

One more thing, I fail to see how from d(e^x )/dx you can get all the rules of logarithms...

Thank you very much,
And i'm sorry if it seems that i'm attacking you, really i'm not. It is just that i fail to get it...


Thanks,

Yatir

I don't mean that people found e by considering derivatives. Most discoveries happen tangentially to the real use of what has been discovered, and only later is it realised the value of the discovery. So it doesn't really matter how e first came about; instead, people found one of the properties of e first and later discovered many others until eventually the real significance was realised.

If you look for a function f(x) which satisfies df/dx=f(x), you'll see that f(x)=e^x (up to multiplication by constants anyway). If you let this be the definition of the number e, you can find the Taylor expansion very easily, and this gives the expression e=1+1/2!+... without needing to know anything about logs or other identities. You don't even need Taylor theory if you treat the integral as an operator; the expansion of e is pretty trivial so long as you know that the differential equation is satisfied.

If you take f(x)=e^x and define log(x) to be the inverse function, you immediately have all rules of logarithms. For example if
log(x)+log(y)=z
then we have
e^z=e^(log(x)+log(y))
=e^(log(x))*e^(log(y))
=xy
so that log(xy)=z=log(x)+log(y).

So long as you know how indices work, you can immediately derive all the rules of logarithms in this way. Of course, log has now been defined as the inverse of a particular function, which may not be what you wanted to do in the first place. But if you look at it from this perspective, it does seem much more natural.

-Dave

Thanks,
it helped a lot.


Yatir