- #1

- 3,121

- 4

e

[inte] dx/x=1, etc."

1

In other words,

__why__choose this function to define e, and how does it most fundamentally relate to other uses of e?

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In summary, e is a constant that shows up in various natural phenomena involving continuous growth or change. It is defined as the limit of (1+1/n)^n as n approaches infinity, and is often used as the base for logarithmic functions due to its relationship with continuously changing values. It is also the solution to the differential equation where the rate of change is proportional to the value itself. Its significance and usefulness lies in its prevalence in nature and its mathematical properties.

- #1

- 3,121

- 4

e

[inte] dx/x=1, etc."

1

In other words,

Mathematics news on Phys.org

- #2

- 502

- 1

That's a good question. I'm sure I could answer it where it not so late, I'll think into it.

- #3

e = lim (1 + x) ^(1/x)

where limit takes x to zero.

Instanton

- #4

- 502

- 1

One thing that I remember is that for the graph of e

- #5

Instanton

- #6

- 152

- 0

it's natural log

integrate from 1 to x for dt/t = ln(x)-ln(1)= ln(x)

because after we define natural logarithm funtion

we fine the base for the ln function

so we defined the exponatial e

hope this can help

- #7

rate of change of X proportional to X

eg X'(t) = c*X(t). For example this very often occurs when X is a number of objects which are independent of one another: reproducing bacteria, decaying atoms. The most mathematically natural case is where c=1; though physically, there is nothing special about that unless the units of X and t are comparable.

It turns out the solution to this differential equation has the form of an exponential, X(t)=e^t, with e itself.

- #8

- #9

In this form it occurs also in other scenarios like the calculation of continuous interest in finance...

As far as considering it as a natural base for logarithms... you realize how natural it is only once you start calculating derivatives I guess...

Another possibility is to say that e is the basis of logarithm when

lim (log(1+x))/x=1

x->0

that is the logarithm that goes to zero as linearly as x does.

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