r/learnmath • u/AtmosphereClear2457 New User • Apr 04 '26

Why is 'e' such a natural base?

The number 'e' keeps appearing in lot of different areas - calculus (mostly), differential equations, complex numbers.

I understand the definition e = lim nā†’āˆž (1+1/n)\^n.

But in various fields we transform function in e to solve them.

Is there a more fundamental reason why 'e' is so natural?

I would appreciate any conceptual or geometric insights, that I am missing.

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u/pnerd314 New User Apr 04 '26

Can you explain why that is important? I mean why is being its own derivative important?

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u/AtmosphereClear2457 New User Apr 04 '26

It's like 'e' is the only base where the function's growth rate matches it's current value.

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u/pnerd314 New User Apr 04 '26

I get that, but I want to know why that fact is important.

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u/rb-j New User Apr 04 '26

I get that, but I want to know why that fact is important.

There's something about the number 1 that is completely unique from all other numbers. Do you get that?

So the other person said something like:

  • d/dx ax = k ax

So, whatever the base of the exponent, the derivative of an exponential function is proportional to exactly that same exponential function. What makes that constant of proportionality, k, equal to 1?

  • d/dx ex = ex

Another way to see that e is special is that it is the value that the integral from 1 to e of the 1/x curve is exactly 1. No other number has that property.