Implications of redefining base natural logarithm constant e

Question

Disclaimer: I'm no math expert!

I understand that the constant $$e$$ is expressed as follows:

$$e = \sum_{n=0}^{\infty} \frac1{n!} = 1 + \frac1{1*1} + \frac1{1*2} + ...$$

What would be the implications of defining it as:

$$e = \sum_{n=2}^{\infty}\frac1{1*2} + \frac1{1*3}...$$

I guess algebraically it wouldn't matter much as it just removes 2 from the equation. But are there other cases where the constant is used in such a way that perhaps a value 'higher than one' might be preferred?

To me it feels more logical to have this value 'between zero and one'.

Answer 1

$e$ is nothing but a name for a number, namely $$ e = 2.71828182845904523536028747135266249775724709369995\ldots $$

This number that we call $e$, has a number of interesting properties:

1. $e^x$ is its own derivative.
2. The inverse of $e^x$, $\ln x$, has the derivative $\frac{1}{x}$.
3. $e = \sum_{n=0}^\infty \frac{1}{n!}$
4. $e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$

If we called a chair a microwave, it wouldn't make it able to cook food. Just the same, naming a different number as $e$ wouldn't give it the same properties as above, it would still just be the same number, with its own (often less interesting) properties. (In fact, $e$ is the only number to satisfy have the four properties above.)