Two questions about Euler's number $e$

Question

I am on derivatives at the moment and I just bumped into this number $e$, "Euler's number" . I am told that this number is special especially when I take the derivative of $e^x$ , because its slope of any point is 1. Also it is an irrational ($2.71828\ldots$) number that never ends, like $\pi$.

So I have two questions, I can't understand

1. What is so special about this fact that it's slope is always 1?
2. Where do we humans use this number that is so useful, how did Mr Euler come up with this number?

and how come this number is a constant? where can we find this number in nature?

Answer 1

You don't take the derivative of a constant. You could, but it's zero.

What you should be talking about is the exponential function, $ e^x $ commonly denoted by $ \exp(\cdot ) $. Its derivative at any point is equal to its value, i.e. $ \frac{d}{dx} e^x \mid_{x = a} = e^a $. That is to say, the slope of the function is equal to its value for all values of $ x $.

As for how to arrive at it, it depends entirely on definition. There are numerous ways to define $ e $, the exponential function, or the natural logarithm. One common definition is to define $$ \ln x := \int\limits_1^x \frac{1}{t} \ dt $$ From here, you can define $ e $ as the sole positive real such that $ \ln x = 1 $ and arrive at it numerically.

Another common definition is $ e = \lim\limits_{n \to \infty}\left(1 + \frac{1}{n}\right)^n $, although in my opinion it is easier to derive properties from the former definition.

Answer 2

Just a slight correction, as Jon Claus notes about the derivative of $e^x$:

what you may be remembering is that "$e$ is the unique real number such that the value of the derivative (slope of the tangent line) of the function $f(x) = e^x$ at the point $x = 0$ is equal to $1$.

See the Wikipedia article on Euler's number $e$ for more fascinating information:

1. The number e is the unique positive real number such that

   $$\frac{d}{dt}e^t = e^t.$$

2. The number e is the unique positive real number such that

   $$\frac{d}{dt} \log_e t = \frac{1}{t}.$$

   The following three characterizations can be proven equivalent:

3. The number e is the limit

   $$e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$$

   Similarly:

   $$e = \lim_{x\to 0} \left( 1 + x \right)^{1/x}$$

4. The number e is the sum of the infinite series

   $$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$

   where $n!$ is the factorial of n.

5. The number e is the unique positive real number such that

   $$\int_1^e \frac{1}{t} \, dt = 1.$$

---

The number $e$ is of eminent importance in mathematics, alongside $0, 1, \pi, \;\text{and}\; i.$ All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity: $$e^{i\pi} + 1 = 0$$ Like the constant $π, e$ is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients.