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People make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never give me examples. Where is the number $e$ used? Also, how did Euler come up with the concept of some number $e$ that has the following property: $$e=\sum\limits_{n=0}^\infty \dfrac{1}{n!}=\dfrac{1}{0!}+\dfrac{1}{1!}+\dfrac{1}{2!}+\cdots$$ I am just curious why people make such a big deal out of $e$.

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    $\begingroup$ Functions of the form $c e^x$ with $c$ constant are the unique functions whose derivative (slope at a given point) is equal to the original function. This is why exponential growth is so useful in many areas. $\endgroup$ – user61527 Mar 11 '14 at 2:59
  • $\begingroup$ The fact that the points on the unit circle are precisely the complex numbers of the form $e^{i\theta}$ for real $\theta$ should be enough to convince anyone that $e$ has very special properties. $\endgroup$ – MPW Mar 11 '14 at 3:09
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    $\begingroup$ If you don't want to use it we won't force you. $\endgroup$ – Will Jagy Mar 11 '14 at 3:30
  • $\begingroup$ Without it, trigonometry and engineering, to name just two, would become unbearably harder to either apply or understand. It simplifies things tremendously, especially when coupled with complex or imaginary numbers. $\endgroup$ – Lucian Mar 11 '14 at 4:14
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    $\begingroup$ All the comments above are basically saying “It’s amazing, trust us” without actually answering the question and showing some applications. $\endgroup$ – STO Apr 2 '20 at 18:05
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$e$ is fundamental in mathematics. Aside from the awesome properties of $e$, such as $e^{i \pi}+1=0$ and the fact that $$\frac{d}{dx} e^x=e^x,$$ it is also found in equations that directly relate to everyday phenomena. For instance, the normal distribution is represented by the probability density function $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}. $$ It also appears in Newton's law of cooling/heating, in the solution to the differential equation $$\frac{dT}{dt}=-k(T-T_0).$$

Alongside these, $e$ appears in the solutions of many differential equation that model anything from electric circuits to spring-mass systems. As for how Euler showed that $$e=\sum_{n=0}^{\infty}\frac{1}{n!}$$ I am not sure.

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  • $\begingroup$ While I never looked up Euler's proof of the last identity, I'd like to remark that it is an easy exercise about Taylor expansions. $\endgroup$ – Stefan Mesken Nov 4 '16 at 2:35
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$e$ is just as important as $\pi$ in mathematics having uses in pretty much every field. For example, $$e=\lim\limits_{x\to \infty}\left(1+\dfrac{1}{x}\right)^x$$ One of the most beautiful examples of its importance would be relating trigonometric functions to hyperbolic functions using the identity: $$e^{ix} = \cos(x)+i\sin(x)$$ For example: $$\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$$ Using these identities can greatly simplify the computation of antiderivatives of rational functions involving trigonometric functions.

One also must note the beauty of: $$e^{i\pi}+1=0$$ $e$ is used to compute the compound interest of a bank account which is compounded continuously.

Many integral transformations such as the Fourier Transform and Laplace Transform make use of e to map a function into different domains in order to make its computation more simple.

$e$ can be used to parameterize the unit hyperbola. $e$ also defines the factorial function or more generally the gamma function which has uses all throughout mathematics. The uses of $e$ are seemingly endless as the number keeps popping everywhere in mathematics in all types of problems.

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Eular's identity pops up everywhere, in calculus, differential equations and even probabilities. For example in elementary probability theory, it shows up in the Poisson distribution general formula which is used to calculate the probability of an event occurring given that we know the rate at which it happens $$P(X=k)=\frac{\lambda^k}{k!} e^{-\lambda}$$

Furthermore, the hyperbolic identities are defined in terms of e.

There are so many applications to e, that I can pretty much write an entire book on it (and someone probably has). It is very important to get comfortable with it as if you're doing math, it will show up everywhere.

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Eular's constant also describes the decay seen in blood concentration after a bolus of intravenous drug is administered: $$C_p=X_0\sum_{i=1}^nA_ie^{-\lambda_i t} $$ Where:$$$$ $C_p$ is drug blood concentration, $X_0$ is dose given, $A_i$ and $\lambda_i$ are hybrid rate constants and $t$ is time. $i$ increases by one for every additional compartment to transform the equation from mono-exponential to poly-exponential.

(Ref: Shafer SL, Gregg KM: Algorithms to rapidly achieve and maintain stable drug concentrations at the site of drug effect with a computer-controlled infusion pump. J. Pharmacokinet. Biopharm. 20:147–69, 1992)

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There are a lot of reasons that $e$ is important, but I think that there are an equal number that show that the exponential function $\exp(t)$ is important. Here is one that I did not see mentioned:

$t \mapsto e^{ta}$ is for every fixed $a \in \mathbb R$ a homomorphism from the additive group on the real numbers onto the multiplicative group, $\phi: (\mathbb R,+) \to (\mathbb R,\cdot)$. In particular, if we require this function to be differentiable, we see that $\phi(x+y)=\phi(x)\phi(y)$ implies by differentiating both sides, $\phi^{\prime}(t+0)=\phi(t)\cdot \phi^{\prime}(0)$. But since it is a homomorphism, $f(0)=1$, so we see that the only solution is $\phi(t)=e^{ta}$. So, every differentiable group homomorphism of this type is the exponential function.

One can generalize this function to see that for a matrix $A \in L_n(\mathbb R)$, we have a function $$e^{tA}=\sum_{k=0}^{\infty}\frac{t^kA^k}{k!}.$$ Then this is a homomorphism from $\mathbb R \to GL_n(\mathbb R)$, a generalization, since $(\mathbb R, \cdot)=GL_1(\mathbb R)$.

This tells us for example that if we associate real numbers to rotations: $$x \mapsto \begin{bmatrix}\cos(x)& \sin(x)\\ -\sin(x) & \cos(x)\end{bmatrix}$$ then since this is a group homomorphism $\mathbb R \to GL_2(\mathbb R)$, it is of the form $e^{tA}$.

This can all be found in Vinberg's book Linear Representations of Groups.

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