#### Euler's number

$$e$$ (2.718281828459045…), also known as Euler’s number, is a mathematical constant that shows up when the rate of change in a quantity is proportional to the current value. Some examples from nuclear medicine are radioactive decay and attenuation of gamma photons.

##### How was $$e$$ discovered?

Value of $$e$$ was discovered by Leonard Euler in 18th century, while trying to solve a problem of compound interest, previously proposed by Jacob Bernoulli. The formula for annual compound interest is $$P \times \left(1 + r/n \right)^n$$, where; $$P$$ is principal, $$r$$ is the interest rate and $$n$$ is the number of times interest is compounded per year. For the sake of simplicity, imagine that \$1 is invested at an annual interest rate of 100% (=1 as a decimal), so that the formula becomes $$\left(1 + 1/n \right)^n$$. What happens if the interest is compounded in shorter and shorter time periods ($$n$$ is increased)?

$$n$$ $$(1 + 1/n)^n$$
1 (annually) 2
2 (biannually) 2.25
4 (quarterly) 2.44140625
12 (monthly) 2.61303529…
52 (weekly) 2.69259695…
365 (daily) 2.71456748…
8,760 (hourly) 2.71812669…
525,600 (minutely) 2.71827924…
31,536,000 (every second) 2.71828178…
$$\infty$$ (continuously) $$e$$

As $$n$$ gets closer to infinity, the result converges around 2.71828…

This transcendental number, called $$e$$, represents the base rate of change, shared by all exponentially changing processes and can be defined as;

$e = \lim\limits_{n\to \infty} \left(1 + \dfrac{1}{n} \right)^n$

Apply binomial theorem on this definition and you’ll get an even fancier one;

$e = \sum\limits_{n=0}^\infty \dfrac{1}{n!} = \dfrac{1}{0!}+\dfrac{1}{1!}+\dfrac{1}{2!}+\cdots$

##### Meaning of $$e^x$$ and $$\ln (x)$$
• $$e^x$$ gives the amount of growth with a certain rate of change, after a certain amount of time, where $$x = rate \cdot time$$.
• $$\ln (x)$$ gives the amount of time (and rate) needed to reach a certain level of growth $$(x).$$
##### Examples from nuclear medicine

$$N_t = N_0 e^{-\lambda t}$$

where;
$$N_t =$$ quantity at time $$t$$,
$$N_0 =$$ initial quantity,
$$\lambda =$$ decay constant $$(0.693\,/\,t_{1/2})$$.

###### Attenuation of gamma photons

$$I = I_0 e^{-\mu x}$$

where;
$$I =$$ intensity of photons after attenuation,
$$I_0 =$$ initial intensity of photons,
$$\mu =$$ linear attenuation coefficient of medium,
$$x =$$ distance travelled in medium.

##### Some pretty aspects of $$e$$
• At any point, the slope of $$e^x$$ is equal to the value of $$e^x$$.
• The area under curve $$e^x$$, from $$-\infty$$ up to any x, is also equal to $$e^x$$.
• The global maximum of $$\sqrt[x]{x}$$ occurs at $$x = e$$.
• Probably the most beautiful equation in the universe: $$e^{i\pi} + 1 = 0$$.