Number e

on 28.10.18


\( e \) (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

  • Radioactive decay

    \( N_t = N_0 e^{-\lambda t} \)

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

  • Attenuation of gamma photons

    \( I = I_0 e^{-\mu x} \)

    where;
    \( I \) is the intensity of photons after attenuation,
    \( I_0 \) is the initial intensity of photons,
    \( \mu \) is the linear attenuation coefficient of medium,
    \( x \) is the distance travelled in medium.

Some cool properties 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: \( e^{i\pi} + 1 = 0 \).