\( e \) (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;

\( N_t = \) 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: \( e^{i\pi} + 1 = 0 \).