e
This number is something I’ve never understood.
However, after watching this video I’ve got a better grasp at the situations where it’s used:
to calculate compound interest
to describe the normal distribution
to model populations
to describe exponential decay
If you increase the number of compounding periods a year, towards infinity, then you’ll get closer and closer to e dollars. (Source)
The common log allows us to talk in multiples of 10. So what does the natural log (the base e) allow us to do?
According to this source, e is referred to as a “magical number.” I agree with that! It’s also called “Euler’s number” and the “irrational constant.” Regardless, e is fundamental to growth in nature.
Here is the definition that video gives for e:
The maximum possible result after continuously compounding 100% growth over one time period.
But Better Explained tells us that we can’t simply focus on this. We have to extrapolate out to the intuition behind the number:
e is the base rate of growth shared by all continually growing processes. [It] lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.
Pat Ballew tells us that a compound number is a number with a whole part and a fractional part put together, i.e. 1 + 1/n. This is exactly what it means to compound in the case of interest, as we add a fraction (the interest) to our principal (the whole).
If you compound at certain discrete time intervals, then I don’t think you’ll approach e. This seems to only happen when a system is growing both exponentially and continuously, that is, over an infinite number of time periods.
Better Explained sums it up as follows:
Just like every number can be considered a scaled version of 1 (the base unit), every circle can be considered a scaled version of the unit circle (radius 1), and every rate of growth can be considered a scaled version of e (unit growth, perfectly compounded).So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate.
I still don’t think I’ve understood it perfectly, but these explanations are a good start.
