Euler’s number and its significance in real life.
In: 7
Let’s say I open a bank and I promise you 100% annual return. If you put in a dollar, at the end of the year I add a dollar to your account. You’d end up with $2.
Now, let’s say I keep that same annual return but I *compound* it biannually. That means, half way through the year I calculate a 50% return, add it to your account then, at the end of the year, I calculate 50% again and add it. But since this second 50% includes the amount added half way through, you get a bit more. Specifically:
Starting with $1 I calculate 50% interest ($0.50) six months in and you now have $1.50. At the end of the year I calculate 50% interest again, but it’s 50% of your $1.50 which is $0.75 leaving you with a total of $2.25.
Now, let’s say I compound it, but at more frequent rates. Instead of 50% twice a year I do 25% four times a year. Now you have $2.44
Five times? $2.48
Six? $2.52
What do you think happens to the final number as we go with higher and higher frequency? It’s clearly going up, but in smaller and smaller amounts each time. Either it just goes up to infinity or it converges on a specific number. Turns out, it converges on a number and we call this number “Eurler’s number” or just e and it’s around 2.71828…
It’s irrational meaning it doesn’t repeat (indefinitely) and never ends.
As above, we can see it has applications in finance. But it also has lots of other applications, especially anything involving what we call exponential growth, which a lot of living systems do, so you can see e crop up a lot in biology. We can use it when modeling radioactivity and it can connect different branches of mathematics as well.
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