Euler’s number and its significance in real life.
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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.
Eulers number was originally derived by a mathematician named Jacob Bernoulli, in essence he was trying to see what happens when your compund interest compounds infinite times. Euler himself defined it later as an infinite series. Some use cases for e include approximation of really large factorials, probability distribution, waves, and calculus. A little explanation on all examples provided, a useful equation called stirlings approximation is used to estimate the value of incredibly large factorials as calculators cant go past certain values due to restrictions in data type sizes. Normal distributions, or standard bell curves are derived using exponentials with base e as well. Waves is probably its second most important use, being particularly useful in calculating AC power as sine and cosine can be rewritten as exponential functions with base e. Lastly, at the time of eulers definition, the big reason we wanted to define e as a value was so we could integrate and derive exponentials and logarithms. When either of these do not have a base of e, you end up with a factor of the natural log of the base leftover. By defining e, we could now do integrals of logarithmic and exponential functions. e has a lot of other uses in stuff like exponential decay, but im not nearly smart enough to talk about radioactive decay. Hope this helped, also sorry im on phone so hopefully this isnt too bad to read.
Euleur’s number (approx 2.718) is a number where any tangent you take along the function e^x, then the slope of that tangent is equal to e^x which is pretty cool imo.
Now that number is used a lot when simulating population growth or anytime you see exponential growth it refers to that.
When you want to simulate a situation where the “then” is calculated using “now” then you would use e somewhere in the equation.
Not really a direct real-world application, but it is very useful in applications of maths:
e^x is its own derivative. As a result, it provides a convenient basis for many applications of exponential growth, where instead of a^x you can write e^(bx) where the growth rate is then equal to b*e^(bx), i.e. b times the amount you have at that moment.
Furthermore, because e^x is its own derivative it is a common solution to differential equations, which are the type equations that most physical models are based on
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