By Kanishk Nishar
Euler’s Number (pronounced ‘oiler’) is a very famous mathematical constant that describes the rate of exponential growth in the universe. I’ve explained how the number is derived and one of its properties in this article.
To understand e, we must first understand compound interest.
You must have seen the formula for compound interest by now:
This formula allows us to calculate our amount at the end of our investment period.
To calculate the amount that an investment will bring, we multiple our initial investment, our principal P, by the rate that the bank is offering and raise it to the amount of time period of our investment.
To calculate how our interest rate, we would divide the rate of interest by the amount of time periods that therate of interest is affected by.
What does that mean? Let’s say that your bank is giving you 10% interest p.a. compounded monthly. That would mean that your interest does not get an interest of 10% every year but instead an interest rate of 10%/12 every month or 0.83% every month.
Finally, we raise the equation to the amount of time that the money is deposited. For example, if you deposited the money for 5 years, then n would be 5 years or if the interest was compounded monthly, 60.
The derivation of e
Before I explain to you why eis important, I want you to imagine that you went to a particularly generous bank that was offering a compound interest of 100% every year. You deposited a dollar and got a dollar as interest. Your money was doubled.
What if the interest rate was compounded half yearly? From the above formula, you could calculate the return as:
You’ve made more money. If the rate of compounding increases, the amount also increases, yes?
When compounded monthly, you get:
When compounded daily, you get:
You notice that you get more money as the rate of compounding increasing. So, what would happen if the interest was compounded every second or every microsecond?
If you were to keep increasing the rate of compounding to infinity, you would approach e, which is roughly equal to 2.71828. Because of the way that eis derived, the number is irrational, i.e. the figures in the decimal never repeat themselves and the number never ends.
This is what we’re going to be talking about.
eis the natural rate of exponential growth that is observed in the universe.
What this means is that what happens when an item keeps growing continuously at a rate that is based upon itself. You see this in the world with regards to its population. The number of children that a billion people can produce against 400 is considerably more. Inversely, the rate of decline or exponential decay can also be inferred from our earlier formula if you were to substitute the addition sign with the subtraction sign. This equation would be applicable in the real world for things like radioactive decay.
Before I explain the derivative of , it is important to understand what derivation even means.
We learned about slopes in primary school. Slope is defined as the rate of change in y against change in x.
Take the graph below of y=2x:
When x is 5, y is 10, when x is 10, y is 20. Therefore, the slope of the function can be calculated as:
What this means is that y is increasing at twice the rate of x and that makes sense too because that is how we defined the function.
The problem with slope however is that it is only applicable when a function has a constant rate of change. The slope can be drawn as a straight line. However, what happens when a function has a shifting rate of change as in non-linear functions (i.e. the function isn’t a straight line) such as ?
The rate of change in the function isn’t constant. At x=2, y is 8 and at x=3, y is 27. The slope of these two points would be:
When x is 4, y=64 and when x is 5, y=125, the slope would then be:
The slope is much steeper at x=4 than at x=3. This shows us that the rate of change varies in non-linear equation. The question that mathematicians then asked is what is the instantaneous rate of change (as Sal would say) for any two points on the function? That is, what is the rate of change between any two points that are only slightly apart? What is the rate of change between 2 and 2.01, between 2 and 2.000001? What is the rate of change as x approaches 0?
Differentiation allows us to answer these questions.
The way that differentiation achieves this is by finding the slope of the tangent line drawn on the point whose slope, or rate of change, that we want to find.
For example, here is a zoomed in version of our earlier graph:
I want to find the rate of change at the marked point of the function. So, I drew a tangent line along the point on the graph. If I were to find the slope of the tangent, I would find the rate of change for that point in that function.
The derivative of e^x
If I were to plot on a graph, it would look like this:
What make unique is that its derivative, or its rate of change, is equal to the function itself. You can understand what I mean from the illustrated graph below.
At x=0, y=1 and it’sslope (or m) = 1, when x=1, y=2.72 and m=2.72, when x=2, y=7.39 and m =7.39, and so on and so forth.
e is unique in being the only constant that can achieve something so remarkable. That is what makes e beautiful to many people.
Thank you for reading my article.