We have talked about many different numbers and their fascinating patterns, but there is one number that is extremely important that we have never touched upon. In fact, it is the most important number in calculus. This number is known as e.
Before I tell you what e is equal to, I want to try something first. Let’s do the following problem:
(1 + 1/10)^10
Multiply it out on your calculator and you will get something like 2.5937... What if we substituted that ten with one hundred? Would it be higher or lower? Let’s see:
(1 + 1/100)^100 = 2.7048...
What about if we use 1000? Then, it becomes:
(1 + 1/1000)^1000 = 2.7169...
What about if we use a million? Well, let’s see how big this gets.
(1 + 1/1000000)^1000000 = 2.7182804...
A billion? That would be giant!
(1 + 1/1000000000)^1000000000 = 2.7182814...
It is getting closer and closer to a number that is about 2.71828. You guessed it. This number is e.
What about if we did this:
(1 + 2/1000000000)^1000000000 = 7.389055...
Turns out that that is getting closer and closer to, or converging to e2. If we were to put a three there, then it would be converging to e3. A basic formula for this would be:
(1 + x/N)^N converges to ex
Now that we know a little background on e, let’s start looking at some applications. Let’s look at compound interest for a moment. Say you had 5000 dollars in the bank and earned 5% interest. After one year, you would have:
5000(1.05) = 5250
Pretty good. Let’s say that instead of just giving you the 5% interest at the end of the year, the bank gave it to you in parts. Say after 6 months, they gave you 2.5% and then another 2.5% at the end of the year. Then, you would have:
5000(1.025)^2 = 5253.12
What if your interest was compounded quarterly, meaning that you would get 1.25% four times throughout the year. Let’s see:
5000(1.0125)^4 = 5254.73
It’s inching up there. What if your interest was compounded monthly. Then, you would have:
5000(1+.05/12)^12 = 5255.81
Okay, you earned an extra buck. How about daily. This would be:
5000(1+.05/365)^365 = 5256.34
Again, just a little bit more. How about secondly. It’s a lot of work for the bank, but it might earn you a little money. It would give you:
5000(1+.05/31536000)^ 31536000 = 5256.35
Just a cent more. It seems like this is getting closer and closer to one number, or converging to one number in the 5256.3 area. Why could this be? What is our formula for ex?
(1 + x/N)^N converges to ex
This is just like what we have been doing! x is acting as the percent interest and N is the amount of pieces we are cutting the year into. So, to find the total amount if your interest was compounded constantly would be:
If r is the rate that we are compounding the interest, this would be your formula if it were just one dollar in the bank. For more money, you would do:
With P being the original amount of money you had in the bank. If you want to know it for 3 years, you just multiply the rate by three. Our universal formula, which is very easy to remember as you will see, is:
This is a very cool formula, but nowhere close to the one I will show you right now. In fact, this formula, or equation I should say, is sometimes titled as “God’s Equation.” Since we have learned about e now, I can present it to you without any confusion. Here it is:
e^iπ + 1 = 0
Look at this for one second. It looks complicated, but we know all of the components very well. We have e, which we know to be 2.718... and the most important number in calculus. We have i, which you may remember back from the cube roots of one post or the geometric addition post. i is the square root of negative one, and is one of the most important numbers in algebra. We have π, which is 3.14... and one of the most important numbers in geometry. We use it for circle and ellipse areas and circumferences, as well as other things in subjects such as trigonometry. We also have one and zero, the foundations of arithmetic and mathematics itself.
If you look closer, you will find three operations: addition, multiplication, and exponentiality, the three most important operations in mathematics. We also have equality, the primary relationship in mathematics. How much better does it get? Next week, you will see that it does get better, but we’ll save that for then. It takes a little bit of trigonometry to prove this, and I don’t want to go into it, but it involves properties of the number i. Anyways, this equation is pretty awesome, and I’m so happy I could share it now that we have covered e.