Let's set our eyes in a very important function, the exponential function. For this, we need to define Euler's number. Euler's number is quite a special number of mathematics. It appears in many branches and it can be defined in many ways. One of them is by the numerical value 2.71828182, etc. The decimal never terminates because a number is irrational, which means it cannot be expressed as a ratio between two integers. Another way to define it is using the expression 1 plus 1 over n to the n. Let's look at this expression for several values of n. For 1, it's 1 plus 1 over 1 to the one which is two. For 10 is 1 plus a tenth to the 10, which is 2.594. For 100 is 2.705. For 1,000 is 2.717. As you notice, the more we increase n, the more these converges to a certain number. The actual number that this gets closer and closer is e, 2.71828182, etc. That's how we're going to define the number e and e has a very particular property. If you consider the function f of x equals e to the x, that function is its own derivative. The derivative of f of x is also f of x. That particular property makes sure that e appears in many places in science, statistics, probability, etc. Now, the way I like to define e is using interest rates. Let's look at the following problem. Let's say that you're trying to find the best bank where to put your money. Bank 1 says the following. Bank 1 says, I'm not going to give you any money for a year. But after a year, I will give you as interest, 100 percent of your money. That means all your money once a year. Means if you put in $5 today in a year, they give you another $5. That's pretty good, but you continue looking. Let's go to bank number 2. Bank number 2 says, well, I'm going to give you 50 percent of your money every six months. If you put in $10 today, in six months, you're going to have $15 because bank 2 gives you half of your money twice a year. Bank 3 is slightly different. This one says, I'm going to give you 1/3 of your money three times a year, so 33 percent every four months. In summary, bank 1 gives you all your money once a year, bank 2 gives you half for money twice a year, and bank 3 gives you a third of your money three times a year. The question is, which bank is better? Let's take five seconds for you to figure out. If you need more time, feel free to pause the video and think about it. This is a very important problem. The answer is that bank 3 is the best bank of the three. Why? Well, let's say you have $1. So bank 1 after a year gives you another one, so you end up with $2. Now, the question is, what are you going to end up with in bank 2 and what are you going to end up with in bank 3? In other words, what's the percentage that you end up getting in a year in bank 2 and bank 3. Let's take a look. Bank 1, you have $1 now. In a year, you're going to have your dollar plus the extra dollar that bank 1 gives you of interests, that's $2. We can express the two as 1 plus 1 to the 1. That's going to be important later. In bank 2, now, you have $1 and in six months, you have $1 plus a half. In one year, you're going to have $8 plus a half of that. In six months, you end up with 1 plus 1/2, which is 1.5. In one year, you end up with 2.25, which is 1 plus 1/2 squared is because you're adding half of your money, which means you're multiplying it by 1 plus 1 half every time and 2.25 is bigger than two because your dollar got doubled. But this amount of money that you earned actually earned more money later. This is called accrued interests into money that gains interests and then that interests gains interests. That's why you did better with bank 2 than with bank 1. Now, let's take a look at bank 3. In bank 3, you start with $1 right now. In four months, you have $1 plus 1/3, which is 1.33. In eight months, you have your 1 and 1/3 plus a 1/3 of that. That's 1.77. In one year you have 1.77 plus 1/3 of that, which is 2.37. Again, you did well because this interest over here accrued interests and accrued more interest. This interest over here accrued more interests. Your money is making more money. At the beginning of the year, you had $1, then you have 1 plus 1/3, then you had 1 plus 1/3 squared, and then you had 1 plus 1/3 cube, and that becomes 237. In short, in bank 1 after one year, you have $2. In bank 2 after one year, you have 2.25. In bank 3 after one year, you have 2.37. That's why the money is better in bank 3. This money actually multiplies a lot. Take a look at what happens in bank 1 after four years, you're going to end up with $16. In bank 2 after four years, you're going to end up with more money. You're going to end up with $25.63. In bank 3, you're going to end up with quite a lot of money. You're going to end up with $31.57. Definitely bank 3 is much better than bank 1 and bank 2. Again, let's recall that after one year, in bank 1 you get 1 plus 1 to the 1, bank 2, you get 1 plus 1/2 to 2 and in bank 3 get 1 plus 1/3 to the 3. This is super important. Now, could you imagine a better bank than these three? Well, of course, the more subdivisions you have, the better. Let's look at bank 12. Bank 12 gives you 1/12 of your money every month. You start with $1 now, in a month, you have 1 plus 1/12, in two months, 1 plus 1/12 squared and so on until in 12 months, you have 1 plus 1/12 to the 12th and that's $2.61. You did much better. In general, bank n, would have n subdivisions. It subdivides the year into n intervals and every time, it gives you one nth of your money. You start with one on the first day. After first interval, 1 plus 1 over n. After the second interval, 1 plus 1 over n squared. If this is not super clear, let me elaborate. 1 plus 1 over n is the money you have and you add an nth of that. You have 1 plus 1 over n plus 1 over n, 1 plus 1 over n and that is the money you have plus the interests you accrue. That factors is 1 plus 1 over n square. After three intervals, you have 1 plus 1 over n cubed. After k intervals, you have 1 plus 1 over n to the k. Eventually, after the end of the year, you have 1 plus 1 over n to the n. N can be really large, so you could potentially accrue money every second or every millisecond. In summary, bank n does the following, it gives you 1 plus 1 over n to the n at the end of the year because it returns one nth of your money and times a year. Now imagine that you go around looking for banks. There's bank 1, there's bank 2, there's 3. Bank 4 gives you a quarter of your money four times a year. Bank 5 gives you a fifth of your money five times a year and so on until the end of the universe. Here's bank 12 that gives you 1/12 of your money every month. Here's bank 365, that gives you 1/365 of your money every day. What's at the very end of the years? What is this bank? Well, this bank, let's call it bank infinity. What bank infinity does is it pretty much gives you instant interests. One infinity of your money infinity times a year. Let's see how much money bank infinity would give you at the end of the year. Bank 1 ends up with 1 plus 1 to the 1. Bank 2 with 1 plus 1/2 to the 2. Bank 3 is 1 plus 1/3 to the 3 and 12 is 1 plus 1/12 to the 12. Bank 365 is 1 plus 1/365 to the 365, which already is 2.7145. Bank infinity gives you, well, this is not a good mathematical expression, 1 plus 1 over infinity to the infinity. But it's basically the number that all these numbers converge to 2.25, 2.37, 2.613, 2.7145, etc. Gets closer and closer to a particular number. That number is 2.7182812 or the number e. So e is the amount of money that you get at the end of the year in bank infinity because bank infinity gives you an infinitesimal if your money every single instance. That is one way to look at the number e.