We have defined exponential functions to be those functions of the type x is mapped to a to the x. The fixed number, a greater than 1, is the base of the exponential function. There is a certain value of the base, a, which gives rise to an exponential function that is, in a sense, more natural than the others, because it arises so inevitably in a great variety of applications. This is the case in which we take for the number a, certain special number, always denoted lowercase e, and a number that is called the Euler number. It was discovered by Leonard Euler in the 18th century.
And the resulting function, the function e to the x, is referred to as the natural exponential function. It is often written with the shorthand exp as well. Now, Euler went on to calculate the number e, this special number, rather mysterious at the moment, to 20 or more decimal places. And I’m going to now look at a simple example of how the number arises just to see how it comes about naturally. Now, this is not an example that Euler himself would have treated. I don’t think he had any interest in money, oddly enough. And this is what the example is in. It’s in the theory of interest. What is the theory of interest?
The theory of interest refers to the fact that you have a capital amount of money invested on which interest is going to be paid, and so the amount of money is going to increase. The interest rate is always given in nominal terms, usually annual. However, there’s also the fact that the interest can be compounded, or not, during this nominal time period. If you have simple interest– that means there is no compounding– for example, at 5%, that means that if you have, say, 100 euros in your account initially at the beginning of the year, then at the end of one year, you will have 105 euros– Simple interest of 5%.
Mathematically, it means that your initial capital, 100, has been multiplied by a certain factor, the factor 1 plus 0.05. the 0.05 is 5 over 100, 5%. However, it’s possible that your nominal rate could be 5% and yet could be compounded every six months. This means that at the end of the first six months, you would get 2.5% credited to your account. And then the resulting six months after that, you would get another 2.5% but of the new amount that you had after six months. That’s what compounding means. To make this precise, after six months, your 100 becomes 100 times the factor 1 plus 0.05 over 2.
And then that same multiplicative factor is applied after the second period of six months. So at the end of one year, you have that multiplicative factor squared. Now, when you calculate that, you will see that it’s 105.0625. You have an extra 6 and 1/4 cents due to the compounding that you don’t have from simple interest of 5%. So compounding is very interesting for saving. The more often you compound, the more interest you get on your interest, and it’s better for the final result. So for example, you often see that money is compounded quarterly. That means every three months, four times a year. Or it’s compounded monthly. That means 12 times a year.
And these formulas will give you the amount you have at the end of the year in those cases, assuming always that the nominal rate is 5%.
Now, in general terms, then, if you call your nominal interest rate 100 r– in my previous example, r was 0.05– then you compound it n times per year, then the initial capital at the end of one year will be multiplied by the factor 1 plus r to the– over n, all to the power n. And a question arises. What would be the effect of continuous compounding, that is, compounding instantaneously at all time? The answer is, it would be the limit to this expression, the multiplicative factor, the limit of this expression as n goes to infinity. Now, this kind of concept, rigorously speaking– and the notation I’ve written here belongs to a calculus course, so we can’t do it in detail.
But let me simply say that this limit can be evaluated. Someday, perhaps, you will do so. And it turns out to be e to the r, where e is just exactly this mysterious Euler number that I introduced before. I remark that when r is equal to 0.05, this turns out to be 1.0513. So you’ve gained 13 cents by continuously compounding the interest rather than simply having simple interest. If you set r equals 1 in the previous proposition and you calculate a few values for a different n on your calculator, you can approximate the number e. And you’ll see that it’s 2.7 and then a 1, and then an 8, and so forth.
Now, what does this natural exponential function look like? Well, its graph is similar to that of our other exponential functions. As you can see, it’s a function that has positive values. It’s defined everywhere. It’s equal to 1 when x is 0. It increases quite fast when x becomes big. It goes down to 0 quite fast when x becomes negative. Also, the function is one to one or injective. That is, any value it takes, it takes only for one x. Furthermore, the function is surjective. That is, any strictly positive value is attained by the function. Now, these terms I’ve just used, you may not be familiar with, or you may have slightly forgotten them.
But we’ll be reviewing them very soon, because they will play an important role.