Euler's Number (e)
Euler's number (not to be confused with Euler's constant, which is something else altogether) is so-called because it was the Swiss mathematician and physicist Leonhard Euler (1707 - 1783) who gave it the label e, although it was not Euler that discovered it. The number is thought to have been discovered by another Swiss mathematician, Jacob Bernoulli (1655-1705), while he was working on the solution to a problem involving compound interest. So what was it exactly that Bernoulli found? Well, suppose we wanted to borrow one hundred pounds (£100) - or whatever currency you prefer to work with - for a period of ten years. Let's assume a rate of interest of ten percent (10%) per year.
If we apply simple interest (interest that is not compounded at all), we would just pay ten percent interest each year on the original loan (called the principal), which over ten years would come to one hundred per cent of the original loan, or one hundred pounds (we would of course have to pay back the original one hundred pounds as well). However, the interest is going to be compounded. This means that at regular intervals during the period of the loan, the amount of interest currently owed is added to the original loan. Each time this happens, the total amount owed increases, and we must thereafter pay the agreed rate of interest on the total amount outstanding. This means we pay more interest (as a percentage of the original loan) as time goes by.
Let's see how this works. We'll assume for the moment that the interest on our ten year loan is compounded annually. The table below shows how the amount of interest paid each year increases. We used a spreadsheet to do the calculations. By the end of the ten-year period, the total amount of money that must be repaid is £259.37, or the original sum of one hundred pounds multiplied by 2.5937.
|Year||Principal + interest||Interest (10% p.a.)||Total owed|
There is actually a standard formula for calculating the total amount A to be repaid:
|A = P × (1 +||r||) nt|
In this formula, P is the principal, r is the annual rate of interest expressed as a decimal, t is the number of years over which the money is borrowed or invested, and n is the number of times per year the interest is compounded. If we apply the formula to our loan (which you will remember is one hundred pounds for ten years at a rate of ten percent), we get the following result:
|A = 100 × (1 +||0.1||) (1 × 10)||= 259.37|
This is the same result we achieved previously. Obviously, we will pay significantly more interest on a loan (or accumulate more interest on an investment) if the interest is compounded. What happens, though, if we compound the interest more frequently? Suppose that instead of compounding the interest once a year, we compound it once a month. If we do that, the value of n becomes twelve (12), and our formula becomes:
|A = 100 × (1 +||0.1||) (12 × 10)||= 270.70|
The total has increased by just over ten pounds. A significant amount, but perhaps not as much as we might have expected given the fact that we are compounding the interest twelve times as often as before. Under these terms, the amount we must repay at the end of the ten-year period is £270.70 - the original sum of one hundred pounds multiplied by 2.7070. Let's make things even more interesting (no pun intended). Instead of compounding the interest monthly, we'll compound it daily. This means we will take a value for n of three hundred and sixty-five (365). For the purposes of this exercise, we will ignore the fact that there would normally be at least two leap years during the period of the loan. Our formula now becomes:
|A = 100 × (1 +||0.1||) (365 × 10)||= 271.79|
So, even though we are now compounding the interest every day, the total amount to be repaid is only one pound and nine pence (£1.09) more than the amount we would have to repay if interest was compounded monthly. This amount is the original sum multiplied by 2.7179. You can probably see by now that, as the frequency with which we compound the interest increases, the value of this multiplier seems to be converging on some fixed point. To confirm this, let's apply the formula one more time. This time we'll set the interest to be compounded every second of every day for ten years! To find the value for n, we must multiply the number of days by the number of seconds in each day (for the sake of accuracy, this time we will take leap years into account and add a quarter of a day to each year):
n = 365.25 × 24 × 60 × 60 = 31,557,600
Plugging this value into our formula, we get:
|A = 100 × (1 +||0.1||) (31,557,600 × 10)||= 271.83|
After all that, the total to be repaid is only four pence more than we got when we were compounding interest every day. It seems like our multiplier is converging on a value of 2.7183. In fact, if you do the above calculation on a calculator you should get a figure for the multiplier of 2.7182818241521875836961248899295 (the exact figure will depend on how many decimal places your calculator will display). Jacob Bernoulli must have recognised the significance of this number - at least in part. The full significance would not become apparent until much later. Bernoulli realised that the number (we'll call it e) approaches a finite limit as n approaches infinity. This limit can be expressed as follows:
|e =||lim||(1 +||1||) n|
There have been many attempts to calculate the value of this number, which we know today as Euler's number or just e. Its exact value cannot be calculated because it is an irrational number (a number that cannot be expressed as the quotient of two integers, and whose decimal representation does not terminate or repeat). Like the mathematical constant Pi (π), it is also a transcendental number, which means that it cannot be a root of a non-zero polynomial equation with rational coefficients. At the time of writing, the value of e is reported to have been calculated to an accuracy of over one trillion digits. Fortunately, we can usually get by with a much less accurate approximation! The known value of e to fifteen decimal places is:
e = 2.718281828459045
From the above, we can derive a formula involving e that gives us the limit (i.e. the maximum value possible) of the total amount to be repaid against a sum of money borrowed for a given period of time, at a specific rate of interest compounded over that period. The formula is:
A = Pe rt
In this formula, as before, P is the principal, r is the annual rate of interest expressed as a decimal, and t is the number of years over which the money is borrowed. Let's suppose that we borrow three hundred and fifty pounds (£350) over seven years, at an annual interest rate of 7.5%, continuously compounded. To discover how much we are going to have to repay at the end of the seven years, we could carry out the following calculation:
A = Pe rt = 350 × 2.7183 (0.075 × 7) = 591.66
So Euler's number, or e, can be very useful for calculations involving compound interest. It turns out, however, that this seemingly innocuous number is also of vital importance in many areas of science, engineering and mathematics. Let's turn our thoughts now to exponential functions. You have no doubt come across functions such as y = x 2. This function takes the variable x as its input, and outputs the value of x 2. Here we have the variable x being raised to the power of the constant value two (2). The variable is the base, and the constant value is the exponent. In an exponential function, the positions are reversed. Suppose we have the exponential function y = 2 x. Here, the constant value is the base, and the variable is the exponent. Consider the following illustration, which shows the graphs of the functions y = 2 x and y = 3 x.
The graphs of the functions y = 2 x and y = 3 x
Both of these exponential functions have similar curves, but whereas the curve for y = 2 x starts off slightly ahead of the curve for y = 3 x, the two curves change places at x = 0. From that point onwards, as the value of x increases, the curve for y = 3 x climbs more steeply. It is interesting to look at how quickly these functions are growing. We'll start by looking at the function y = 2 x. The function that describes the growth rate of this function is y = ln 2 · 2 x. Here are the graphs of these two functions side by side:
The graphs of the functions y = 2 x and y = ln 2 · 2 x
As you can see, the graph of function y = ln 2 · 2 x follows a similar path to that of y = 2 x. However, it gradually falls behind as the value of y = 2 x increases. Now let's see what happens when we carry out the same exercise for the function y = 3 x. The function that describes the growth rate of this function is y = ln 3 · 3 x. Here are the graphs of the two functions:
The graphs of the functions y = 3 x and y = ln 3 · 3 x
Again, you can see that the graph of function y = ln 3 · 3 x follows a similar path to the graph of function y = 3 x. The graphs are closer together, but they still diverge. This time, the growth rate is slightly ahead of the function itself. You might be starting to suspect that there is a constant number c such that the graph of the function y = c x and the graph of the function y = ln c · c x follow exactly the same path. You might feel that if this number does indeed exist, it lies somewhere between two and three - maybe a little bit closer to three than two. Given what you have learned so far, you might even hazard a guess that the number you are looking for is e. Let's test this theory. Here are the graphs of the functions y = e x and y = ln e · e x.
The graphs of the functions y = e x and y = ln e · e x
As you can see, the graphs are indeed identical. The function y = e x is known as the exponential function, and the function y = ln e · e x describes its growth rate. A function that describes the rate at which another function is changing is called a derivative function. If you already have a working knowledge of differential calculus, you will be familiar with the concept of functions and their derivatives. If not, don't worry too much about the details. It is enough to understand that for any differentiable function ƒ(x), the derivative function ƒ′(x) will give the instantaneous rate of change of that function for any value of x. The exponential function is unique in that its derivative function is the exponential function itself (ln e · e x = e x). In other words, the instantaneous rate of change of the exponential function for any value of x matches its output for that value of x.
The instantaneous rate of change (or derivative) of a function at some point P on the graph of the function can be found by measuring the slope of the graph at that point, i.e. the slope of the line that is tangent to the graph at point P. For the exponential function e x, the slope of the graph at point P will always be equal to the y coordinate of point P. This property is unique to the exponential function, and is why mathematicians, scientists and engineers find the exponential function so useful.
The slope of the tangent at e x = 1 is one (1)
The exponential function e x - sometimes written exp(x) - is of particular interest in the branch of mathematics known as calculus. This is because we use calculus to study things that change in a non-linear way. We have already looked at how the value of a loan or investment grows when the rate of interest is compounded. We can find many more examples of things that change in a non-linear way throughout the natural world. Examples include population growth, the spread of a disease, and the decay of a radioactive material. These are all quantities that increase or decrease at a rate that is proportional to their current value. The exponential function is particularly useful for describing such phenomena, because as we have seen, its growth rate always matches its current value.
The inverse function of the exponential function e x is the natural logarithm, ln(x) or loge(x). You might wonder why we use natural logarithms, which are to the base e, rather than common logarithms, which are to the base ten (log10). After all, it might seem more logical to use a number base with which we are familiar. On the other hand, the natural logarithm is directly related to the exponential function, so there is reason to suspect that it might be just as useful. Let's have a look at another graph to try and get a feel for what makes the natural logarithm so special. The following illustration shows the graph of the logarithmic function y = ln(x). By definition, the natural logarithm of e is one, and we have indicated this on the graph. We have also included the tangent to the graph at the point (1, 0). If you look closely at the tangent, you can see that it also passes through the point (0, -1), so the slope of the graph (remember "rise over run") must be one (1).
The graph of the logarithmic function ln(x)
It turns out that the slope of the graph of ln(x) will be 1/x for any value of x. As we have seen, this means that the instantaneous rate of change (or derivative) of the function ln(x) will also be 1/x for any value of x. This property is unique to the natural logarithm function, and can often greatly simplify calculations involving logarithmic quantities. There is another property of natural logarithms that is rather interesting, especially if you are studying integral calculus (the implications will be discussed in the section dealing with that subject). Consider the graph of the function y = 1/x (the derivative of the natural logarithm function):
The graph of the function y = 1/x
Note that we have shaded the area under the graph between x = 1 and x = e. If you were able to measure this area accurately, you would find that it is exactly one (1), which is the natural logarithm of e. In fact, the natural logarithm of any positive number n will be equal to the area under the graph between x = 1 and x = n. If n is one, the area will be zero - this makes perfect sense, since e 0 = 1. For any value of n greater than zero but less than one, the value of ln(n) is considered to be negative.
We have already seen the following definition of Euler's number, which allows us to calculate the value of e by plugging in a suitably large value for n:
|e =||lim||(1 +||1||) n|
There are a number of other ways to calculate the value of e. The following definition can also be useful in some situations:
|e =||lim||(1 + x) 1/x|
One of the more frequently used methods involves the representation of e as the sum of an infinite series, as shown here:
|e = 1 +||1||+||1||+||1||+||1||+||1||+||1||+||1||+ · · ·|
In this representation, we start with one, and then add the reciprocals of successive factorials. The factorial n! is simply the product of all positive integers less than or equal to n. For example, the factorial 5! is a shorthand way of writing 1 × 2 × 3 × 4 × 5, which is equal to one hundred and twenty (120). The number of terms used for the calculation will determine the accuracy of the resulting approximation. You should be able to see that each successive fraction represents a significantly smaller increment than its predecessor.