What is ‘e’

Just like pie, ‘e’ is widespread in mathematics and shows up in many different places. In this post I want to explore a little about what ‘e’ is and look at how you might draw a graphical representation of it using the R programming language.

Try this.

Before we begin exploring ‘e’ try this simple exercise using a calculator:
1. Enter a memorable 7-digit number on your calculator.
2. Take the reciprocal of that number, by pressing the 1/x button.
3. Add 1 to your answer.
4. Now raise the number to the power of the original 7-digit number.

Does your answer begin 2.718? If the answer is yes then you have found ‘e’!

What is ‘e’

$$e=lim_{n\to\infty}(1+\frac{1}{n})^{n}$$ We define ‘e’ to be the number that $(1+\frac{1}{n})^{n}$ is getting closer and closer to as ‘n’ gets larger and larger. $$ e=2.718281828459045... $$ If we replace 1/n with x/n we get the following equation: $$e^{n}=lim_{n\to\infty}(1+\frac{x}{n})^{n}$$ This formula $e^{n} $ has lots of interesting properties and applications.

R code to Plot ‘e’

With the simple programme below, we can show you what the growth of $e^{n}$ looks like.

x <- seq(-5, 5, by = 0.1)
y <- exp(x)
plot(x, y, type = "l", xlab = "x", ylab = "e^x")

This produces the following graph:

$$lim_{n\to\infty} e^{n}=\infty \\ lim_{n\to-\infty} e^{n}=0 \\ lim_{n\to0} e^{n}= 1 $$ The graph shows at minus Infinity the y axis approaches 0 and at Infinity the y axis approaches Infinity. But the y axis always passes through a value of 1 when x is 0.

R code to Plot inverse ‘e’

We can also plot the graph for the inverse of ‘e’ as follows:

x <- seq(-5, 5, by = 0.1)
y <- exp(-x)
plot(x, y, type = "l", xlab = "x", ylab = "e^-x")

$$ lim_{n\to\infty} e^{n}= 0\\ lim_{n\to-\infty} e^{n}= \infty \\ lim_{n\to0} e^{n}= 1$$