2 + 3[1] 53 R as a Calculator
In this chapter we will learn how to use R as a calculator.
3.1 The R Console
The R console is where you provide commands in the R programming language to be executed by the computer. It is possible to access the R console without RStudio using a very basic command-line interface that looks like this:1
But RStudio also has this R console, as well as many other useful features. Therefore we will stick to using RStudio for the rest of this course.
If we want to calculate 2+3, we simply go to the Console tab in RStudio and type 2+3, just like in the screenshot below:
We then press Enter to see the answer. What we see then is R giving the following output:
The [1] before the 5 here essentially means the 5 is the first number in the output. This is obvious here, but this feature will be more useful later when we do operations on more numbers. But for now, we can just ignore the [1].
In this book, I won’t always show screenshots like this. Instead I will show code snippets in boxes like this:
The part that is code will be in color and the output will be in a separate gray box below it. In these code boxes there is a small clipboard icon on the right which you can use to copy the code to be able to experiment with it in RStudio yourself.
We will now go through some different operations. We will also learn about functions and their arguments along the way, which we will be be using again and again throughout the rest of this course.
3.2 Addition, Subtraction, Multiplication and Division
Addition, subtraction, multiplication and division are given by the standard +, -, * and / operators that you would use in other programs like Excel, or even in an internet search engine. For example:
2 + 3[1] 55 - 3[1] 22 * 3[1] 63 / 2[1] 1.5It is also possible to do multiple operations at the same time using parentheses. For example, suppose we wanted to calculate: \frac{2+4}{4\times 2} = \frac{6}{8} = 0.75 We can calculate this in R as follows:
(2 + 4) / (4 * 2)[1] 0.753.3 Troubleshooting: “Escaping” in R
Suppose by accident you left out the closing parentheses and you see the following:
> (2 + 4) / (4 * 2
+R didn’t run the command, but it also didn’t give an Error. What happened is that Enter moved to a new line instead of executing the command. To “Escape” this situation, you just need to press the Esc button. In general, if anything strange happens in R and you get stuck, you can always press the Esc button to return to “normal”.
3.4 Exponentiation (Taking Powers of Numbers)
x^n multiplies x by itself n times. For example, 2^3=2\times 2\times 2 = 8. In R we use the ^ operator to do this:
2^3[1] 83.5 Absolute value
Taking the absolute value turns a negative number into the same number without a minus sign. It has no effect on positive numbers.
In mathematics notation we write |x| for the absolute value of x. The formal definition is:
|x| = \begin{cases} x & \text{ if } x \geq 0 \\ -x & \text{ otherwise} \end{cases}
Here are some examples:
- |-2|=2
- |3|=3.
This is what the function looks like when we plot it for different x:
Show code generating the plot below
if (!require(ggplot2)) install.packages("ggplot2")
library(ggplot2)
x <- seq(-2, 2, length.out = 1000)
ggplot(data.frame(x, y = abs(x)), aes(x, y)) +
geom_line() +
ylab("abs(x)")
We’ll learn how to make plots like this later in Chapter 16.
In R we can calculate these with:
abs(-2)[1] 2abs(3)[1] 3Taking the absolute value in R involves using what is called a function. Functions are used by calling their names and giving the arguments to the function in parentheses. When we do abs(-2), abs is the name of the function and -2 is the argument.
In many ways the functions in R work a lot like the functions in Excel, just they might have different names or be used a bit differently. For example, in Excel you write =ABS(-2) to take the absolute value of -2. The argument is the same, and the function name only differs in that in Excel you need to use capital letters whereas in R you use lowercase letters (in addition, Excel requires you to put an = before the function name).
When using functions it is helpful to read their help pages. You can look at this by typing help(abs) or ?abs in the Console and pressing Enter. The help page then pops up in the Help tab, like in the screenshot below:
absWe can see that it says abs(x) computes the absolute value of x. So we are told that x is the argument.
We will be using many different functions and it’s a good habit of to look at their help pages. The help pages will be available to you in the Exam.
3.6 Square and Cubed Roots
The square root of a number x is the y that solves y^2=x. For example, if x=4, both y=-2 and y=2 solve this. The principal square root is the positive y from this.
Here is what the square root function looks like for different x:
Show code generating the plot below
if (!require(ggplot2)) install.packages("ggplot2")
library(ggplot2)
x <- seq(0, 10, length.out = 1000)
ggplot(data.frame(x, y = sqrt(x)), aes(x, y)) +
geom_line() +
ylab("sqrt(x)")
We take the principal square root in R using the sqrt() function:
sqrt(9)[1] 3It is also possible to take a square root by exponentiating a number by \frac{1}{2}:
9^(1/2)[1] 3With this approach we can also take the cubed root of a number: \sqrt[3]{8}=8^\frac{1}{3}=2:
8^(1/3)[1] 23.7 Exponentials
A very important function in mathematics and statistics is the exponential function. The definition of \exp\left(x\right), or e^x, is given by: e^x = \underset{n\rightarrow \infty}{\lim} \left(1 + \frac{x}{n}\right)^n
Note: you don’t need to know or remember this definition for the exam. You only need to know how to use the exponential function in R.
This is what the function looks like:
Show code generating the plot below
if (!require(ggplot2)) install.packages("ggplot2")
library(ggplot2)
x <- seq(-2, 2, length.out = 1000)
ggplot(data.frame(x, y = exp(x)), aes(x, y)) +
geom_line() +
ylab("exp(x)")
In R we can use the exp() function to calculate the exponential of any number:
exp(1)[1] 2.7182823.8 Logarithms
Another common mathematical function is the logarithm, which is like the reverse of exponentiation.
The log of a number x to a base b, denoted \log_b\left(x\right), is the number of times we need to multiply b by itself to get x. For example, \log_{10}\left(100\right) = 2, because 10\times 10=100. We need to multiply the base b=10 by itself twice to get to x=100.
A special logarithm is the natural logarithm, \log_e(x), which is the logarithm to the base \exp(1)=e^1\approx 2.7183. This is also written as \ln(x).
This is what the function looks like:
Show code generating the plot below
if (!require(ggplot2)) install.packages("ggplot2")
library(ggplot2)
x <- seq(0.01, 10, length.out = 1000)
ggplot(data.frame(x, y = log(x)), aes(x, y)) +
geom_line() +
ylab("ln(x)")
In R we use the log() function to calculate the natural logarithm:
log(1)[1] 0What if we want to calculate the logarithm to a base other than e? If we look at the help page for log() using help(log) or ?log we can see that the log function has 2 arguments:
x: the number we want to take the log of.base: “the base with respect to which the logarithms are computed. Defaults to e=exp(1)”.
This is the first time that we have seen a function with more than one argument. Earlier when we used the log() function to calculate the natural logarithm we only used one argument because we used the default setting for the base. But when we want to use a base other than e, we need to specify it.
How we calculate \log_{10}\left(100\right) in R is as follows:
log(100, base = 10)[1] 2We write both arguments into the log() function, separated by commas.
This is just like how we used functions with more than one argument in Excel, for example the VLOOKUP function. We separated the arguments there by commas as well.
3.9 Examples from Mathetmatics for E&BI
Below are some examples to show that we can use R to calculate the answers to some of the problems you face in the course Mathematics for E&BI.
3.9.1 Example 1: Future Value
A manager invests €200,000 for a period of 15 years. The interest rate is 6% per year. What is the future value of this investment?
Solution
In general, the future value of an investment K after T years with an interest rate i is K\left(1+\frac{i}{100}\right)^T. The future value of this investment is therefore 200000\times \left(1.06\right)^{15}. We can calculate this with:
200000 * 1.06^15[1] 479311.6If the manage invests €200,000 today, then in 15 years this amount will grow to €479,311.64.
3.9.2 Example 2: Present Value
A manager wants to have an amount of €200,000 after 15 years. The interest rate is 6% per year. How much does the manager need to invest today?
Solution
In general, the present value of investment K in T years with an interest rate i is K\left(1+\frac{i}{100}\right)^{-T}. The present value is therefore 200000\times\left(1.06\right)^{-15}. We can calculate this with:
200000 * 1.06^(-15) [1] 83453.01If the manage invests €83,453.01 today, then in 15 years this amount will grow to €200,000.
3.9.3 Example 3: Doubling Time
If the interest rate is 6%, how long would it take for your investment to double in value?
Solution
After T years, the investment of K will grow to K\times\left(1.06\right)^T. We need to know what the value of T is to make the investment grow to 2K (double). This is the T that solves:
K\times\left(1.06\right)^T=2K Canceling the Ks from both sides: \left(1.06\right)^T=2 Taking the natural logarithm of both sides: T\ln\left(1.06\right) = \ln\left(2\right) Solving for T: T= \frac{\ln\left(2\right)}{\ln\left(1.06\right)} We can calculate this with:
log(2) / log(1.06)[1] 11.89566So at a 6% interest rate, it takes just under 12 years to double your investment.
We can also use a formula for changing the base, which is \frac{\ln(a)}{\ln(b)}=\log_b (a). Applied to this case: T= \log_{1.06}\left(2\right)
We can calculate the answer using the log of 2 to the base 1.06 and see that it gives the same result:
log(2, base = 1.06)[1] 11.89566On Windows you can access this by going to
C:\Program Files\R\R-4.3.1\binand running the fileR.exe. On Mac or Linux you can go to the terminal and typeRandEnter.↩︎