Chapter 4 Mathematical Operations

4.1 Addition

You can use R as a calculator. Type your mathematical expression in the console, and get the result instantaneously.

A better way to use R, is to write your code in R-scripts.

Most often, you will assign values to objects.

Probably the most basic mathematical operation is adding two or more numbers.

Below we assign the value 2 to a, and 5 to b. You can then assign the addition of \(a+b\) to another object, \(c\).

2+5

## [1] 7

a = 2; b = 5
c = a + b
cat(a,"+",b,"=",c,"\n")

## 2 + 5 = 7

# if c = a + b, then b = c - a
c-a

## [1] 5

## [1] 5

4.2 Multiplication and Division

Multiplication is like adding a number several times.

Adding 5 four times, is equivalent to multiplying 5 by 4.

Often, in data science, you will multiply broken numbers. Like \(4.5*3.88\). The analogy (adding 4.5 3.88 times) is somewhat harder to envision.

The operator for multiplication is the asterisk (*).

Although in mathematical textbooks, you may find \(xy\) as shorthand for x times y, that doesn’t work in R, and other software. You have to use \(x*y\) in your code!

5+5+5+5

## [1] 20

4*5

## [1] 20

\(a+a+a+a = 4*a\) (or 4a, for short)

\(4a = 20\)

We can divide both sides by 4 to find a:

\(4a/4 = 20/4\) \(\Rightarrow\) \(a=5\)

4.3 Exponentiation

Exponentiation is equivalent to multiplying by the same number, several times. For instance, \(5*5\) is the same as 5 raised to the power 2, or \(5^{2}\). Exponentiation in R uses the operator ^, like in the example below.

Exponents do not have to be integers (1, 2, …), but can be broken numbers (e.g. 1.2, 2.8, …).

A special case is an exponent of \(0.5\). Exponentiation by \(0.5\) is called the square root.

Taking the square root of a number, is the reverse of taking the square.

If \(x^{2} = y\), then \(y^{0.5} = x\). For example, the square of 5 is \(5*5 = 25\); reversely, the square root of 25 is 5.

Other special cases are exponents of 0 and 1.

\(x^{0} = 1\)

\(x^{1} = x\)

5*5*5*5

## [1] 625

5^4

## [1] 625

5*5

## [1] 25

5^2

## [1] 25

sqrt(25)

## [1] 5

25^(1/2)

## [1] 5

5^0

## [1] 1

5^1

## [1] 5

Exponentiation has the following structure:

\(a^b=c\)

In this formula:

a is the base
b is the exponent
c is the power

4.4 Rooting and Logarithms

There is a relationship between exponentiation, rooting and logarithms.

In a simple example, 10 squared (or \(10^{2}\)) is \(10*10 = 100\).

That is:

\(10^2 = 100\)

Rooting is:

\(\sqrt(100) = 10\) (the square root of 100)

Logarithm:

\(log(100) = 2\) (using 10 as the base for the logarithm).

Note that the three numbers (2; 10; and 100) keep coming back, in different settings!

In R this would look like:

cat("The square of 10, or 10*10, equals",10^2,"\n")

## The square of 10, or 10*10, equals 100

cat("The square root of 100 equals",sqrt(100),"\n")

## The square root of 100 equals 10

cat("The logarithm of 100 (base 10) equals", log10(100),"\n")

## The logarithm of 100 (base 10) equals 2

Since 10 is an exceptional base, and squaring and square roots are special cases, we can use a more general version.

Suppose we do the same for \(2^3 = 2*2*2 = 8\).

cat("2 raised to the power 3 equals",2^3,"\n")

## 2 raised to the power 3 equals 8

cat("The cubic root of 8 equals",8^(1/3),"\n")

## The cubic root of 8 equals 2

cat("The logarithm of 8 (base 2) equals", log(8, 2),"\n")

## The logarithm of 8 (base 2) equals 3

For an easy explanation of the links between roots and exponents, have a look at this video.

4.5 The Order of Operations

The order of operations is governed by the principle of PEMDAS.

Parentheses
Exponents
Multiplication and Division
Addition and Subtraction

As a general rule, in programming for data science and statistics it is best to use parentheses (brackets) in order to avoid confusion.

Some examples:

2+5*8

## [1] 42

2+(5*8)

## [1] 42

(2+5)*8

## [1] 56

3+2^2

## [1] 7

(3+2)^2

## [1] 25

12-24-34+12

## [1] -34

12-(24-34)+12

## [1] 34

12-24-(34+12)

## [1] -58

12-(24-34+12)

## [1] 10

As you see, formulas are prone to errors!

4.6 Negative Numbers in Multiplication

As a rule:

A positive number times a positive number gives a positive number
A positive number times a negative number gives a negative number
A negative number times a negative number gives a positive number

Examples:

8*7

## [1] 56

8*-7

## [1] -56

-8*7

## [1] -56

-8*-7

## [1] 56

4.7 Rules for Operations

Commutative law

\(a+b = b+a\)

Associative law

\((a+b)+c = a+(b+c)\)

This holds for addition, not for subtraction!

(8+2)+3

## [1] 13

8+(2+3)  # Same as above

## [1] 13

(8-2)-3

## [1] 3

8-(2-3)  # Not the same as above!!

## [1] 9

Distributive law

\(a*(b+c) = (a*b) + (a*c)\)

8*(2+3)

## [1] 40

8*2 + 8*3

## [1] 40

We can use one command line to evaluate if the latter two expressions are identical.

For these evaluations, you have to use the \(a == b\) format (double =), which evaluates whether \(a\) and \(b\) are the same. \(a = b\) (single =) would allocate the value of \(b\) to \(a\), which is not what we want.

8*(2+3) == 8*2 + 8*3 # Evaluate if the expressions are identical

## [1] TRUE

Another rule:

\((b+c) / a = (b/a)+(c/a)\)

(8+7)/3

## [1] 5

(8/3) + (7/3)

## [1] 5

(8+7)/3 == (8/3) + (7/3)

## [1] TRUE

But: \((a) / (b+c) \neq (a/b) + (a/c)\)

15/(2+3)

## [1] 3

15/2 + 15/3

## [1] 12.5

15/(2+3) == 15/2 + 15/3

## [1] FALSE

4.8 Rules for Fractions

Rule 1: \((a/p) + (b/p) = (a+b)/p\)

8/4 + 3/4 == (8 + 3)/4

## [1] TRUE

Rule 2: \((a/p)*(b/q) = (a*b)/(p*q)\)

(8/4)*(6/3) == (8*6)/(4*3)

## [1] TRUE

Rule 3: \((a/p)/(b/q) = (a/b)*(q/b)\)

(8/4)/(6/3) == (8/4)*(3/6)

## [1] TRUE

4.9 Rules for Exponentiation

Rule 1: \(a^n = a * a * a * ...\) (n times)

Rule 2: \(a^n\) is positive if \(a>0\)

Rule 3: \(a^n\) is positive if \(a<0\) and n is an even number

Rule 4: \(a^n\) is negative if \(a<0\) and n is an odd number

Examples:

8^2    # Rule 2

## [1] 64

(-8)^2 # Rule 3

## [1] 64

-8^3   # Rule 4

## [1] -512

-8^2   # Is the outcome what you expected??

## [1] -64

In the expression \(-8^2\), the PEMDAS rule forces R to first evaluate \(8^2\) (E, for exponentiation), before multiplying (M) by -1!!

If you intend to square -8 (\(-8*-8=64\)), then the code should read:

(-8)^2   # Is the outcome what you expected??

## [1] 64

Parentheses make all the difference!