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# Built-in Mathematical Functions in R

Like other programming languages, R programming language also has various built-in mathematical functions to perform mathematical calculations. All these functions are vectorised. In this tutorial you will learn what are the built-in mathematical functions in R and how to use mathematical functions in R.

## Built-in Mathematical Functions in R

Some commonly used built-in mathematical functions in R are as follows:

Function Operation Performed
max(x) Maximum of the elements of x
min(x) Minimum of the elements of x
sqrt(x) Square root of x
abs(x) Absolute value of x
exp(x) Exponential value of x
log(x,base) Logarithm value of x with given base
log10(x) Logarithm of x (base 10)
ceiling(x) Closest integer not less than x
floor(x) Closest integer not greater than x
trunc(x) Closest integer towards zero
round(x,n) Round x to n significant places
rev(x) reverse the elements of x
sort(x) sort the elements of x in increasing order
rank(x) assign ranks to the elements of x

### Examples of Some Mathematical Functions

#### min() function in R

The min() function returns the minimum of the elements of a vector.

# Create a vector x
x <- c(10,-20,45,30)
y <- c(10,-20,45,30,NA)
min(x)
[1] -20
min(y)
[1] NA

#### max() function in R

The max() function returns the maximum of the elements of a vector.

# Create a vector x
x <- c(10,-20,45,30)
y <- c(10,-20,45,30,NA)
max(x)
[1] 45
max(y,na.rm = TRUE)
[1] 45

Note that while using max or min function, if the vector contains NA values the result will be NA because the default value of argument na.rm=FALSE. To remove the missing values while computing min or max use the argument na.rm=TRUE.

#### abs() function in R

The abs() function returns the absolute values of the elements of a vector.

# Create a vector x
x <- c(10,-20,45,30)
# Compute absolute of x
abs(x)
[1] 10 20 45 30

#### sqrt() function in R

The sqrt() function returns the principal square root of the elements of a vector.

# compute square root of x
sqrt(x)
Warning in sqrt(x): NaNs produced
[1] 3.162278      NaN 6.708204 5.477226

### Examples of logarithmic and exponent functions in R

#### log() function in R

The log() function returns the natural logarithm of the elements of a vector.

# Create a vector y
y <- c(5,6.5,8)
# compute natural logarithm of y
log(y)
[1] 1.609438 1.871802 2.079442

Note that in log(x,base) function, if you do not specify the base argument, R will use the default base as e (i.e., logarithm of x to the base e).

Suppose we need to find the logarithm of elements of y vector with base 2, we can use same log() function with argument base=2.

log(y,base=2)
[1] 2.321928 2.700440 3.000000

#### log10() function in R

The log10() function returns the logarithm of the elements of a vector to the base 10.

# compute logarithm of y to the base 10
log10(y)
[1] 0.6989700 0.8129134 0.9030900

#### exp() function in R

The exp() function returns the exponential of the elements of a vector.

# compute exponent of y
exp(y)
[1]  148.4132  665.1416 2980.9580

### Examples of Rounding functions in R

R has number of rounding functions available in the base package like ceiling(), floor(), trunc(), round() and signif().

# Create a vector x
x<-c(10.6787,9.456,12.554,-10.67)

#### ceiling() function in R

The ceiling(x) function is used to return the closest integer not less than the corresponding elements of x.

# return closest integer not less than x
ceiling(x)
[1]  11  10  13 -10

#### floor() function in R

The floor(x) function is used to return the closest integer not greater than the corresponding elements of x.

# return closest integer not greater than x
floor(x)
[1]  10   9  12 -11

#### trunc() function in R

The trunc(x) function returns a numeric vector containing the integers formed by truncating the values in x towards zero.

trunc(x)
[1]  10   9  12 -10

#### round() function in R

The round(x,digits) function is used to return the rounded values to the specified number of decimal places.

# round x to 2 significant places
round(x,digits=2)
[1]  10.68   9.46  12.55 -10.67

#### signif() function in R

The signif(x,digits) function rounds the values in its first argument to the specific number of significant digits.

g <- exp(-10:-7)
g
[1] 4.539993e-05 1.234098e-04 3.354626e-04 9.118820e-04
signif(g,digits = 3)
[1] 4.54e-05 1.23e-04 3.35e-04 9.12e-04

### Examples of rev, rank and sort function in R

#### rev() function in R

The rev(x) function is used to get the vector x in reverse order, i.e., the rev(x) function reverse the elements of the vector x.

# create a vector x
x <- c(10,12,9,13,8,17,13)
# reverse the elements of x
rev(x)
[1] 13 17  8 13  9 12 10

#### rank() function in R

The rank() function is used to assign the ranks to the values in a vector. The ranks are given from smallest to largest. In case of ties in the observations, the ranks are given as an average rank they jointly occupy (by default R uses ties.method="average").

x
[1] 10 12  9 13  8 17 13
# assign ranks to the elements of x
avg_ties <- rank(x)
avg_ties
[1] 3.0 4.0 2.0 5.5 1.0 7.0 5.5

Assign the rank 1 to the smallest value 8, the rank 2 to the next value 9 and so on. Since the value 13 occurs twice, R assign the average rank which they jointly occupy (i.e., $\frac{5+6}{2}=5.5$).

For other ties.method see the R code and its output.

first_ties <- rank(x,ties.method="first")
first_ties
[1] 3 4 2 5 1 7 6

The first value of the rank (which they jointly occupy) is assigned to the values when ties encountered.

last_ties <- rank(x,ties.method="last")
last_ties
[1] 3 4 2 6 1 7 5

The last value of the rank (which they jointly occupy) is assigned to the values when ties encountered.

random_ties <- rank(x,ties.method="random")
random_ties
[1] 3 4 2 5 1 7 6

The random value of the rank (which they jointly occupy) is assigned to the values when ties encountered.

max_ties <- rank(x,ties.method="max")
max_ties
[1] 3 4 2 6 1 7 6

The maximum value of the rank (which they jointly occupy) is assigned to the values when ties encountered.

min_ties <- rank(x,ties.method="min")
min_ties
[1] 3 4 2 5 1 7 5

The minimum` value of the rank (which they jointly occupy) is assigned to the values when ties encountered.

Ranks by different ties.method for comparison

x avg_ties first_ties last_ties random_ties max_ties min_ties
10 3.0 3 3 3 3 3
12 4.0 4 4 4 4 4
9 2.0 2 2 2 2 2
13 5.5 5 6 5 6 5
8 1.0 1 1 1 1 1
17 7.0 7 7 7 7 7
13 5.5 6 5 6 6 5

#### rank() function in R with NA

If the vector contains NA values, then by default NA values are ranked last (default argument is na.last=TRUE).

data_1 <- c(10,12,9,13,NA,8,17,13)
rank(data_1)
[1] 3.0 4.0 2.0 5.5 8.0 1.0 7.0 5.5

To assign NA the first rank, use the argument na.last=FALSE.

rank(data_1,na.last = FALSE)
[1] 4.0 5.0 3.0 6.5 1.0 2.0 8.0 6.5

The NA values can be removed while ranking using the argument na.last=NA.

rank(data_1,na.last=NA)
[1] 3.0 4.0 2.0 5.5 1.0 7.0 5.5

#### sort() function in R

The sort() function sorts the elements of vector in increasing order of magnitude by default. To sort the elements in decreasing order, use the argument decreasing=TRUE in sort() function.

# sort elements of x in increasing order
sort(x)
[1]  8  9 10 12 13 13 17
# sort elements of x in decreasing order
sort(x,decreasing=TRUE)
[1] 17 13 13 12 10  9  8

If the data contains NA value(s), by default the sort function ignore the NA value(s) and sort the remaining data.

sort(data_1)
[1]  8  9 10 12 13 13 17

Use the argument na.last=TRUE to put the NA values last while sorting.

sort(data_1,na.last=TRUE)
[1]  8  9 10 12 13 13 17 NA

Use the argument na.last=FALSE to put the NA values first while sorting.

sort(data_1,na.last=FALSE)
[1] NA  8  9 10 12 13 13 17

## Build-in Functions for Complex numbers

Some built-in function for complex numbers in R are as follows:

Function Operation Performed
Re(x) Real part of complex number x
Im(x) Imaginary part of complex number x
Mod(x) Modulus of complex number x
abs(x) Modulus of complex number x
Arg(x) Angle in radian of complex number x
Conj(x) Complex conjugate of complex number x

### Examples of Built-in function for complex numbers

Suppose the given complex number is z=3+4i.

# Store a complex number to z
z<-3+4i

#### Re() or Im() function for complex numbers

For the complex number $z=a+bi$, the real part of $z$ is $a$ and the imaginary part of $z$ is $b$. The function Re(z) and Im(z) returns real and imaginary part of $z$ respectively.

## Returns real part of z
Re(z)
[1] 3
## Returns imaginary part of z
Im(z)
[1] 4

#### Mod() or abs() function for complex numbers

To get the modulus of complex number, use the function Mod(z) or abs(z). The modulus of complex number $z=a+bi$ is given by $|z|=\sqrt{a^2+b^2}$.

## Returns modulus of z
Mod(z)
[1] 5
## Returns modulus of z
abs(z)
[1] 5

#### Arg() function for complex numbers

The argument of complex number $z=a+bi$ is the angle made by the complex number $z$ with the positive real axis. To get the real part of complex number, use the function Re(z).

## Returns angle in radians of z
Arg(z)
[1] 0.9272952

#### conj() function for complex numbers

The conjugate of the complex number $z=a+bi$ is $\overline{z}=a-bi$. To get the complex conjugate of the complex number, use the function Conj(z).

## Return complex conjugate of z
Conj(z)
[1] 3-4i

## Endnote

In this tutorial you learned about some commonly used built-in mathematical functions in R and how to use these functions in R.