2.6 The math module

Learning objectives

By the end of this section you should be able to

Distinguish between built-in functions and math functions.
Use functions and constants defined in the math module.

Importing modules

Python comes with an extensive standard library of modules. A module is previously written code that can be imported in a program. The import statement defines a variable for accessing code in a module. Import statements often appear at the beginning of a program.

The standard library also defines built-in functions such as print, input, and float. A built-in function is always available and does not need to be imported. The complete list of built-in functions is available in Python's official documentation.

A commonly used module in the standard library is the math module. This module defines functions such as sqrt (square root). To call sqrt, a program must import math and use the resulting math variable followed by a dot. Ex: math.sqrt(25) evaluates to 5.0.

The following program imports and uses the math module, and uses built-in functions for input and output.

Example 1 Calculating the distance between two points

import math

x1 = float(input("Enter x1: "))
y1 = float(input("Enter y1: "))
x2 = float(input("Enter x2: "))
y2 = float(input("Enter y2: "))

distance = math.sqrt((x2-x1)**2 + (y2-y1)**2)
print("The distance is", distance)

Mathematical functions

Commonly used math functions and constants are shown below. The complete math module listing is available in Python's official documentation.

**Table 2.3** Example constants in the math module.
Constant	Value	Description
`math.e`	$e equals 2.71828 and so on$	Euler's number: the base of the natural logarithm.
`math.pi`	$π equals 3.14159 and so on$	The ratio of the circumference to the diameter of a circle.
`math.tau`	$τ equals 6.28318 and so on$	The ratio of the circumference to the radius of a circle. Tau is equal to 2π.

**Table 2.4** Example functions in the math module.
Function	Description	Examples
Number-theoretic
`math.ceil(x)`	The ceiling of `x`: the smallest integer greater than or equal to `x`.	`math.ceil(7.4)` $\to$ `8` `math.ceil(-7.4)` $\to$ `-7`
`math.floor(x)`	The floor of `x`: the largest integer less than or equal to `x`.	`math.floor(7.4)` $\to$ `7` `math.floor(-7.4)` $\to$ `-8`
Power and logarithmic
`math.log(x)`	The natural logarithm of `x` (to base e).	`math.log(math.e)` $\to$ `1.0` `math.log(0)` $\to$ `ValueError: math domain error`
`math.log(x, base)`	The logarithm of `x` to the given base.	`math.log(8, 2)` $\to$ `3.0` `math.log(10000, 10)` $\to$ `4.0`
`math.pow(x, y)`	`x` raised to the power `y`. Unlike the `**` operator, `math.pow` converts `x` and `y` to type float.	`math.pow(3, 0)` $\to$ `1.0` `math.pow(3, 3)` $\to$ `27.0`
`math.sqrt(x)`	The square root of `x`.	`math.sqrt(9)` $\to$ `3.0` `math.sqrt(-9)` $\to$ `ValueError: math domain error`
Trigonometric
`math.cos(x)`	The cosine of `x` radians.	`math.cos(0)` $\to$ `1.0` `math.cos(math.pi)` $\to$ `-1.0`
`math.sin(x)`	The sine of `x` radians.	`math.sin(0)` $\to$ `0.0` `math.sin(math.pi/2)` $\to$ `1.0`
`math.tan(x)`	The tangent of `x` radians.	`math.tan(0)` $\to$ `0.0` `math.tan(math.pi/4)` $\to$ `0.999` (Round-off error; the result should be 1.0.)

Note

Quadratic formula

In algebra, a quadratic equation is written as $a x^{2} + b x + c = 0$ . The coefficients a, b, and c are known values. The variable x represents an unknown value. Ex: $2 x^{2} + 3 x - 5 = 0$ has the coefficients $a equals 2$ , $b equals 3$ , and $c = - 5$ . The quadratic formula provides a quick and easy way to solve a quadratic equation for x:

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

The plus-minus symbol indicates the equation has two solutions. However, Python does not have a plus-minus operator. To use this formula in Python, the formula must be separated:

$x_{1} = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}$

$x_{2} = \frac{- b - \sqrt{b^{2} - 4 a c}}{2 a}$

Write the code for the quadratic formula in the program below. Test your program using the following values for a, b, and c:

**Table 2.5**
Provided input			Expected output
`a`	`b`	`c`	`x1`	`x2`
`1`	`0`	`-4`	`2.0`	`-2.0`
`1`	`2`	`-3`	`1.0`	`-3.0`
`2`	`1`	`-1`	`0.5`	`-1.0`
`0`	`1`	`1`	`division by zero`
`1`	`0`	`1`	`math domain error`

# TODO: Add the import statement.

a = float(input("Value of a? "))
b = float(input("Value of b? "))
c = float(input("Value of c? "))

x1 = 0  # TODO: Replace with plus formula.
x2 = 0  # TODO: Replace with minus formula.

print("Solutions:", x1, "and", x2)

Note

Cylinder formulas

In geometry, the surface area and volume of a right circular cylinder can be computed as follows:

$A = 2 π r h + 2 π r^{2}$

$V = π r^{2} h$

Write the code for these two formulas in the program below. Hint: Your solution should use both math.pi and math.tau. Test your program using the following values for r and h:

**Table 2.6**
Provided input		Expected output
`r`	`h`	`area`	`volume`
`0`	`0`	`0.0`	`0.0`
`1`	`1`	`12.57`	`3.14`
`1`	`2`	`18.85`	`6.28`
`2.5`	`4.8`	`114.67`	`94.25`
`3.1`	`7.0`	`196.73`	`211.33`

If you get an error, try to look up what that error means.

# TODO: Add the import statement.

r = float(input("radius? "))
h = float(input("height? "))

area = 0    # TODO: Replace with area formula.
volume = 0  # TODO: Replace with volume formula.

print("Area:", round(area, 2))
print("Volume:", round(volume, 2))