Udayan Das; Aubrey Lawson; Chris Mayfield; Narges Norouzi

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 2.1

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)

Checkpoint

Importing math in a Python shell

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Concepts in Practice

Built-in functions and math module

1.

In the above example, when evaluating math, why did the interpreter raise a NameError?

The math module was not available.
The variable pie was spelled incorrectly.
The math module was not imported.

2.

Which of these functions is builtin and does not need to be imported?

log()
round()
sqrt()

3.

Which expression results in an error?

math.abs(1)
math.log(1)
math.sqrt(1)

Mathematical functions

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

Constant	Value	Description
`math.e`	$e = 2.71828 \dots$	Euler's number: the base of the natural logarithm.
`math.pi`	$π = 3.14159 \dots$	The ratio of the circumference to the diameter of a circle.
`math.tau`	$τ = 6.28318 \dots$	The ratio of the circumference to the radius of a circle. Tau is equal to 2π.

Table 2.3 Example constants 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.)

Table 2.4 Example functions in the math module.

Concepts in Practice

Using math functions and constants

4.

What is the value of math.tau/2?

approximately 2.718
approximately 3.142
approximately 6.283

5.

What is the value of math.sqrt(100)?

the float 10.0
the integer 10
ValueError: math domain error

6.

What is πr² in Python syntax?

pi * r**2
math.pi * r**2
math.pi * r*2

7.

Which expression returns the integer 27?

3 ** 3
3.0 ** 3
math.pow(3, 3)

Try It

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 $a = 2$ , $b = 3$ , 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:

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

Table 2.5

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Try It

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:

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

Table 2.6

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

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2.6 The math module

Learning objectives

Importing modules

Calculating the distance between two points

Importing math in a Python shell

Built-in functions and math module

Mathematical functions

Using math functions and constants

Quadratic formula

Cylinder formulas