3  Variables

A variable is a way of naming a value that can be stored and referenced later.

3.1 Assignment versus Equality

In mathematics, the equals sign, (=), means something profound. It is a statement that the left-hand expression is equal to the right-hand expression.

In programming, the equals sign is used to assign a value to a variable. The syntax for this is:

variable_name = value

where the variable name must be on the left-hand side. The expression

x = 22.7

assigns the value of 22.7 to the variable x. While this looks a lot like an equation, it is an assignment. This is a subtle distinction which can cause a lot of confusion later on if you are not aware of it. In Section 11.1, we will see how to test if two expressions are equal.

3.2 Variables

When we enter commands into Python, and later when we write scripts, the code is interpreted from top to bottom. Consider the sequence of commands below:

a = -3
b = 15
print( a*b )   # Prints -45

b = -6
print( a*b )   # Prints 18

The first line creates a variable called a and assigns it the value \(-3\). The second line creates a variable called b and assigns it the value \(15\). The expression a*b is then evaluated with those values of a and b. When we enter b = -6 on the next line, we are overwriting the value of b to now be \(-6\) instead of \(15\). Finally, the second execution of a*b results in \(18\) reflecting the updated value of b.

Variables are particularly useful when plugging values into formulas. Consider Newton’s law of universal gravitation, \[ F = G \frac{m_{1} m_{2}}{r^{2}} \] where \(F\) is the force of gravity between masses \(m_{1}\) and \(m_{2}\) separated by a distance \(r\), and \(G\) is Newton’s constant of gravitation. Anyone who has plugged into this formula while taking physics knows how combersome of a calculation this can be, particularly when dealing with scientific notation on a calculator. Let’s calculate the force of gravity between a mass of 32000 kg and a mass of 250000 kg that are separated by 50 meters.

m1 = 32000
m2 = 250000
r = 50
G = 6.67 * 10**-11
F = G*m1*m2/r**2
print(F)
0.00021344

3.3 Iterating with Variables

Variables can be overwritten to update their value. This is convenient for algorithms that iterate, repeating some process over and over.

Consider the ancient algorithm for calculating the square root of a number. Given a number \(n\) that we want to find the square root of, we start with a first guess \(g_1\). We then improve our guess to get \(g_2\) with: \[g_2 = \frac{g_1 + \frac{n}{g_1}}{2}\] We then get an improved third guess with \[g_3 = \frac{g_2 + \frac{n}{g_2}}{2}\] This calculation pattern continues so that \[g_{i+1} = \frac{g_i + \frac{n}{g_i}}{2}\]

Let’s use this pattern to find the square root of \(2800\), using only one guess variable, g, and overwriting our guess as we go:

n = 2800
g = 100
g = (g+n/g)/2
print(g)
64.0
g = (g+n/g)/2
print(g)
53.875
g = (g+n/g)/2
print(g)
52.92357888631091
g = (g+n/g)/2
print(g)
52.915026912364496
g = (g+n/g)/2
print(g)
52.91502622129182

Our guess g is converging quickly. We can define a stopping condition to be when the difference between adjacent guesses is smaller than some threshold. Depending on what we are calculating, 52.91502622129182 might just be “good enough”!

3.4 Strings

Numbers are one type of value that we can store. We can also store strings of characters.

greeting = "Good day!"
print( greeting )
Good day!

Note that strings are defined between quotes. Common errors include neglecting quotes around strings or putting quotes around variables which are not strings.

We often concatenate (connect together) two or more strings. For example, we can combine multiple variables into a single message with:

greeting = "Hello"
name = "Atticus"
print( greeting + name )
HelloAtticus

The + symbol here is not acting like addition of numbers, rather it is concatenating the string "Hello" together with the string Atticus.

There are many useful string methods, some of which are:

Python Description Example Result
len(s) Length of a string len("hello") 5
s.lower() Convert to lowercase "HELLO".lower() "hello"
s.upper() Convert to uppercase "hello".upper() "HELLO"
s.replace(a, b) Replace a substring "cat".replace("c","b") "bat"
s.count(x) Occurrences of substring "Mississippi".count("ss") 2

3.5 Type

type is a built-in function that returns the type of the object passed to it. Some of the basic types we’ve seen include:

Value Type Description
42 int Integer (whole number)
2.718 float Floating point (decimal number)
“Data” str String of characters

Experiment by making new variables with integers, floats, and strings and calling the function type:

num = 3.14159
type(num)
float
name = "Atticus"
type(name)
str

3.6 User Input

We can prompt the user for input with the input() function. Even better, we can store that input in a variable.

user_name = input("Enter your name: ")
print("Hello " + user_name)

The first line accomplishes multiple things. It prints the string "Enter your name: ", waits for the user to type something and press [Enter], and finally stores the user’s input in the variable called user_name. Then next line concatinates "Hello " with the user’s input and the prints the result.

Something interesting happens if we prompt the user for a number. Try the following:

user_number = input("Enter your favorite number: ") 
type(user_number)

You will see that user_number has type <class 'str'>. The user’s input is stored as a string of characters, not a number. This means that you can use it like a string, but you cannot do math with it, yet. You can change a number from a string to a int or float. For example, the following should work as long as the user enters a valid number:

user_number = input("Enter your favorite number: ") 
actual_number = float(user_number)   # Convert the number from str to float
twice = 2 * actual_number            # Double the number
print( "Your number doubled is: " + str(twice) )

Note that we converted the number twice back to a str to be able to concatenate it with another string for printing.

Exercises

  1. Coulomb’s Law calculates the force between two charges \(q_1\) and \(q_2\) separated by a distance \(d\): \[F_e = k \frac{q_1 q_2}{d^2}\] where \(k = 9 \cdot 10^9\) is a constant. Using variables, calculate the force between a charge of 0.3 C separated by 345 m from a charge of 0.08 C.

  2. Use the square root algorithm: \[g_{i+1} = \frac{g_i + \frac{n}{g_i}}{2}\]

    with an initial guess of \(g_0=10\), to find the approximation \(g_5\) for the following values:

    1. \(\sqrt{64}\)
    2. \(\sqrt{63}\)
    3. \(\sqrt{99}\)
    4. \(\sqrt{200}\)
    5. \(\sqrt{1000000}\)

  3. Use the input() function to prompt the user for two numbers, a and b, then print out the product and ratio of those two numbers.

    Hint: The input() function returns a string, so you will have to convert that string to a float so it can be used in a calculation.

  4. Using the input() function,

    1. prompt a user for a height in meters, then
    2. tell them how long it would take for a bowling ball to fall that far when dropped from rest.

    Recall from Physics that: \[\Delta x = v_{i} \Delta t + \frac{1}{2} a \Delta t^{2}\]