10 Functions

10.1 Objective

Understand what functions are and how they’re useful
Be able to use built-in functions
Understand what function arguments are
Be able to write your own functions
Be able to write good functions
Understand the difference between positional arguments and keyword arguments

10.2 Chapter Project Specification

Allow the user to enter the locations of several starships in 3D space.

These should be entered as x,y,z triplets when prompted. When the user enters “done”, stop entering ship locations.

Enter ship location x,y,z (or "done"): 10,20,30
Enter ship location x,y,z (or "done"): -17,16,50
Enter ship location x,y,z (or "done"): 0,13,30
Enter ship location x,y,z (or "done"): 5,20,-40
Enter ship location x,y,z (or "done"): done

Then print out a grid of the distances between them. The grid’s top row and left column should indicate the ship number (starting with 0).

Crossing a column with a row should give you the distance between the ships.

Distances should be printed to 2 decimal places in fields of width 8.

We’ll use a variant of the Pythagorean Theorem ⁹⁰ to find the distance between two 3D points.

$d=\sqrt{(x_0-x_1)^2+(y_0-y_1)^2+(z_0-z_1)^2}$

For each pair of 3D points, we take the difference in the X coordinates squared, plus the difference in the Y coordinates squared plus the difference in the Z coordinates squared, and then we take the square root of that whole thing. And that’s the distance between the two points.

Example output (corresponding to the example input, above):

               0       1       2       3
       0    0.00   33.84   12.21   70.18
       1   33.84    0.00   26.42   92.74
       2   12.21   26.42    0.00   70.53
       3   70.18   92.74   70.53    0.00

So we can see ship #2 (along the top) is distance 26.42 from ship #1 (along the left). And notice the diagonal is all 0.00, which makes sense because every ship is zero distance from itself.

Keep this project in mind as we go through the chapter.

10.3 What Are Functions?

You’re about to seriously level up in programming by learning how to do this, and it’s not even that difficult. Functions are the key to getting away from the toy programs we’ve been doing so far and doing real programs.

So what are they?

Problem-solving step: Understanding the Problem.

Functions are self-contained pieces of code that you can call that do a specific thing.

Does this sound at all familiar? Because we’ve been doing this the whole time with the built-in print() function.

Not to be confused with statements like for and if. Know a function because it has parenthesis right after the name that you can use to pass arguments to the function⁹¹.

The print() function has built-in functionality to print things on the screen. Thankfully (really, really) we don’t have to write that code ourselves. We can just say:

print(23 + 34)

and have print() do all the dirty work of getting us the answer printed to the screen.

We’ve also used the input() function to get a string entered from the keyboard.

name = input("Enter your name: ")

This turns out to be a great way to simplify and organize code. Can you imagine if you had to put all the code in to print out something on the screen every time you wanted to print something? Much easier to define the print() function once, and then use it over and over again by calling it.

We have an important principle in computer programming called the DRY principle ⁹² (Don’t Repeat Yourself). If you can remove as much repetitive code as you can and move it to a function, that makes your code easier to read and maintain. DRY code is happy code.

Not only can we use functions to make DRY code, we can also use them to organize our code into logical sections, even if a function is called only one time.

It is clearer to have your functionality in discrete sections that you call in sequence rather than just having a single huge block of code that does everything

10.4 Using Built-In Functions

Problem-solving step: Understanding the Problem.

Python has a lot of built-in functions that you can use. It’s not necessary to memorize the usage in detail (you can always look up the details), but it’s a good idea to skim their descriptions just so you can recall that they exist.

We’ve already used print() and input() quite a bit, but there are plenty more. Look them up online ⁹³.

Here are some common ones, though this is not an exhaustive list:

Built-in Function	Description
`abs()`	Return the absolute value of a number.
`bin()`	Return a string representing the number in binary.
`chr()`	Return a character for the Unicode (and ASCII) value passed in.
`dir()`	Return a list of methods on this object.
`enumerate()`	Return an iterator over a list of (index, value) pairs.
`filter()`	Repeatedly run a function on items of a list filtering some out.
`float()`	Convert a number or string to a floating-point value.
`help()`	Get help on a data type.
`hex()`	Return the hexadecimal representation of a number.
`input()`	Get input from the keyboard and return that string.
`int()`	Convert a floating point or string to an integer value.
`isinstance()`	Return true if an object is an instance of a class.
`len()`	Return the length of an iterable, like a list or string.
`map()`	Run a function on every value in an iterable, returning the results.
`max()`	Return the maximum of the arguments or of an iterable.
`min()`	Return the minimum of the arguments or of an iterable.
`ord()`	Return the Unicode value (code point) for a given character.
`pow()`	Return a base value raised to an exponent.
`print()`	Print to the console.
`range()`	Return an iterator that runs over a range of values.
`reversed()`	Return an iterator over a reversed version of the argument.
`round()`	Round a number to the nearest whole number.
`sorted()`	Return a sorted version of an iterable.
`str()`	Convert a value to a string.
`sum()`	Return the sum of all arguments and/or of an iterable.
`super()`	Get access to an object’s superclass.
`zip()`	Merge parallel lists into a dictionary.

These are all available for use at any time in your program.

10.5 Arguments

“Oh look, this isn’t an argument.”

“Yes it is.”

“No it isn’t. It’s just contradiction.”

“No it isn’t.”

—Excerpt from Monty Python’s Argument Clinic sketch

Problem-solving step: Understanding the Problem.

When you call a function, the things you put in the parentheses are called the arguments to the function. Arguments are separated by commas.

print(2)        # 1 argument
print(2, 3)     # 2 arguments
print(2, 3, 4)  # 3 arguments
print()         # 0 arguments

The arguments are the way you can pass data into a function to have it operate on that data to produce a result.

We refer to them by number, as well. “Pass the number of goats in as the second argument to the function.”

Functions often take a specific number of arguments. But, as we see above with print(), they can take variable numbers of arguments, too.

10.6 Writing Your Own Functions

“What do you like most about hacking?”

“The power.”

—A good friend of mine trolling a reporter at DEFCON 2 ⁹⁴

The power. This is where we fly. Writing your own functions. This is where we get to DRY our code, organize it better, and make it more readable overall.

Problem-solving step: Understanding the Problem.

All programmers use their own functions heavily, and to great effect.

Let’s say we want to write a program that adds 13 to a number, and then divides the resultant sum by 7.

No problem. We got this. Let’s do it with 30.

print((30 + 13) / 7)

Great! Oh, and we also need it for 8, 19, 21, 37, 402, 516, 1024, and 3490.

OK. Still no problem.

print((30 + 13) / 7)
print((8 + 13) / 7)
print((19 + 13) / 7)
print((21 + 13) / 7)
print((37 + 13) / 7)
print((402 + 13) / 7)
print((516 + 13) / 7)
print((1024 + 13) / 7)
print((3490 + 13) / 7)

Great! Oh, did I say “divide by 7”? I’m sorry, that should be “multiply by 7”.

And here we get a taste of why DRY code is good code. Since the spec changed⁹⁵, we have to go back and modify all those lines of code to make them right.

If only we hadn’t repeated ourselves, we might have been able to only change it in one place. This would also be better because we wouldn’t be taking the risk of missing a place we should have changed the code.

But how to do it?

With… FUNCTIONS! Let’s write a function to do that math and return the result. Then we can just call it over and over, like this:

print(do_the_math(30))
print(do_the_math(8))
print(do_the_math(19))
print(do_the_math(21))
print(do_the_math(37))
print(do_the_math(402))
print(do_the_math(516))
print(do_the_math(1024))
print(do_the_math(3490))

Sure, we’re repeating the function name do_the_math(), but we’re not repeating the actual mathematical expression, itself, which is what matters.

But how do we define the function do_the_math()?

We use the def statement, like this:

def do_the_math(x):
    result = (x + 13) * 7

    return result

answer = do_the_math(30)
print(answer)

The indented stuff after the def is called the function body. That’s where all the work gets done.

There’s a lot of stuff to unpack here, so let’s take it nice and slow.

Line 1: defining the function

When we say def do_the_math, we’re telling Python, “Hey, I’m making a brand new function from scratch called do_the_math.

The part in the parentheses is a list of what we call parameters. This describes what can be passed to the function. The actual values we pass to the function are called arguments.

In this case, we can pass a single argument to our function. It will be represented inside the function by the variable x.

When we call the function on line 6, x will be initialized with the argument, just as if it had been assigned to. So in this example, x will be assigned 30, because that’s the argument we passed to the function for that call.

In particular, whatever you pass as an argument will also be referred to by the corresponding variable in the parameter list.

Remember: you pass in arguments that get assigned into parameters.

A parameter is a special type of local variable, one that comes pre-initialized with whatever value we passed in as an argument. We’ll get into their story on line 2.

Line 2: Compute the result

You can see on line 2 that we take the parameter x and feed it into the equation our boss tasked us with. (They told us to change it to multiply by 7, remember?)

And then Python does that math and stores that into the variable result.

Now that variable result is special. It’s what we call a local variable. Because it’s used inside the function, it’s only visible inside the function. Once the function returns (more on that below), the value in the local variables vanish forever.

You can name the local variables anything you want, even if you’ve used that same name in another function! Each local variable is isolated from the rest of the world and only exists within its own function.

And as I mentioned above, the parameters are also local variables. They just have the added property that they are assigned the values of the arguments automatically.

Line 4: Return the result

A function can optionally return a value. You know how arguments are a way to pass data into a function? The return value is a way to pass data back out of the function to the caller.

Back where the function is called, you can think of the function call as having the return value. Then you can do things like assign it into a variable.

Line 6: Capture the returned result in a variable

We call our function, and we take the return value from that function and assign it into the variable answer. Then we can do things with it, like print it out on line 7.

Problem-solving step: Looking Back.

In the function as written, above, we use a lot of variables to help demonstrate a number of concepts and ideas surrounding functions.

But in reality, a dev would probably simplify the above code to:

def do_the_math(x):
    return (x + 13) * 7

print(do_the_math(30))

Compare the two until you are convinced they are equivalent.

10.7 Multiple Return Values

Problem-solving step: Understanding the Problem.

You can always return a list or a dict or anything from a function, but you can also return two values and assign them to different variables.

# Function that returns the argument times 10, and divided by 10

def timesdiv10(x):
    return x * 10, x / 10

a, b = timesdiv10(100)

print(a, b)   # 1000 10

Or you can assign the result to a single variable. The result will be a tuple, which is a read-only list that you can access with square bracket notation:

a = timesdiv10(100)

print(a[0], a[1])   # 1000 10

Magic!

10.8 What Makes a Good Function

Problem-solving step: Understanding the Problem.

The biggest rule is “do one thing and do it well”.

It’s more of a guideline than a rule because you’re the boss when it comes to programming.

But it’s a great guideline that should be followed when you can.

The biggest things you’ll use functions for are:

Mathematical equations
Anything that happens repeatedly
Breaking up code into smaller, self-contained sections

Let’s talk about that last item more, since it’s less obvious.

Very frequently—almost always—you’ll start with a series of steps you need to perform to solve the problem, like this:

# While we still have data to process
    # Process the data
    # Output the data

And then you go in and fill in all the code for processing the data and outputting the data.

But that could make for a bulky, hard-to-read while loop. It might be better to code it up like this:

while d in data:
    d = process(d)
    output(d)

and the write the functions process() and output() to operate on the data that is passed into them.

This makes the logic of the loop really easy to read. And being easy-to-read is king (or queen, if you prefer).

If you find you have some large amounts of code that are getting deeply-nested, it might be time to break them out into functions, even if you only call those functions from that single place.

Knowing when to break up code into functions is more of an art than a science. If you start feeling like your code is remotely unwieldy, consider what it might look like split into different functions. If you like it more, do it!

10.9 Positional Arguments versus Keyword Arguments

Problem-solving step: Understanding the Problem.

We’ll talk more about this in detail later, but function arguments can be split into these two broad classes in Python: positional and keyword.

Positional arguments are the arguments that have to occur at certain positions in a function call.

For instance, if we want to compute $14^{12}$, we need to pass those two arguments in a specific order to the math.pow() function, which expects the base to be first and the exponent to be second:

math.pow(14, 12)   # 56693912375296.0

If you put the 12 first, you get a different answer. These are positional arguments.

But for some functions, after the positional arguments, you can specify additional keyword arguments. These are arguments that are identified by a certain name.

For example, normally print() puts a newline at the end of the string. But you can override this behavior with the end keyword argument by telling print() to put nothing (an empty string) at the end of the line:

# Together, prints "Beej" on a single line

print("Be", end="")
print("ej")

Notice how we had to identify the keyword argument by name.

If you’re looking through documentation, you might find something that looks like this (the documentation for print()):

print(*objects, sep=' ', end='\n', file=sys.stdout, flush=False)

Ignoring the *objects for now, look at all those keyword arguments… in the docs, they have = followed by some value. This indicates the default value if you don’t specify one otherwise. In other words, they’re optional keyword arguments.

You can see that print() ends each line with a newline \n unless we tell it to end the line with some other string.

For now, it’s enough to know that these exist and how to call them. Later on, we’ll talk about how to write our own.

10.10 Interlude: Evaluation Strategies

If you don’t want to read this section, it’s OK to skip. But it’s more fun if you read it.

Different languages have different behaviors when it comes to passing arguments to functions.

Some of them make copies of the values. Some of them just make it so the parameter name is an alias for the argument name. Some of them give you the option through some keyword.

How the programming language does this is called it’s evaluation strategy⁹⁶.

Three very common ones are:

Call by Sharing: This is what Python does. When you call a function, the parameters become references to the same objects as their corresponding arguments. This means that if, inside the function, you change the thing the parameter refers to, you’ll see that change reflected out in the caller. However, if you assign a new value into a parameter, you will not see the change.
```
def foo(x):
    # Modify the passed-in object. Caller will see this change.
    x[1] = 99

    # Assign a new list into x. The caller will _not_ see this change.
    x = [4, 5, 6]

a = [1, 2, 3]

foo(a)
print(a)   # [1, 99, 3]
```
Call by Value: a copy of the argument is made and stored in the parameter. The C programming language ⁹⁷, among others, uses this one. (Sometimes people say Python uses Call by Value, but this is technically not accurate since parameters are not copied when they are passed in; only the reference to the argument is copied. You can make Python simulate Call by Value by making a copy of the parameter when calling the function.)
```
# Simulating call-by-value in Python

def foo(x):
    # Modify the passed-in object. The caller would normally see this
    # change, but if they called it with a copy, only the copy will
    # be affected.
    x[1] = 99

a = [1, 2, 3]

# Simulate call-by-value by making a copy
foo(a.copy())

print(a)   # [1, 2, 3]
```
- Call by Reference: the parameter effectively becomes an alias for the variable name passed in as the argument. Anything you do to the parameter you effectively do to the argument, including assigning it some other value. Example languages Fortran ⁹⁸, C++⁹⁹ (usually by value, but can be made by reference). (Sometimes people say Python uses Call by Reference, but this is technically not accurate since you can’t assign new values to parameters and see that reflected in the argument.)

Here’s a quick summary between them.

Note: In the following table, “Can modify item in caller” means that if you have a variable in the caller that refers to a thing, changes to the thing in the function will be reflected in the caller’s variable.

“Parameter reassignment affects caller” means that if you assign into a parameter variable (i.e. change the parameter to refer to a completely different thing) the argument variable will also change.

	Can modify item in caller	Parameter reassignment affects caller
Call by Sharing	Yes	No
Call by Value	No	No
Call by Reference	Yes	Yes

Now, I’ve seen people be fast and loose with these descriptions when it comes to Python, so don’t hold them to it. In my opinion, from the above three items, Call by Sharing is the most accurate. Call by Reference is the second-most.

10.11 The Chapter Project

Problem-solving step: Understanding the Problem

It’s that time! If you need a refresher, pop back up to the top and check out the spec.

So we want to do a couple of things:

Read in a list of starship positions.
Print out a grid of distances between each ship.

The positions are entered as X, Y, Z coordinates (until the user enters “done”).

The distances are computed with the distance formula variation of the Pythagorean Theorem.

A grid of distances is printed. There should be ship numbers along the axes.

Problem-solving step: Devising a Plan

We already have a bit of a plan, don’t we? We listed it right up there.

Read in a list of starship positions.
Print out a grid of distances between each ship.

Don’t look now, but these would make a great couple of functions. Sure, we could stick them all in one big main routine, but it’s way more logical and easier to read if we make them their own functions.

We could have one function read the list of positions and return it.

And we could take that returned list and pass it into another function to print the grid.

locations = get_ship_locations()
print_grid(locations)

We’re taking a _ top-down_ approach, here. Starting with the big pieces of logic and then implementing them as we go down.

In fact, let’s simplify the problem, and just start like this:

locations = get_ship_locations()
print(locations)
#print_grid(locations)

That way we can just do one bit and make sure it’s working.

Fun Motto Fact: Get something working as quickly as possible, no matter how much a piece of the project it is.

Problem-solving step: Understanding the Problem

For getting the ship locations, looks like we want to have the user enter an X, Y, Z coordinate. Then we want to split that into a list of numbers. And we need to make sure they are int type—remember that input() returns a string, so we’ll have to convert them to int.

And we repeat that until the user enters “done”.

Problem-solving step: Devising a Plan

Looping until the user enters “done” is a solved problem. We’ve done that a few times now. Just make a variable for the while() condition, and then set it when the user enters “done”.

Otherwise, we have to split the string up into a list of 3 numbers, splitting on the comma. Remember how to do that?

Yes, that’s right (or no, not right, if you don’t remember), it’s the split() string method. Example:

"1,2,3".split(',')  # produces list ["1","2","3"]

But that’s not enough. As we noted, we need to convert that list from a list of strings to a list of numbers. Remember how to convert a string to a number? Yes, with the built-in int() function!

int("2") == 2  # True!

So we can loop through our list of strings and convert them to ints.

And then we’ll have a list of 3 ints, representing the X, Y, Z coordinates of a single ship. Of course, the user is going to enter any number of ships, and we’ll have to keep all those lists of coordinates somewhere… where?

It’s almost like we’ll need a list for all those lists. A list of lists! Why not? We made some of those in the lists chapter, right?

So we add the new X, Y, Z list to the end of a master list that holds all the coordinates.

Then we can return that master list to be used later when we print the grid.

Let’s code!

Wait—you’re right—we haven’t finished the entire plan for the whole project. True. But that’s okay. We completed enough devising of a plan to do the first part of the project.

And this is one of the beautiful things about functions. We’ve split the problem up so nicely that we can implement different parts of it completely independently! In fact, if you have a programming pal, you could get them to write the print_grid() function at the same time you were writing the get_ship_locations() function!

Problem-solving step: Carrying Out the Plan

Let’s get coding!

def get_ship_locations():
    """
    Ask the user to enter a number of X, Y, Z ships coordinates.
    Returns a list of all the coordinates.
    """

    done = False     # True when the user asks to be done
    locations = []   # Master list of all ship positions

    while not done:
        xyz = input('Enter ship location x,y,z (or "done"): ')

        if xyz == "done":
            done = True
        else:
            # Get a list of the x,y,z coordinates
            xyz_list = xyz.split(',')

            # Convert to integers
            for i, v in enumerate(xyz_list):
                xyz_list[i] = int(v)

            # Build the master list
            locations.append(xyz_list)

    return locations

locations = get_ship_locations()
print(locations)
#print_grid(locations)

If we give that a run, we see something like this:

Enter ship location x,y,z (or "done"): 1,2,3
Enter ship location x,y,z (or "done"): 4,5,6
Enter ship location x,y,z (or "done"): done
[[1, 2, 3], [4, 5, 6]]

There’s our list of lists holding the ships’ coordinates! It’s ready to feed into the print_grid() function. But first, we’d better think about that for a bit.

One more thing: if you were looking closely, you saw the big multiline string at the beginning of the function describing what the function does. This is called a doc string and it’s a comment that gives overall information about the function. Automatic documentation generators can extract these and build documentation for you, just like you can get with the help() function in the REPL.

Problem-solving step: Understanding the Problem

All righty. What do we need to do for this second part of printing out the grid of distances between the ships?

There are sort of three big pieces here.

We need to print out a grid.
We need the first row and first column to list out ship numbers so we can cross-reference.
We need to compute the distance and print that.

Problem-solving step: Devising a Plan

Let’s simplify a bit first. Instead of worrying about computing the distance, let’s just concentrate on the grid. We’ll just put the bogus number of 99.99 in for all the inter-ship distances.

Protip: When putting in bogus data, make sure it’s obviously bogus so that obviously people obviously know they must obviously replace this with real data before the product ships.

And to simplify even further, let’s forget about the ship numbers in the first row and first column. We can add those in later. Remember: it’s good to identify the minimum independent piece you can implement next and test that it’s working.

We know how many ships we have—it’s the length of the master list of ship coordinates.

If we have n ships, we’ll need and n by n grid to be displayed to show all the distances between all of them. But how? Think back to the loops chapter…

You can do it with a nested loop. The outer one goes for n rows, and the inner one goes for n columns.

For printing the number with 2 decimal places in a field of width 8, you can use a f-string with some special formatting specifiers, for example:

distance = 99.99
print(f'{distance:8.2f})   # prints "   99.99"

All righty! Let’s print some stuff!

Problem-solving step: Carrying Out the Plan

def print_grid(locations):
    """Print a grid of ship-to-ship distances."""

    num_ships = len(locations)
    
    for i in range(num_ships):
        for j in range(num_ships):
            dist = 99.99
            print(f'{dist:8.2f}', end="")
        print()

locations = get_ship_locations()
print_grid(locations)

Running that gives:

Enter ship location x,y,z (or "done"): 1,2,3
Enter ship location x,y,z (or "done"): 4,5,6
Enter ship location x,y,z (or "done"): 7,8,9
Enter ship location x,y,z (or "done"): done
   99.99   99.99   99.99
   99.99   99.99   99.99
   99.99   99.99   99.99

Hey, a nice 3x3 grid for our 3 ships!

Next, let’s put the ship row number just to the left of each row. Just editing the for-loop here:

    for i in range(num_ships):
        print(f'{i:8}', end="")   # <-- add this
        for j in range(num_ships):
            dist = 99.99
            print(f'{dist:8.2f}', end="")
        print()

We printed it in field width 8 just for consistency with the distance numbers. Output is now:

       0   99.99   99.99   99.99
       1   99.99   99.99   99.99
       2   99.99   99.99   99.99

Which is good! Now we just need a row on the top. How about another loop before everything else to print out that row?

    # Add this loop:
    for i in range(num_ships):
        print(f'{i:8}', end="")
    print()

    for i in range(num_ships):
        print(f'{i:8}', end="")
        for j in range(num_ships):
            dist = 99.99
            print(f'{dist:8.2f}', end="")
        print()

And we get this:

       0       1       2
       0   99.99   99.99   99.99
       1   99.99   99.99   99.99
       2   99.99   99.99   99.99

Er, well, that’s almost right. The top row is shifted one column left. We need to stick 8 blank spaces in before it to scooch it over. So let’s just do it, using string multiplication to make us 8 spaces:

    print(" " * 8, end="")   # <-- Add this

    for i in range(num_ships):
        print(f'{i:8}', end="")
    print()

And now we have this:

               0       1       2
       0   99.99   99.99   99.99
       1   99.99   99.99   99.99
       2   99.99   99.99   99.99

which is looking right on. Except that all the distances are listed as 99.99. Let’s get that out of there and replace it with the real distance between the ships.

Problem-solving step: Devising a Plan

Distance! We see the distance formula up there at the top, repeated here:

$d=\sqrt{(x_0-x_1)^2+(y_0-y_1)^2+(z_0-z_1)^2}$

How do we turn that into code?

We could use math.sqrt() to get the square root, and math.pow() to square the numbers.

Another option is to compute, say, the difference in the X coordinates and then multiply that result by itself to square it (since $x\times{x}=$x^2$).

Let’s code them both and then use the one you like best.

Problem-solving step: Carrying Out the Plan

Here’s computing the distance between points p0 and p1 (both of which are lists of X,Y,Z triples) using math.pow():

d = math.sqrt(
    math.pow(p0[0] - p1[0], 2) +  # difference in Xs
    math.pow(p0[1] - p1[1], 2) +  # difference in Ys
    math.pow(p0[2] - p1[2], 2)    # difference in Zs
)

And here’s option two:

dx = p0[0] - p1[0]   # difference in the Xs
dy = p0[1] - p1[1]   # difference in the Ys
dz = p0[2] - p1[2]   # difference in the Zs

d = math.sqrt(dx*dx + dy*dy + dz*dz)

Which do you prefer? Both should give equivalent answers.

Now, sure, we could just throw that equation in the middle of our code, but this is actually ripe to be made into its own function! If we do that, we can reuse it easily later, should we ever want to. Let’s wrap it up and add it to our file:

def dist3d(p0, p1):
    """Return the Euclidean distance between 2 3D points."""

    # Compute the difference in the Xs, Ys, and Zs
    dx = p0[0] - p1[0]
    dy = p0[1] - p1[1]
    dz = p0[2] - p1[2]

    # Compute the distance. (Remember that multiplying a number by
    # itself is the same as squaring the number.)
    return math.sqrt(dx*dx + dy*dy + dz*dz)

And then we can mod our code where we hardcoded that 99.99 and have it call the distance formula instead! We just need to pass in the two ships’ locations—one is represented by the column number, and the other by the row number:

    for i in range(num_ships):
        print(f'{i:8}', end="")
        for j in range(num_ships):
            dist = dist3d(locations[i], locations[j]])  # <-- Mod here
            print(f'{dist:8.2f}', end="")
        print()

And there we have it!

(Solution ¹⁰⁰.)

10.12 Exercises

Remember: to get your value out of this book, you have to do these exercises. After 20 minutes of being stuck on a problem, you’re allowed to look at the solution.

Use any knowledge you have to solve these, not only what you learned in this chapter.

Always use the four problem-solving steps to solve these problems: understand the problem, devise a plan, carry it out, look back to see what you could have done better.

Write a program to input numbers repeatedly until the user types “done”. Keep track of all the numbers in a list.

Print out the maximum value the user entered using built-in functions.

(Solution ¹⁰¹.)
Write a function that takes an integer between 0 and 9 as an argument. It should return a string that corresponds to the English word for that number. For example, if the argument is 3, the function should return "three".

(Solution ¹⁰².)
Write a function that accepts the mass of two planets and the distance between them (for a total of 3 arguments) and returns the force between them.

Use Newton’s Universal Law of Gravitation to calculate the force $F$:

$F=G\cfrac{m_1 m_2}{r^2}$

In mathematical notation, to variables next to each other like $m_1 m_2$, above, indicate multiplication.

And for $r^2$, a simple equivalent is $r\times{r}$.

So we can convert that whole equation to:
```
F = G * (m1 * m2) / (r * r)
```
where m1 is the mass of one planet, m2 is the mass of the other planet and r is the distance between them.

So far so good.

But what’s G? It’s the gravitational constant:

$G=6.67430\times10^{-11}$

That’s written in scientific notation, but we can do the same thing in Python:
```
G = 6.67430e-11
```
That should be enough to go on. Write the function and call it with a variety of different values. Here are some sample values so you can see if your code is working:

m1 m2 r result

10 20 30 1.48317e-11

10 40 30 2.96635e-11

100 5 10 3.33714e-10

(Solution ¹⁰³.)

m1	m2	r	result
`10`	`20`	`30`	`1.48317e-11`
`10`	`40`	`30`	`2.96635e-11`
`100`	`5`	`10`	`3.33714e-10`

10.13 Summary

Functions are a super-important part of programming and a highly valuable tool to have at your disposal.

You can use them to break up code into smaller, more manageable pieces. You can also use them to create standalone, reusable pieces of code.

Python comes with a pile of built-in functions, and it pays to know what they are (so you don’t reinvent them yourself!)

Finally, you also learned that function arguments come in two flavors: positional and keyword. We’ll revisit more of that topic later.

But now: well-deserved break time!

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