Python

Scientific Computing

Author

Prof. Calvin

Why Python?

On Python

Python is free,
Python is very widely used,
Python is flexible,
Python is relatively easy to learn,
and Python is quite powerful.

Why not Python?

Python is a general purpose language used for Scientific Computing.
Not to build apps, build software, manage databases, or develop user interfaces.
Solve scientific and mathematical problems.

Python libraries

Scientists often use the following:

We’ll get to these.

Running Example

Diving In

Taking Python as a given, we’ll:
- Use an example of something I helped a student with recently
- Show step-by-step how to use Python
- Introduce a number of Python features to solve the problem more easily.
This was from an introductory physics class I believe; I don’t know the context.

Motivating Example

Recently, I helped a student stuck on this:

\[ f(x) = \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} \]

Find $f(x)$ for each of the following $x$ values: \[ \{-1, 4, 5, 8, 11\} \]

Is this “real”?

Models income tax brackets, one of the most important drivers of human behavior in the largest economies in the world.
We use a simpler contrived example for now…

Marginal Tax Rate	Single Taxable Income	Married Filing Jointly or Qualified Widow(er) Taxable Income	Married Filing Separately Taxable Income	Head of Household Taxable Income
10%	$0 – $9,275	$0 – $18,550	$0 – $9,275	$0 – $13,250
15%	$9,276 – $37,650	$18,551 – $75,300	$9,276 – $37,650	$13,251 – $50,400
25%	$37,651 – $91,150	$75,301 – $151,900	$37,651 – $75,950	$50,401 – $130,150
28%	$91,151 – $190,150	$151,901 – $231,450	$75,951 – $115,725	$130,151 – $210,800
33%	$190,151 – $413,350	$231,451 – $413,350	$115,726 – $206,675	$210,801 – $413,350
35%	$413,351 – $415,050	$413,351 – $466,950	$206,676 – $233,475	$413,351 – $441,000
39.6%	$415,051+	$466,951+	$233,476+	$441,001+

How to solve?

Think about how you would solve such a problem.
What steps would you take?
What would making solving it hard?
- Keep track of details?
- Performing the arithmetic?
- Anything else?
Python, in my view, is a way to solve these problems.

Python in action

My preferred way to do calculation as an experienced Python user is writing code:

Mathematical expression \[ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} \]

Python expression (code)

9 * -1 * -1 + 5

I write x * x for $x^2$ because it’s non-obvious how to write “squared” yet.

Arithmetic Operations

In Python, we can write many of the same arithmetic operations we use in our math and science classes.

6 + 3

6 - 3

6 * 3

6 / 3

2.0

Wait?

Use Python? How?
To use Python to make this calculation, first we must install Python.
- It is a program, like Firefox, MS Word or Fortnite

Installing Python

Warning!

There is a very important box to check while installing!
- Be careful working ahead!
- We’ll teach how to fix not checking this box but not for a while

Installation

We go to the offial webpage to get an installer for our computer:
- https://www.python.org/downloads/
Be careful about starting it!
- You can open it and check the next slide.

Add Python to PATH

Why?

The benefit of adding Python to path is that we can use it at
<dramatic music>
The Command Line
- A text-based interface to computing resources, including its ability to do e.g. arithmetic and save notes.

Terminal

On MS Windows
- I press Windows key, type “terminal” then press enter.
On MacOS
- I open Launchpad, type “terminal” then press enter.

MacOS

Windows

Expression evaluation

Returning to the terminal, we can type at the “prompt”.
On MacOS, perhaps a line that begins with $ and a flashing cursor
On Window, perhaps PS C:\Users\calvin>
- PS stands for “powershell” - more latter.
- C:\Users\calvin is the name of a folder - more latter
- > is the prompt, with a flashing cursor.

“Run” Python

In the following examples, I remove line numbers to denote they are not Python code snippets.

On Windows, type python

Windows

PS C:\Users\calvin> python

On MacOS, type python3

MacOS

$ python3

On both, press the ↵ᴇɴᴛᴇʀ key.

See Python

You’ll see something like this:

Python 3.12.5 (tags/v3.12.5:ff3bc82, Aug  6 2024, 20:45:27) [MSC v.1940 64 bit (AMD64)] on win32
Type "help", "copyright", "credits" or "license" for more information.
>>>

Take note of the prompt!
>>>
Those three are how you know it is Python, and not the Terminal, that you are working in.

Example

If I type python at the command line
Then 9 * -1 * -1 + 5 within Python
It will look like this:

PS C:\Users\calvin> python
Python 3.12.5 (tags/v3.12.5:ff3bc82, Aug  6 2024, 20:45:27) [MSC v.1940 64 bit (AMD64)] on win32
Type "help", "copyright", "credits" or "license" for more information.
>>> 9 * -1 * -1 + 5
14
>>>

This is how Python shows that 9 * -1 * -1 + 5 is 14

Evaluating Expressions

Recall

\[ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} \] \[ \{-1, 4, 5, 8, 11\} \]

>>> 9 * -1 * -1 + 5
14
>>> 9
9
>>> 9
9
>>> 9
9
>>> 2 - 11
-9

This is still quite tedious and annoying!

(In)equality testing

Like + or - which we use to calculate numbers…
We can use < or > to calculate inequalities.
Specifically, we see whether an inequality is True or False

-1 < 4

True

4 < 4

False

Accomodating Keyboards

Some things aren’t super easy to type.
I don’t have a “$\leq$” key on my keyboard.
Combine with = as <= for “less than or equal” or “$\leq$”

4 <= 4

True

We can “chain” inequalities as well - one after another.

4 <= 4 <= 8

True

A note

We always put the equal sign = second.

4 <= 5

True

5 >= 4

True

A (somewhat confusing) error if we try =>

>>> 4 => 5
  File "<stdin>", line 1
    4 => 5
    ^
SyntaxError: cannot assign to literal

`=` and `==`

New Topics

We have now touch on two new topics:
- True and False
  - Expressions which don’t evaluate to a number
  - Called “booleans”
- “Assignment”
  - Associated with the = sign
  - Different from inequality testing!
We’ll explore both!

Booleans

Sometimes, a Python expression is a numerical value.

2 + 2

But it doesn’t have to be!

2 < 2

False

If

Booleans are mostly useful for writing if statements.
- These are multiline expressions in Python.
To see the result of multi-line expression, we have to print the result…

if (1 <= 1):
    print(2 + 2)

if (1 < 1):
    print(2 + 2)

Piecewise

We can see the immediate use of this in a piecewise function!

\[ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} \] \[ \{-1, 4, 5, 8, 11\} \]

if (-1 < 4):
    print(9 * -1 * -1 + 5)

Or least part of one…

if (4 < 4):
    print(9 * 4 * 4 + 5)

Uh oh!

Else

Oftentimes, we use if with else

if (1 < 1):
    print("1 is less than 1")
else:
    print("1 is not less than 1")

1 is not less than 1

Elif

If we have more than two options, we can place a special elif in the middle.

if (1 < 1):
    print("1 is less than 1")
elif (1 > 1):
    print("1 is greater than 1")
else:
    print("1 is equal to 1")

1 is equal to 1

if (2 < 1):
    print("2 is less than 1")
elif (2 > 1):
    print("2 is greater than 1")
else:
    print("2 is equal to 1")

2 is greater than 1

By the way, it is extremely obnoxious to manually type 2 in 5 different places

Assignment

We can also assign variables!
- I call this single-equals assignment
- Use a single equals sign = and some variable name, like x

x = 1
if (x < 1):
    print("x is less than 1")
elif (x > 1):
    print("x is greater than 1")
else:
    print("x is equal to 1")

x is equal to 1

x = 2
if (x < 1):
    print("x is less than 1")
elif (x > 1):
    print("x is greater than 1")
else:
    print("x is equal to 1")

x is greater than 1

Double equals equality

I call it single-equals assignment because sometimes we check if a variable is precisely equal to some value.
That is somewhat confusingly done with == double equals equality test.

x = 1
if (x == 1):
    print("x is equal to 1")
else:
    print("x is not equal to 1")

x is equal to 1

x = 2
if (x == 1):
    print("x is equal to 1")
else:
    print("x is not equal to 1")

x is not equal to 1

Double ** Exponentation

Doubling is used in a few other cases.
Double asterisk ** is exponentiation.

5 ** 2

We note that ^ - sometimes used for exponentiation - is something different (and confusing!) in Python.

5 ^ 2

Always use **

Piecewise

We can finally write the piecewise expression!

\[ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} \] \[ \{-1, 4, 5, 8, 11\} \]

x = 11
if (x < 4):
    print(9 * x ** 2 + 5)
elif (4 <= x <= 8):
    print(9)
elif (x > 8):
    print(2 - x)

-9

Code Reuse

Recycle

It is still extremely tedious to either:
- Copy-paste, or
- Type more than once.
Also very annoying/difficult to write them many lines without error!
We introduce the def keyword to define functions
A way to reuse code we have already written.

`def`

We recall we use print in a multi-line.

def double(x):
    print(x + x)

def + some function name + ( + some variable name + ):

double(7)

“Call” functions the same way we call print - with parens.

Metaphor

def is like asking a friend for something.
“Hey can you add x and x together?”
print is how your friend tells you the result.
“Can you print out the result of x + x for me?”

`return`

It is more common to use return than print in a function.
When we call a function, we can think of it as expression with some value.
That value is defined by the value that is return.

def six_gt_five_print():
    print(True)

if (six_gt_five_print()):
    print(123)

True

def six_gt_five_print():
    return True

if (six_gt_five_print()):
    print(123)

Example

Suppose an interest rate doubles our savings every ~7 years.
How much savings in 14 years?

savings = 10
double(savings)

Use return

def double(x):
    return x + x

Now, double(savings) becomes equal to x + x

savings = 10
savings = double(savings)
savings

double(savings)

Or even

double(double(10))

Piecewise

We can finally write the piecewise function (was: expression)!
- Function: use many times in many expressions
- Expression: evaluate once and get one answer

\[ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} \] \[ \{-1, 4, 5, 8, 11\} \]

def piecewise(x):
    if (x < 4):
        return 9 * x ** 2 + 5
    elif (4 <= x <= 8):
        return 9
    elif (x > 8):
        return 2 - x

piecewise(4)

Iteration

Tedium

This is still quite tedious.

piecewise(-1)

piecewise(4)

piecewise(5)

piecewise(8)

piecewise(11)

-9

Sets

This is a set:

\[ \{-1, 4, 5, 8, 11\} \]

A set is an unordered collection of elements.
- In this case, elements are integers - whole numbers.
Python can also recognize sets!

{-1, 4, 5, 8, 11}

{-1, 4, 5, 8, 11}

Type

To be sure that is, in fact, a set, we use the helpful built-in type() to ask Python.

s = {-1, 4, 5, 8, 11}
type(s)

set

We have also worked with integers.

type(1)

int

And booleans - True and False values.

type(1 == 1)

bool

Loops

With a set, do something to each element
In Python, use a for loop:
- The for keyword
- The name to refer to an element, like e or x
- The in keyword
- The set/collection (or its variable name)
- The : colon special character
- An indented new line of code

Example

for element in collection
- do thing

for e in {-1, 4, 5, 8, 11}:
    print(piecewise(e))

What do you see?

Example

What if we just try to print the elements?
- Easier to think about!

for e in {-1, 4, 5, 8, 11}:
    print(e)

What do you see?

Ordering

Recall: “A set is an unordered collection of elements”.
We need to put things in order!
The most common way to do this in Python is with a list.
Almost the same as sets, just use “boxy brackets” []

type([1,2,3])

list

Example

for element in collection
- do thing

for e in [-1, 4, 5, 8, 11]:
    print(piecewise(e))

What do you see?

Seeing lists

We can also use lists to print multiple values!
- Very helpful for keeping track of things!

for e in [-1, 4, 5, 8, 11]:
    print([e, piecewise(e)])

[-1, 14]
[4, 9]
[5, 9]
[8, 9]
[11, -9]

Updating lists

Each element is a lot like a variable.
- They just have a name and number, use []
- The initial element is number 0 (not 1)

xs = [-1, 4, 5, 8, 11]
xs[0]

-1

And like variables, we can assign a value with =

xs[0] = -2
xs

[-2, 4, 5, 8, 11]

Metaphor

Think of a list like a block in a neighborhood.
- Perhaps to “400” block or “8000”
The house on the corner has the block number.
- 400 1st St, 8000 9th Ave
Latter houses have higher numbers.
- 404 1st St, 8019 9th Ave

Adding lists

Lists also helpfully support addition with +

[-1] + [4, 5, 8, 11]

[-1, 4, 5, 8, 11]

We often use this to make new lists.

result = []
for e in [-1, 4, 5, 8, 11]:
    result = result + [piecewise(e)]
result

[14, 9, 9, 9, -9]

This pattern also works with numbers!

x = 0 
x = x + 1
x

Careful!

We cannot add numbers directly to lists
- [1] + 2 doesn’t work!
I think of this as adding a house to block without having an address for it.
- Where would the house go?
To add a house to block, we need to have a new “lot” for the house to be on.

Exercise

Income tax

Recall the example of a piecewise function:

Marginal Tax Rate	Single Taxable Income	Married Filing Jointly or Qualified Widow(er) Taxable Income	Married Filing Separately Taxable Income	Head of Household Taxable Income
10%	$0 – $9,275	$0 – $18,550	$0 – $9,275	$0 – $13,250
15%	$9,276 – $37,650	$18,551 – $75,300	$9,276 – $37,650	$13,251 – $50,400
25%	$37,651 – $91,150	$75,301 – $151,900	$37,651 – $75,950	$50,401 – $130,150
28%	$91,151 – $190,150	$151,901 – $231,450	$75,951 – $115,725	$130,151 – $210,800
33%	$190,151 – $413,350	$231,451 – $413,350	$115,726 – $206,675	$210,801 – $413,350
35%	$413,351 – $415,050	$413,351 – $466,950	$206,676 – $233,475	$413,351 – $441,000
39.6%	$415,051+	$466,951+	$233,476+	$441,001+

Singles only…

Rate	From
10%	0
15%	9275
25%	37650
28%	91150
33%	190150
35%	413350
39.6%	415050

Motivating example

How much would a single making 400k pay?

10% on 9275
15% on 37650 - 9275
25% on 91150 - 37650
28% on 190150 - 91150
33% on 400000 - 190150

Rate	From
10%	0
15%	9275
25%	37650
28%	91150
33%	190150
35%	413350
39.6%	415050

Sum it up!

Watch out for order-of-operations!

10% on 9275
15% on 37650 - 9275
25% on 91150 - 37650
28% on 190150 - 91150
33% on 400000 - 190150

x = .10 * 9275
y = .15 * (37650 - 9275)
z = .25 * (91150 - 37650)
r = .28 * (190150 - 91150)
s = .33 * (400000 - 190150)
x + y + z + r + s

115529.25

Add as you go

We may set a variable to an expression over that variable
- I think of this as an “old” version of the variable on the right side of the equal sign.

x = .10 * 9275
x = x + .15 * (37650 - 9275)
x = x + .25 * (91150 - 37650)
x = x + .28 * (190150 - 91150)
x + .33 * (400000 - 190150)

115529.25

x = .10 * 9275
y = .15 * (37650 - 9275)
z = .25 * (91150 - 37650)
r = .28 * (190150 - 91150)
s = .33 * (400000 - 190150)
x + y + z + r + s

115529.25

Assign-update

We may use an “assignment operator” +=
- Reassign a variable based on the result of an arithmetic operation.

x = .10 * 9275
x = x + .15 * (37650 - 9275)
x = x + .25 * (91150 - 37650)
x = x + .28 * (190150 - 91150)
x + .33 * (400000 - 190150)

115529.25

x = .10 * 9275
x += .15 * (37650 - 9275)
x += .25 * (91150 - 37650)
x += .28 * (190150 - 91150)
x + .33 * (400000 - 190150)

115529.25

Exercise

Write function
def single_tax(pay):
Return tax cost.
- Return not print!
Bonus: Also write single_tax_rate which returns the percent tax rate at some income level.

Rate	From
10%	0
15%	9275
25%	37650
28%	91150
33%	190150
35%	413350
39.6%	415050

Solution

Spoiler alert!

Code

def single_tax(pay):
    tax = 0
    if pay > 415050:
        tax += (pay - 415050) * .396
        pay = 415050
    if pay > 413350:
        tax += (pay - 413350) * .35
        pay = 413350
    if pay > 190150:
        tax += (pay - 190150) * .33
        pay = 190150
    if pay > 91150:
        tax += (pay - 91150) * .28
        pay = 91150
    if pay > 37650:
        tax += (pay - 37650) * .25
        pay = 37650
    if pay > 9275:
        tax += (pay - 9275) * .15
        pay = 9275
    return tax + pay * .1

The solution gives the following results!

[
    single_tax(100000),
    single_tax(100000),
    single_tax(200000),
    single_tax(300000),
    single_tax(400000),
]

[21036.75, 21036.75, 49529.25, 82529.25, 115529.25]

115529.25 matches our calculations.

Challenge Problem

Calculate at what income does the tax rate reach the second highest marginal tax rate of 35%?
- That is, at what pay does tax == pay * .35

Code

m = .396
b = 0
b += (415050 - 413350) * .35
b += (413350 - 190150) * .33
b += (190150 - 91150) * .28
b += (91150 - 37650) * .25
b += (37650 - 9275) * .15
b += 9275 * .1
(m * -415050 + b)/(.35 - m)

952827.173913043

Bonus Solution

single_tax with loops

Code

def single_tax(pay):
    tax_policy = [
        [415050, .396], 
        [413350, .35],
        [190150, .33],
        [91150, .28],
        [37650, .25],
        [9275, .15]
    ]
    tax = 0
    for bracket in tax_policy:
        if pay > bracket[0]:
            tax += (pay - bracket[0]) * bracket[1]
            pay = bracket[0]
    return tax + pay * .1

--- title: Python --- # Why Python? ## On Python * Python is free, * Python is very widely used, * Python is flexible, * Python is relatively easy to learn, * and Python is quite powerful. ## Why not Python? - Python is a general purpose language used for Scientific Computing. - _Not_ to build apps, build software, manage databases, or develop user interfaces. - Solve scientific and mathematical problems. ## Python libraries Scientists often use the following: * [Python](https://www.python.org/), * [`numpy` (numerical Python)](https://www.numpy.org/), * [`matplotlib` (a suite of plotting tools)](https://matplotlib.org/), * [`scipy` (scientific Python)](https://www.scipy.org/), and * [`sympy` (symbolic Python)](https://www.sympy.org/en/index.html). We'll get to these. # Running Example ## Diving In - Taking Python as a given, we'll: - Use an example of something I helped a student with recently - Show step-by-step how to use Python - Introduce a number of Python features to solve the problem more easily. - This was from an introductory physics class I believe; I don't know the context. ## Motivating Example - Recently, I helped a student stuck on this: $$ f(x) = \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} $$ - Find $f(x)$ for each of the following $x$ values: $$ \{-1, 4, 5, 8, 11\} $$ ## Is this "real"? - Models income tax brackets, one of the most important drivers of human behavior in the largest economies in the world. - We use a simpler contrived example for now... <table style="font-size:12px"> <th>Marginal Tax Rate</th> <th>Single Taxable Income</th> <th>Married Filing Jointly or Qualified Widow(er) Taxable Income</th> <th>Married Filing Separately Taxable Income</th> <th>Head of Household Taxable Income</th> </tr> <tr> <th>10%</th> <td>$0 – $9,275</td> <td>$0 – $18,550</td> <td>$0 – $9,275</td> <td>$0 – $13,250</td> </tr> <tr> <th>15%</th> <td>$9,276 – $37,650</td> <td>$18,551 – $75,300</td> <td>$9,276 – $37,650</td> <td>$13,251 – $50,400</td> </tr> <tr> <th>25%</th> <td>$37,651 – $91,150</td> <td>$75,301 – $151,900</td> <td>$37,651 – $75,950</td> <td>$50,401 – $130,150</td> </tr> <tr> <th>28%</th> <td>$91,151 – $190,150</td> <td>$151,901 – $231,450</td> <td>$75,951 – $115,725</td> <td>$130,151 – $210,800</td> </tr> <tr> <th>33%</th> <td>$190,151 – $413,350</td> <td>$231,451 – $413,350</td> <td>$115,726 – $206,675</td> <td>$210,801 – $413,350</td> </tr> <tr> <th>35%</th> <td>$413,351 – $415,050</td> <td>$413,351 – $466,950</td> <td>$206,676 – $233,475</td> <td>$413,351 – $441,000</td> </tr> <tr> <th>39.6%</th> <td>$415,051+</td> <td>$466,951+</td> <td>$233,476+</td> <td>$441,001+</td> </table> ## How to solve? - Think about how you would solve such a problem. - What steps would you take? - What would making solving it hard? - Keep track of details? - Performing the arithmetic? - Anything else? - Python, in my view, is a way to solve these problems. ## Python in action - My preferred way to do calculation as an experienced Python user is writing *code*: :::: {.columns} ::: {.column width="50%"} - Mathematical expression $$ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} $$ ::: ::: {.column width="50%"} - Python expression (code) ```{python} 9 * -1 * -1 + 5 ``` ::: :::: - I write `x * x` for $x^2$ because it's non-obvious how to write "squared" yet. ## Arithmetic Operations - In Python, we can write many of the same arithmetic operations we use in our math and science classes. ```{python} 6 + 3 ``` ```{python} 6 - 3 ``` ```{python} 6 * 3 ``` ```{python} 6 / 3 ``` ## Wait? - Use Python? How? - To use Python to make this calculation, first we must install Python. - It is a program, like Firefox, MS Word or Fortnite # Installing Python ## Warning! - There is a very important box to check while installing! - Be careful working ahead! - We'll teach how to fix not checking this box *but not for a while* ## Installation - We go to the offial webpage to get an installer for our computer: - [https://www.python.org/downloads/](https://www.python.org/downloads/) - Be careful about starting it! - You can open it and check the next slide. ## Add Python to PATH ![](img/add-python-to-path.png) ## Why? - The benefit of adding Python to path is that we can use it at - <dramatic music> - *The Command Line* - A text-based interface to computing resources, including its ability to do e.g. arithmetic and save notes. ## Terminal - On MS Windows - I press Windows key, type "terminal" then press enter. - On MacOS - I open Launchpad, type "terminal" then press enter. ## MacOS ![](img/zsh.png) ## Windows ![](img/ps.webp) ## Expression evaluation - Returning to the terminal, we can type at the "prompt". - On MacOS, perhaps a line that begins with `$` and a flashing cursor - On Window, perhaps `PS C:\Users\calvin>` - `PS` stands for "powershell" - more latter. - `C:\Users\calvin` is the name of a folder - more latter - `>` is the prompt, with a flashing cursor. ## "Run" Python *In the following examples, I remove line numbers to denote they are not Python code snippets.* - On Windows, type `python` ```{.ps filename="Windows" code-line-numbers="false"} PS C:\Users\calvin> python ``` - On MacOS, type `python3` ```{.zsh filename="MacOS" code-line-numbers="false"} $ python3 ``` - On both, press the **↵ᴇɴᴛᴇʀ** key. ## See Python - You'll see something like this: ```{.email code-line-numbers="false"} Python 3.12.5 (tags/v3.12.5:ff3bc82, Aug 6 2024, 20:45:27) [MSC v.1940 64 bit (AMD64)] on win32 Type "help", "copyright", "credits" or "license" for more information. >>> ``` - Take note of the prompt! - `>>>` - Those three are how you know it is Python, and not the Terminal, that you are working in. ## Example - If I type `python` at the command line - Then `9 * -1 * -1 + 5` *within* Python - It will look like this: ```{.email code-line-numbers="false"} PS C:\Users\calvin> python Python 3.12.5 (tags/v3.12.5:ff3bc82, Aug 6 2024, 20:45:27) [MSC v.1940 64 bit (AMD64)] on win32 Type "help", "copyright", "credits" or "license" for more information. >>> 9 * -1 * -1 + 5 14 >>> ``` - This is how Python shows that `9 * -1 * -1 + 5` is `14` # Evaluating Expressions ## Recall :::: {.columns} ::: {.column width="50%"} $$ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} $$ $$ \{-1, 4, 5, 8, 11\} $$ ::: ::: {.column width="50%"} ```{.python code-line-numbers="false"} >>> 9 * -1 * -1 + 5 14 >>> 9 9 >>> 9 9 >>> 9 9 >>> 2 - 11 -9 ``` ::: :::: - This is still quite tedious and annoying! ## (In)equality testing - Like `+` or `-` which we use to calculate numbers... - We can use `<` or `>` to calculate inequalities. - Specifically, we see whether an inequality is `True` or `False` ```{python} -1 < 4 ``` ```{python} 4 < 4 ``` ## Accomodating Keyboards - Some things aren't super easy to type. - I don't have a "$\leq$" key on my keyboard. - Combine with `=` as `<=` for "less than or equal" or "$\leq$" ```{python} 4 <= 4 ``` - We can "chain" inequalities as well - one after another. ```{python} 4 <= 4 <= 8 ``` ## A note - We always put the equal sign `=` second. ```{python} 4 <= 5 ``` ```{python} 5 >= 4 ``` - A (somewhat confusing) error if we try `=>` ```{.email code-line-numbers="false"} >>> 4 => 5 File "<stdin>", line 1 4 => 5 ^ SyntaxError: cannot assign to literal ``` # `=` and `==` ## New Topics - We have now touch on two new topics: - `True` and `False` - Expressions which don't *evaluate* to a number - Called "booleans" - "Assignment" - Associated with the `=` sign - Different from inequality testing! - We'll explore both! ## Booleans - Sometimes, a Python *expression* is a numerical value. ```{python} 2 + 2 ``` ```{python} 7 ``` - But it doesn't have to be! ```{python} 2 < 2 ``` ## If - Booleans are mostly useful for writing `if` statements. - These are *multiline* expressions in Python. - To see the result of multi-line expression, we have to print the result... ```{python} if (1 <= 1): print(2 + 2) ``` ```{python} if (1 < 1): print(2 + 2) ``` ## Piecewise - We can see the immediate use of this in a piecewise function! :::: {.columns} ::: {.column width="50%"} $$ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} $$ $$ \{-1, 4, 5, 8, 11\} $$ ::: ::: {.column width="50%"} ```{python} if (-1 < 4): print(9 * -1 * -1 + 5) ``` - Or least part of one... ```{python} if (4 < 4): print(9 * 4 * 4 + 5) ``` - Uh oh! ::: :::: ## Else - Oftentimes, we use `if` with `else` ```{python} if (1 < 1): print("1 is less than 1") else: print("1 is not less than 1") ``` ## Elif - If we have more than two options, we can place a special `elif` in the middle. :::: {.columns} ::: {.column width="50%"} ```{python} if (1 < 1): print("1 is less than 1") elif (1 > 1): print("1 is greater than 1") else: print("1 is equal to 1") ``` ::: ::: {.column width="50%"} ```{python} if (2 < 1): print("2 is less than 1") elif (2 > 1): print("2 is greater than 1") else: print("2 is equal to 1") ``` ::: :::: - By the way, it is extremely obnoxious to manually type `2` in *5 different places* ## Assignment - We can also assign variables! - I call this *single-equals assignment* - Use a single equals sign `=` and some variable name, like `x` :::: {.columns} ::: {.column width="50%"} ```{python} x = 1 if (x < 1): print("x is less than 1") elif (x > 1): print("x is greater than 1") else: print("x is equal to 1") ``` ::: ::: {.column width="50%"} ```{python} x = 2 if (x < 1): print("x is less than 1") elif (x > 1): print("x is greater than 1") else: print("x is equal to 1") ``` ::: :::: ## Double equals equality - I call it *single-equals assignment* because sometimes we check if a variable is precisely equal to some value. - That is somewhat confusingly done with `==` *double equals equality* test. :::: {.columns} ::: {.column width="50%"} ```{python} x = 1 if (x == 1): print("x is equal to 1") else: print("x is not equal to 1") ``` ::: ::: {.column width="50%"} ```{python} x = 2 if (x == 1): print("x is equal to 1") else: print("x is not equal to 1") ``` ::: :::: ## Double ** Exponentation - Doubling is used in a few other cases. - Double asterisk `**` is exponentiation. ```{python} 5 ** 2 ``` - We note that `^` - sometimes used for exponentiation - is something different (and confusing!) in Python. ```{python} 5 ^ 2 ``` - Always use `**` ## Piecewise - We can finally write the piecewise expression! :::: {.columns} ::: {.column width="50%"} $$ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} $$ $$ \{-1, 4, 5, 8, 11\} $$ ::: ::: {.column width="50%"} ```{python} x = 11 if (x < 4): print(9 * x ** 2 + 5) elif (4 <= x <= 8): print(9) elif (x > 8): print(2 - x) ``` ::: :::: # Code Reuse ## Recycle - It is still extremely tedious to either: - Copy-paste, or - Type more than once. - Also very annoying/difficult to write them many lines without error! - We introduce the `def` keyword to define `functions` - A way to reuse code we have already written. ## `def` - We recall we use `print` in a multi-line. ```{python} def double(x): print(x + x) ``` - `def` + *some function name* + `(` + *some variable name* + `):` ```{python} double(7) ``` - "Call" functions the same way we call `print` - with parens. ## Metaphor - `def` is like asking a friend for something. - "Hey can you add `x` and `x` together?" - `print` is how your friend tells you the result. - "Can you print out the result of `x + x` for me?" ## `return` - It is more common to use `return` than `print` in a function. - When we call a function, we can think of it as expression with some value. - That value is defined by the value that is return. :::: {.columns} ::: {.column width="50%"} ```{python} def six_gt_five_print(): print(True) if (six_gt_five_print()): print(123) ``` ::: ::: {.column width="50%"} ```{python} def six_gt_five_print(): return True if (six_gt_five_print()): print(123) ``` ::: :::: ## Example - Suppose an interest rate doubles our savings every ~7 years. - How much savings in 14 years? ```{python} savings = 10 double(savings) ``` ## Use return ```{python} def double(x): return x + x ``` - Now, `double(savings)` becomes equal to `x + x` ```{python} savings = 10 savings = double(savings) savings ``` ```{python} double(savings) ``` - Or even ```{python} double(double(10)) ``` ## Piecewise - We can finally write the piecewise function (was: expression)! - Function: use many times in many expressions - Expression: evaluate once and get one answer :::: {.columns} ::: {.column width="50%"} $$ \begin{cases} 9x^2 + 5 & x < 4 \\ 9 & 4 \leq x \leq 8 \\ 2 - x & x > 8 \end{cases} $$ $$ \{-1, 4, 5, 8, 11\} $$ ::: ::: {.column width="50%"} ```{python} def piecewise(x): if (x < 4): return 9 * x ** 2 + 5 elif (4 <= x <= 8): return 9 elif (x > 8): return 2 - x ``` ```{python} piecewise(4) ``` ::: :::: # Iteration ## Tedium - This is still quite tedious. ```{python} piecewise(-1) ``` ```{python} piecewise(4) ``` ```{python} piecewise(5) ``` ```{python} piecewise(8) ``` ```{python} piecewise(11) ``` ## Sets - This is a *set*: $$ \{-1, 4, 5, 8, 11\} $$ - A set is an *unordered* collection of *elements*. - In this case, elements are integers - whole numbers. - Python can also recognize sets! ```{python} {-1, 4, 5, 8, 11} ``` ## Type - To be sure that is, in fact, a set, we use the helpful built-in `type()` to ask Python. ```{python} s = {-1, 4, 5, 8, 11} type(s) ``` - We have also worked with **int**egers. ```{python} type(1) ``` - And ***bool**eans* - `True` and `False` values. ```{python} type(1 == 1) ``` ## Loops - With a set, do something *to each element* - In Python, use a `for` loop: - The `for` keyword - The name to refer to an element, like `e` or `x` - The `in` keyword - The set/collection (or its variable name) - The `:` colon special character - An indented new line of code ## Example - for element in collection - do thing ```{python} for e in {-1, 4, 5, 8, 11}: print(piecewise(e)) ``` - What do you see? ## Example - What if we just try to print the elements? - Easier to think about! ```{python} for e in {-1, 4, 5, 8, 11}: print(e) ``` - What do you see? ## Ordering - Recall: "A set is an *unordered* collection of *elements*". - We need to put things in order! - The most common way to do this in Python is with a *list*. - Almost the same as sets, just use "boxy brackets" `[]` ```{python} type([1,2,3]) ``` ## Example - for element in collection - do thing ```{python} for e in [-1, 4, 5, 8, 11]: print(piecewise(e)) ``` - What do you see? ## Seeing lists - We can also use lists to print multiple values! - Very helpful for keeping track of things! ```{python} for e in [-1, 4, 5, 8, 11]: print([e, piecewise(e)]) ``` ## Updating lists - Each element is a lot like a variable. - They just have a name *and* number, use `[]` - The initial element is number `0` (*not* 1) ```{python} xs = [-1, 4, 5, 8, 11] xs[0] ``` - And like variables, we can assign a value with `=` ```{python} xs[0] = -2 xs ``` ## Metaphor - Think of a list like a block in a neighborhood. - Perhaps to "400" block or "8000" - The house on the corner has the block number. - 400 1st St, 8000 9th Ave - Latter houses have higher numbers. - 404 1st St, 8019 9th Ave ## Adding lists - Lists also helpfully support addition with `+` ```{python} [-1] + [4, 5, 8, 11] ``` - We often use this to make new lists. ```{python} result = [] for e in [-1, 4, 5, 8, 11]: result = result + [piecewise(e)] result ``` - This pattern also works with numbers! ```{python} x = 0 x = x + 1 x ``` ## Careful! - We cannot add numbers directly to lists - `[1] + 2` doesn't work! - I think of this as adding a house to block without having an address for it. - Where would the house go? - To add a house to block, we need to have a new "lot" for the house to be on. # Exercise ## Income tax - Recall the example of a piecewise function: <table style="font-size:12px"> <th>Marginal Tax Rate</th> <th>Single Taxable Income</th> <th>Married Filing Jointly or Qualified Widow(er) Taxable Income</th> <th>Married Filing Separately Taxable Income</th> <th>Head of Household Taxable Income</th> </tr> <tr> <th>10%</th> <td>$0 – $9,275</td> <td>$0 – $18,550</td> <td>$0 – $9,275</td> <td>$0 – $13,250</td> </tr> <tr> <th>15%</th> <td>$9,276 – $37,650</td> <td>$18,551 – $75,300</td> <td>$9,276 – $37,650</td> <td>$13,251 – $50,400</td> </tr> <tr> <th>25%</th> <td>$37,651 – $91,150</td> <td>$75,301 – $151,900</td> <td>$37,651 – $75,950</td> <td>$50,401 – $130,150</td> </tr> <tr> <th>28%</th> <td>$91,151 – $190,150</td> <td>$151,901 – $231,450</td> <td>$75,951 – $115,725</td> <td>$130,151 – $210,800</td> </tr> <tr> <th>33%</th> <td>$190,151 – $413,350</td> <td>$231,451 – $413,350</td> <td>$115,726 – $206,675</td> <td>$210,801 – $413,350</td> </tr> <tr> <th>35%</th> <td>$413,351 – $415,050</td> <td>$413,351 – $466,950</td> <td>$206,676 – $233,475</td> <td>$413,351 – $441,000</td> </tr> <tr> <th>39.6%</th> <td>$415,051+</td> <td>$466,951+</td> <td>$233,476+</td> <td>$441,001+</td> </table> ## Singles only... | Rate | From | |--------|--------:| | 10% | 0 | | 15% | 9275 | | 25% | 37650 | | 28% | 91150 | | 33% | 190150 | | 35% | 413350 | | 39.6% | 415050 | ## Motivating example - How much would a single making 400k pay? :::: {.columns} ::: {.column width="50%"} - 10% on 9275 - 15% on 37650 - 9275 - 25% on 91150 - 37650 - 28% on 190150 - 91150 - 33% on 400000 - 190150 ::: ::: {.column width="50%"} | Rate | From | |--------|--------:| | 10% | 0 | | 15% | 9275 | | 25% | 37650 | | 28% | 91150 | | 33% | 190150 | | 35% | 413350 | | 39.6% | 415050 | ::: :::: ## Sum it up! - Watch out for order-of-operations! :::: {.columns} ::: {.column width="50%"} - 10% on 9275 - 15% on 37650 - 9275 - 25% on 91150 - 37650 - 28% on 190150 - 91150 - 33% on 400000 - 190150 ::: ::: {.column width="50%"} ```{python} x = .10 * 9275 y = .15 * (37650 - 9275) z = .25 * (91150 - 37650) r = .28 * (190150 - 91150) s = .33 * (400000 - 190150) x + y + z + r + s ``` ::: :::: ## Add as you go - We may set a variable to an expression *over that variable* - I think of this as an "old" version of the variable on the right side of the equal sign. :::: {.columns} ::: {.column width="50%"} ```{python} x = .10 * 9275 x = x + .15 * (37650 - 9275) x = x + .25 * (91150 - 37650) x = x + .28 * (190150 - 91150) x + .33 * (400000 - 190150) ``` ::: ::: {.column width="50%"} ```{python} x = .10 * 9275 y = .15 * (37650 - 9275) z = .25 * (91150 - 37650) r = .28 * (190150 - 91150) s = .33 * (400000 - 190150) x + y + z + r + s ``` ::: :::: ## Assign-update - We may use an "assignment operator" `+=` - Reassign a variable based on the result of an arithmetic operation. :::: {.columns} ::: {.column width="50%"} ```{python} x = .10 * 9275 x = x + .15 * (37650 - 9275) x = x + .25 * (91150 - 37650) x = x + .28 * (190150 - 91150) x + .33 * (400000 - 190150) ``` ::: ::: {.column width="50%"} ```{python} x = .10 * 9275 x += .15 * (37650 - 9275) x += .25 * (91150 - 37650) x += .28 * (190150 - 91150) x + .33 * (400000 - 190150) ``` ::: :::: ## Exercise :::: {.columns} ::: {.column width="50%"} - Write function - `def single_tax(pay):` - *Return* tax cost. - Return not print! - Bonus: Also write `single_tax_rate` which returns the percent tax rate at some income level. ::: ::: {.column width="50%"} | Rate | From | |--------|--------:| | 10% | 0 | | 15% | 9275 | | 25% | 37650 | | 28% | 91150 | | 33% | 190150 | | 35% | 413350 | | 39.6% | 415050 | ::: :::: ## Solution :::: {.columns} ::: {.column width="50%"} - Spoiler alert! ```{python} #| code-fold: true def single_tax(pay): tax = 0 if pay > 415050: tax += (pay - 415050) * .396 pay = 415050 if pay > 413350: tax += (pay - 413350) * .35 pay = 413350 if pay > 190150: tax += (pay - 190150) * .33 pay = 190150 if pay > 91150: tax += (pay - 91150) * .28 pay = 91150 if pay > 37650: tax += (pay - 37650) * .25 pay = 37650 if pay > 9275: tax += (pay - 9275) * .15 pay = 9275 return tax + pay * .1 ``` ::: ::: {.column width="50%"} - The solution gives the following results! ```{python} [ single_tax(100000), single_tax(100000), single_tax(200000), single_tax(300000), single_tax(400000), ] ``` - `115529.25` matches our calculations. ::: :::: ## Challenge Problem - Calculate at what income does the tax rate reach the second highest marginal tax rate of 35%? - That is, at what `pay` does `tax == pay * .35` ```{python} #| code-fold: true m = .396 b = 0 b += (415050 - 413350) * .35 b += (413350 - 190150) * .33 b += (190150 - 91150) * .28 b += (91150 - 37650) * .25 b += (37650 - 9275) * .15 b += 9275 * .1 (m * -415050 + b)/(.35 - m) ``` ## Bonus Solution - `single_tax` with loops ```{python} #| code-fold: true def single_tax(pay): tax_policy = [ [415050, .396], [413350, .35], [190150, .33], [91150, .28], [37650, .25], [9275, .15] ] tax = 0 for bracket in tax_policy: if pay > bracket[0]: tax += (pay - bracket[0]) * bracket[1] pay = bracket[0] return tax + pay * .1 ```

Why Python?

On Python

Why not Python?

Python libraries

Running Example

Diving In

Motivating Example

Is this “real”?

How to solve?

Python in action

Arithmetic Operations

Wait?

Installing Python

Warning!

Installation

Add Python to PATH

Why?

Terminal

MacOS

Windows

Expression evaluation

“Run” Python

See Python

Example

Evaluating Expressions

Recall

(In)equality testing

Accomodating Keyboards

A note

= and ==

New Topics

Booleans

If

Piecewise

Else

Elif

Assignment

Double equals equality

Double ** Exponentation

Piecewise

Code Reuse

Recycle

def

Metaphor

return

Example

Use return

Piecewise

Iteration

Tedium

Sets

Type

Loops

Example

Example

Ordering

Example

Seeing lists

Updating lists

Metaphor

Adding lists

Careful!

Exercise

Income tax

Singles only…

Motivating example

Sum it up!

Add as you go

Assign-update

Exercise

Solution

Challenge Problem

Bonus Solution

`=` and `==`

`def`

`return`