 Lecture 2:¶

• This lecture will cover the basics of arithmetic and basic datatypes in python
• Integers (int)
• Real numbers (float)
• Functions in python

Integers¶

• The set of integers, denoted mathematically as $\mathbb{Z}$, is the collection of all positive and negative whole numbers.
• In particular, $$\mathbb{Z} := \{...,-3,-2,-1,0,1,2,3,...\}$$
• Note: this notation is imprecise, but the important thing is we have an intuitive understanding of what an integer is.
• In python, integers are implemented in the integer class (int). Unlike the $\mathbb{Z}$ or $\mathbb{R}$ which has no maximum, computers have finite memory and as a result are bounded. Later in the course we will talk about precision, overflow and underflow.
• As usual, we can add, subtract, multiply, divide and exponentiate integers as we would on paper
• Example: We know that $2+2=4$ but we can also have python give us the answer by running $2+2$

Real Numbers¶

• The set of real numbers, denoted mathematically as $\mathbb{R}$, is a bit tricky to define rigorously, but can be thought of as the collection of all decimal sequences. This includes the integers, the rationals, and irrational numbers.

Operators on $\mathbb{Z}$ and $\mathbb{R}$¶

• Addition $+$
• Example: 2+2 = 4
• Subtraction $-$
• Example: 3-1=2
• Multiplication $*$
• 3*3=9
• Division $/$
• Example: 7/2=3.5}
• Integer Division $//$ (division but you drop the remainder)
• Note that integer division always rounds to closest non-zero value -i.e. 1//-10 = -1
• Example: 7//2=3
• Exponentiation $**$
• Example 1: 3**2 = 9 is computing $3^2 = 9$
• Example 2: 4**.5 = 2 is computing $4^{\frac{1}{2}} = \sqrt{4} = 2$
• ## Recall that you can double click on this cell to see how these expressions are created using MathJax

Functions¶

• In Python, a function is a a block of code that takes an input (potentially empty) and returns an output (also potentially empty)
• Function blocks begin with the keyword def followed by the function name and parentheses ( ): followed by a colon.
• Any input parameters or arguments should be placed within these parentheses. You can also define parameters inside these parentheses.
• The first statement of a function can be an optional statement - the documentation string of the function or docstring.
• The code block within every function starts with a colon (:) and is indented.
• The statement return [expression] exits a function, optionally passing back an expression to the caller. A return statement with no arguments is the same as return None.
• Source: https://www.tutorialspoint.com/python/python_functions.htm
• Notice that this is different from what a mathematical function is!
• That said, we can use python functions to build mathematical functions!

Mathematical Functions¶

• A function is a mathematical relation between a set of inputs and a set of outputs were each input is associated to exactly 1 output
• Example: $$f(x)=x^2 +1$$
• Non Example: $$g(x)= \pm \sqrt{x}$$
• You might have heard the expression "vertical line test" in the past. We explore this graphically in the coming days
In :
def f(x):
"""
Arguments: x -- an int or a float
Computes $x^2 + 1$
"""
return x **2 + 1

In :
print("f(3) = ", f(3))
print("f(4) = ", f(4))
print("f(5) = ", f(5))

f(3) =  10
f(4) =  17
f(5) =  26


PEMDAS¶

• Python uses the usual mathematical order of operations
• Parentheses come before Exponentiation which comes before Multiplication and division which comes before addition and subtraction
In :
# Example: run this cell to try your hand at a question!
x = 2 + 1 * 3
# input grabs an input from the user/standard input
# int() coerces an object into an integer.
answer = int(input("What do you think x is equal to? "))
# int(input()) converts the input to an integer
print("Succsess!")

What do you think x is equal to? 5
Succsess!


Python Functions are not Mathematical Functions¶

• As said above, python functions are not necessarily mathematical functions.
• We can build functions that have multiple outputs!
In :
def g(x):
"""returns the plus/minus square root of a value"""
return x**.5, -x**.5

In :
print("The positive/negative square root of 4 is ", g(4))

The positive/negative square root of 4 is  (2.0, -2.0)

In :
def print_g(x):
"""Prints g(x)"""
print("The positive/negative square root of ", x, " is ",x**.5, -x**.5)

In :
print_g(1)
print_g(2)
print_g(3)
print_g(4)
print_g(5)

The positive/negative square root of  1  is  1.0 -1.0
The positive/negative square root of  2  is  1.4142135623730951 -1.4142135623730951
The positive/negative square root of  3  is  1.7320508075688772 -1.7320508075688772
The positive/negative square root of  4  is  2.0 -2.0
The positive/negative square root of  5  is  2.23606797749979 -2.23606797749979


Functions of functions!¶

• You can pass anything you want into a function-- including another function!
• For example, what if we wanted to build a general print function?
• We could set up into multiple pieces: first we have a string that describes what that function does on an input x.
• Next we have the function called function that actually computes a value on x
• Finally we tie it all together below
In :
def print_function(function_string, function, x):
f_x = function(x)
print(function_string, x, " is ",f_x)

In :
f_string = "x^2 + 1 for the given input x ="
print_function(f_string, f, 3)


x^2 + 1 for the given input x = 3  is  10


What just happened?¶

• First we called print_function with the inputs "x^2 + 1 for the given input x =", f, 3
• the function then computes f(3) and finally prints out the result
• See python tutor code execution Recap¶

• Overview of operations and numbers
• Overview functions ## Next Time:
• Lists
• Plotting in python