The following is part of a on-going collection of Jupyter notebooks. The goal being to have a library of notebooks as an introduction to Mathematics and technology. These were all created by Gavin Waters. If you use these notebooks, be nice and throw up a credit somewhere on your page.

def

Alot of times when programing it is good to call something and pass varaible to that thing. An example in mathematics is defining a mathematical function $f(x)$

The syntax for setting up, creating, your definition is simple.

def name(variable):

do stuff
do more stuff
__return something else__

def f(x):
    x +=4
    return x

f(2)

6

You can also use composition of functions for this

f(f(2))

10

lets do something silly with this to illustrate the power.

def silly_sum(x,n):
    x += float(1/n)
    return x

y=0
for count in range(1,11):
    y=silly_sum(y,count)
    print(y)
    

1.0
1.5
1.8333333333333333
2.083333333333333
2.283333333333333
2.4499999999999997
2.5928571428571425
2.7178571428571425
2.8289682539682537
2.9289682539682538

The above code was simply $\sum_{n=1}^{10}\frac{1}{n}$. If you try this for very large numbers then it would be nice to not print all of the numbers and just the last one

y=0
for count in range(1,100001):
    y=silly_sum(y,count)
print(y)

12.090146129863335

We know that this series, the harmonic series, is divergent. One of the proofs collects the terms in groups, so that they always add up to a half. Could we write a script to give the number of terms for each half? Lets do the first 7 "halfs"

y=0
half_steps = 1
sub_total = 0
n=1
total =0
while half_steps<=7:
    count = 0
    while sub_total<=0.5:
        count +=1
        sub_total = silly_sum(sub_total,n)
        n+=1
    total = sub_total + total
    print( "half-steps  #"+str(half_steps) + ' , takes '+str(count)+' terms and reaches a total of '+str(total)+'. The sub_total was : '+str(sub_total))    
    
    sub_total = 0
    half_steps +=1
    

half-steps  #1 , takes 1 terms and reaches a total of 1.0. The sub_total was : 1.0
half-steps  #2 , takes 2 terms and reaches a total of 1.8333333333333333. The sub_total was : 0.8333333333333333
half-steps  #3 , takes 3 terms and reaches a total of 2.45. The sub_total was : 0.6166666666666667
half-steps  #4 , takes 5 terms and reaches a total of 3.019877344877345. The sub_total was : 0.5698773448773449
half-steps  #5 , takes 8 terms and reaches a total of 3.547739657143682. The sub_total was : 0.5278623122663371
half-steps  #6 , takes 13 terms and reaches a total of 4.05849519543652. The sub_total was : 0.5107555382928382
half-steps  #7 , takes 22 terms and reaches a total of 4.575430393744269. The sub_total was : 0.5169351983077494

definitions as functions

def f(x):
    return x**2 -3*x -28

f(2)

-30

To illustrate the function in a graphical way, we will use a package called numpy and matplotlib. See Numpy and Matplotlib post to see more about these.

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

The above code loads the packages, the below code sets up a grid of numbers that we can feed into f(x). That will give us our y values and then we can plot x-y graph.

X=np.linspace(-100,100,500)

plt.plot(f(X))

[<matplotlib.lines.Line2D at 0x10e068320>]