#!/usr/bin/env python
# coding: utf-8

# Thanks to unicode, we may use Greek letters directly in our code.  
# 
# In this Jupyter Notebook, lets use θ (theta), π (pi) and τ (tau) with τ = 2 * π.  
# 
# Then we'll plot the graph of Euler's Formula, e to the i θ over the range 0 to τ.
# 
# In Python we signify i, the root of -1, as 1j for readability, so lets bind i to 1j as well.

# In[69]:


from math import e, pi as π
τ = 2 * π
i = 1j
result = e ** (i * τ)
print ("{:1.5f}".format(result.real))


# Below we import some industrial grade tools used for plotting with Python.  The same greek letter names remain active and guide the construction of a domain t and range s.  Then we label the graph and generate a picture.  plt.show(), if used, produces a plot in its own window.

# In[71]:


import matplotlib.pyplot as plt
import numpy as np
t = np.arange(0.0, τ, 0.01)
s = np.array([(np.e ** (i * θ)).real for θ in t])
plt.plot(t, s)
plt.xlabel('radians')
plt.ylabel('real part')
plt.title('Euler\'s Formula from 0 to tau')
plt.grid(True)
plt.savefig("euler_test.png") # uploaded to Flickr for display below


# ![euler_test.png](https://c2.staticflickr.com/2/1562/26157050065_392d24fb62_z.jpg "euler_test.png")