'''Complex Numbers from the book'''

from math import sqrt, degrees,atan2, sin, cos,radians

def cAdd(a,b):
 return [a[0]+b[0],a[1]+b[1]]

def cMult(u,v):
 '''Returns the product of two complex numbers'''
 return [u[0]*v[0]-u[1]*v[1],u[1]*v[0]+u[0]*v[1]]
 

def theta(z):
 '''Calculates the angle of rotation of a complex number'''
 return degrees(atan2(z[1],z[0]))

def magnitude(z):
 return sqrt(z[0]**2 + z[1]**2)

def power(z,n):
 r = magnitude(z)
 angle = (1/3)*radians(theta(z)+360) 
 return [r**n*cos(n*angle),r**n*sin(n*angle)]

def synthDiv(divisor,dividend):
 '''divides a polynomial by a constant and returns a lower-degree polynomial. Enter divisor as a constant: (x - 3) is 3
 Enter dividend as a list of coefficients:
 x**2 – 5*x + 6 becomes [1,-5,6]'''
 quotient = [] #empty list for coefficients of quotient
 row2 = [0] #start the second row
 for i in range(len(dividend)):
 quotient.append(dividend[i]+row2[i]) #add the ith column
 row2.append(divisor*quotient[i]) #put the new number in row 2
 print(quotient)

def quad(a,b,c): 
 '''Returns the solutions of an equation
 of the form a*x**2 + b*x + c = 0'''
 x1 = (-b + sqrt(b**2 - 4*a*c))/(2*a)
 x2 = (-b - sqrt(b**2 - 4*a*c))/(2*a)
 return x1,x2

def f(x):
 return x**3 - 15*x - 4

def average(a,b):
 return (a + b) / 2

def guess():
 lower = -3
 upper = -4
 for i in range(20):
 midpt = average(lower,upper)
 if f(midpt) == 0:
 return midpt
 elif f(midpt) < 0:
 upper = midpt
 else:
 lower = midpt
 return midpt

def arange(start,stop,step):
 '''Returns a list of numbers from 
 start to stop by step'''
 output = []
 x = start
 while x < stop:
 output.append(x)
 x += step
 return output