In his wonderful book “Road to Reality” Roger Penrose mentions that the Euler rule is “almost mystical” because it relates the 5 fundamental numbers 0, 1, i, π and e to each other.
The Euler rule
is a special case of the Cotes-Euler identity
if then the Cotes-Euler identity would be written as
we know that and ; the Euler rule follows.
The Cotes-Euler identity is a powerful tool. It makes the multiplication and division of complex numbers much easier. Instead of using trigonometry which involves awkward computations we use the simple algebra of exponential numbers. For example
C1 = r1[cos(φ1) + i sin (φ1)]
C2 = r2[cos(φ2) + i sin (φ2)]
We multiply these two complex numbers easily
C1 C2 = r1 eiφ1 r2 eiφ2 = r1r2ei(φ1+φ2)
The division operation is very easy too
C1 / C2 = r1 eiφ1 / r2 eiφ2 = (r1/r2) ei(φ1-φ2)
In the multiplication operation we add the phases, and in the division operation we subtract the phases in the exponent.
Computing the roots of a complex number becomes very easy too
Square root of C= (r)1/2 e(iφ/2)
Cubic root of C= (r)1/3 e(iφ/3)
N’th root of C= (r)1/N e(iφ/N)
In the N’th root operation we divide the phase by N.
Similarly, computing the powers of a complex number is very easy
N’th power of C= (r)N e(iNφ)
In the N’th power operation we multiply the phase by N.
In multiplication, division, root and power operations we simply applied the rules of exponential numbers
ea eb = e(a+b)
ea/eb = e(a-b)
The (natural) logarithm of a complex number is easily computed as well
ln(C)= ln(r eiφ) = ln(r) + iφ
Here we simply applied the ‘multiplication-to-addition’ rule of logarithms.
ln(ab) = ln(a) + ln(b)